# Subregular $J$-rings of Coxeter systems via quiver path algebras

Ivan Dimitrov  Department of Mathematics and Statistics, Queen’s University
<EMAIL_ADDRESS>,  Charles Paquette  Department of Mathematics and
Computer Science, Royal Military College of Canada
<EMAIL_ADDRESS>,  David Wehlau  Department of Mathematics and
Computer Science, Royal Military College of Canada<EMAIL_ADDRESS>and
Tianyuan Xu  Department of Mathematics, University of Colorado Boulder
<EMAIL_ADDRESS>
###### Abstract.

We study the subregular $J$-ring $J_{C}$ of a Coxeter system $(W,S)$, a
subring of Lusztig’s $J$-ring. We prove that $J_{C}$ is isomorphic to a
quotient of the path algebra of the double quiver of $(W,S)$ by a suitable
ideal that we associate to a family of Chebyshev polynomials. As applications,
we use quiver representations to study the category mod-$A_{K}$ of finite
dimensional right modules of the algebra $A_{K}=K\otimes_{\mathbb{Z}}J_{C}$
over an algebraically closed field $K$ of characteristic zero. Our results
include classifications of Coxeter systems for which mod-$A_{K}$ is
semisimple, has finitely many simple modules up to isomorphism, or has a bound
on the dimensions of simple modules. Incidentally, we show that every group
algebra of a free product of finite cyclic groups is Morita equivalent to the
algebra $A_{K}$ for a suitable Coxeter system; this allows us to specialize
the classifications to the module categories of such group algebras.

###### Key words and phrases:

Coxeter systems, asymptotic Hecke algebras, Kazhdan–Lusztig cells, quiver
representations

###### 1991 Mathematics Subject Classification:

Primary: 20C08, 16G20; Secondary: 16D60, 20C07, 20E06.

##  1\. Introduction

We study a subring of the $J$-ring of an arbitrary Coxeter system $(W,S)$. The
$J$-ring was first introduced by Lusztig in [Lus87] in the case where $W$ is a
Weyl or affine Weyl group to help study the Kazhdan–Lusztig cells in $W$.
Later, Lusztig showed in [Lus14b] that the same construction of the $J$-ring
is valid for arbitrary Coxeter systems, at least in the so-called “equal-
parameter” case. In the “unequal-parameter” case, the validity of the
construction relies on what has come to be known as Lusztig’s conjectures
P1-P15; see [Bon17, Section 14.2]. We only deal with the equal-parameter case
in this paper.

By definition, the $J$-ring equals the free abelian group $J=\oplus_{w\in
W}\mathbb{Z}t_{w}$ as a group, and products in $J$ are given by the formula

(1) $t_{x}t_{y}=\sum_{z\in W}\gamma_{x,y,z^{-1}}t_{z}$

where each coefficient $\gamma_{x,y,z^{-1}}$ is a certain nonnegative integer
obtained via the Kazhdan–Lusztig basis of the Hecke algebra of $(W,S)$. The
formula endows $J$ with the structure of an associative (but not necessarily
unital) ring. Moreover, for each _two-sided Kazhdan–Lusztig cell_ $E$ of $W$,
the subgroup $J_{E}:=\oplus_{w\in E}\mathbb{Z}t_{w}$ of $J$ is a subring of
$J$. In this paper, we focus on the ring $J_{C}$ where $C$ is a particular
two-sided cell of $W$ called the _subregular cell_. This cell consists of all
the non-identity elements in $W$ that are _rigid_ in the sense that they each
have a unique reduced expression; see [Lus83] and [Xu19]. We call $J_{C}$ the
_subregular $J$-ring_ and study the structure and representations of $J_{C}$.

Also called the _asymptotic Hecke algebra_ , the $J$-ring may be viewed as a
limit of the Hecke algebra of $W$ in the sense of [Lus95]. As such, the
$J$-ring has been an important tool for studying Hecke algebras and reductive
groups; see, for example, [Lus89], [Gec98], [Gec07] and [Lus18]. Besides its
applications, the structure of the $J$-ring itself has also been studied
extensively. Notable results include the following: Bezrukavnikov, Finkelberg,
and Ostrik studied a categorical version of the $J$-ring in [BFO09] and used
it to compute explicitly the structure of the ring $J_{E}$ for each two-sided
cell $E$ in $W$; Braverman and Kazhdan showed in [BK18] that $J$ is isomorphic
to a certain subalgebra of the Harish-Chandra Schwartz algebra of a reductive
group; by using a generalization of the Robinson–Schensted algorithm called
the _affine matrix-ball construction_ , Kim and Pylyavskyy gave a canonical
presentation for the $J$-ring in the special case where $W$ is an (extended)
affine symmetric group in [KP19], extending the work of Xi in [Xi02] for the
same case.

It is worth noting that the results on the structure of the $J$-ring mentioned
above are all restricted to Weyl or (extended) affine Weyl groups. On the
other hand, the $J$-ring makes sense for an arbitrary Coxeter system, so it is
natural to wonder what the structure of the $J$-ring is for more general
Coxeter systems. Indeed, in Kazhdan–Lusztig theory it can often be interesting
to study Coxeter systems in the full generality. One such indication comes
from the proof of the famous “positivity conjecture” of Kazhdan and Lusztig,
which states that all coefficients of so-called _Kazhdan–Lusztig polynomials_
are nonnegative integers. After its first appearance in [KL79] in 1979, the
conjecture was proved along with other related deep results for Weyl and
affine Weyl groups in the next two years by Kazhdan–Lusztig [KL80],
Beilinson–Bernstein [BB81] and Brylinski–Kashiwara [BK81], via geometric
methods involving local intersection cohomology of Schubert varieties,
$D$-modules and perverse sheaves. The proof for the general case came much
later: building upon the work of Soergel in [Soe90] [Soe92] [Soe07], Elias and
Williamson proved the positivity conjecture for arbitrary Coxeter systems in
[EW14] in 2014. In their work, they introduced a graphical calculus and a type
of Hodge theory for the Soergel category, each of which is interesting in its
own right; see [EW16] and [Wil18].

As was the case for the positivity conjecture, a disparity exists between what
is known about the $J$-rings of Weyl or affine Weyl groups and the $J$-rings
of other Coxeter systems. With the exception of Alvis’ work on the Coxeter
group of type $H_{4}$ in [Alv08], the structures of the $J$-rings of non-Weyl
Coxeter groups remain largely unexplored. One obstacle to understanding
$J$-rings in general is the diffculty in computing the structure constants of
Hecke algebras with respect to their Kazhdan–Lusztig bases, which are
necessary for obtaining the coefficients $\gamma_{x,y,z^{-1}}$ in Equation
(1). As we will show, however, it is possible to circumvent this obstacle if
we restrict from the $J$-ring to the subregular $J$-ring. In [Xu19], the last
named author gave a description of products of the form $t_{x}t_{y}$ in
$J_{C}$ that does not involve Kazhdan–Lusztig theory. In the present paper, we
use this description to show that $J_{C}$ is isomorphic to certain quotients
of the path algebra of a quiver, then use quiver representations to study
representations of $J_{C}$. Roughly speaking, the reason why we can understand
$J_{C}$ in full generality, in contrast to the entire $J$-ring, is that the
rigidity of the elements of $C$ makes $C$ and $J_{C}$ more amenable to
combinatorial analysis. It seems interesting that a similar contrast is also
visible in the book [Bon17] by Bonnafé, where he singles out the subregular
cell in Chapters 12 and 13 (he calls the cell the _submaximal cell_) and
exploits its rich combinatorics in his investigation of various
Kazhdan–Lusztig objects attached to the cell, including the so-called _cell
module_ of $C$ and its connection to the reflection representation of the
Hecke algebra of $(W,S)$.

Let us elaborate on how $J_{C}$ relates to quivers. Recall that every Coxeter
system $(W,S)$ corresponds to a unique Coxeter diagram $G$ with vertex set $S$
and edge set $\\{a-b:a,b\in S,m(a,b)\geq 3\\}$. We define the _double quiver_
of $(W,S)$ to be the directed graph $Q=(Q_{0},Q_{1})$ with vertex set
$Q_{0}=S$ and edge set $Q_{1}=\\{a\rightarrow b:a,b\in S,m(a,b)\geq 3\\}$,
where we have a pair of arrows $a\rightarrow b$ and $b\rightarrow a$ arising
from each edge $a-b$ in $G$. Next, we consider the path algebra $\mathbb{Z}Q$
of $Q$ over $\mathbb{Z}$ and associate to each suitable family
$\\{f_{n}:n\in\mathbb{Z}_{\geq 2}\\}$ of polynomials an ideal
$\mathcal{I}_{f}^{\mathbb{Z}}$ in $\mathbb{Z}Q$ called an _evaluation ideal_
of $\\{f_{n}:n\in\mathbb{Z}_{\geq 2}\\}$ (Definition 3.4). Our first main
result, Theorem 3.6, establishes an algebra isomorphism between $J_{C}$ and
the quotient $\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}$ where
$\mathcal{I}_{u}^{\mathbb{Z}}$ is the evaluation ideal of a family
$\\{u_{n}\in\mathbb{Z}[x]:n\geq 2\\}$ of “Chebyshev polynomials”.

Fixing an algebraically closed field $K$ of characteristic zero, we extend the
result that $J_{C}\cong\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}$ in two ways
(see Remark 4.17 for a discussion about assumptions on the field $K$). First,
in Theorem 3.7, we show that upon an extension of scalars we may alter the
family $\\{u_{n}\\}$ without changing the isomorphism type of the quotient of
the path algebra by the evaluation ideal. More precisely, we show that for any
two _uniform_ families of polynomials $\\{f_{n}\\},\\{g_{n}\\}$ over $K$
(Definition 3.3), we have $KQ/\mathcal{I}_{f}\cong KQ/\mathcal{I}_{g}$ where
$KQ$ is the path algebra of $Q$ and $\mathcal{I}_{f},\mathcal{I}_{g}$ are
evaluation ideals of $KQ$ constructed from $\\{f_{n}\\},\\{g_{n}\\}$. The
Chebyshev polynomials $\\{u_{n}\\}$ form a uniform family, and the result
enables us to realize the algebra $A_{K}:=K\otimes_{\mathbb{Z}}J_{C}$ as a
quotient $KQ/\mathcal{I}_{f}$ where the ideal $\mathcal{I}_{f}$ is generated
by elements which can take very simple forms; see § 4.2. Together, Theorems
3.6 and 3.7 generalize Example 6.10 of Diaz-Lopez’s paper [DL15]. That paper
cites the example as its main motivation and remarks that the example suggests
a stronger connection between path algebras and asymptotic Hecke algebras. We
hope our result can be viewed as further support for such a connection.

In our second extension of Theorem 3.6, we develop a procedure to modify a
quiver $Q$ to a new quiver $\bar{Q}$ such that the algebras
$KQ/\mathcal{I}_{f}$ and $K\bar{Q}/\bar{\mathcal{I}}_{f}$ are Morita
equivalent for any uniform family of polynomials $\\{f_{n}\\}$, where
$\bar{\mathcal{I}}_{f}$ is the evaluation ideal of $K\bar{Q}$ associated to
$\\{f_{n}\\}$; see Theorem 4.5. We call the procedure a _quiver contraction_
and will often apply it iteratively, starting from the double quiver of a
Coxeter diagram. Quiver contractions reveal certain interesting algebras that
are Morita equivalent to algebras $A_{K}$ associated to Coxeter systems, such
as the Laurent polynomial ring $K[t,t^{-1}]$ or group algebras of free
products of finite cyclic groups; see Examples 4.10 and 4.12. In addition, we
use quiver contractions to justify certain assumptions on Coxeter systems in
the study of representations of $A_{K}$. For example, for any Coxeter system
whose Coxeter diagram $G$ is a tree, quiver contractions allow us to assume
that $G$ contains no simple edges when studying representations of $A_{K}$;
see Example 4.8.

Theorem 3.6 and its extensions allow us to study representations of the
subregular $J$-ring via quivers. More precisely, we use the double quiver $Q$
to study the category mod-$A_{K}$ of finite dimensional right modules of the
algebra $A_{K}$. Representations of the $J$-ring and of rings of the form
$J_{E}$ (where $E$ is a two-sided cell) are not only interesting on their own
but also intimately related to representations of $W$ and its Hecke algebra;
see [Lus14b], [Lus14a], [Lus18], [Gec07] and [Pie10]. On the other hand,
quivers arise naturally in many areas of mathematics and have close
connections to the representation theory of finite dimensional algebras,
Kac–Moody algebras, quantum groups, and so on; see [Sav05] and [Sch14].

To study mod-$A_{K}$ via $Q$, we use the well-known fact that for each ideal
$\mathcal{I}$ in $KQ$, the category of modules of the quotient
$KQ/\mathcal{I}$ is equivalent to the category of _representations of $Q$_
that _satisfy_ the relations in $\mathcal{I}$ (see § 2.4). Our main results
are Theorems 5.1 and 5.2, which characterize in terms of the Coxeter diagram
$G$ when the category mod-$A_{K}$ is semisimple, contains finite many simple
modules, or has a bound on the dimensions of simple modules. In a sense, the
chacterizations are similar to those of the representation types of quivers
given by the celebrated Gabriel’s Theorem (see [DDPW08]). Since we can use
quiver contractions to show that every group algebra of a free product of
finite cyclic groups is Morita equivalent to the algebra $A_{K}$ for a
suitable Coxeter system (Example 4.12), Theorems 5.1 and 5.2 lead to similar
characterizations for the module categories of such group algebras, which may
be of independent interest as they are stated without mention of Coxeter
systems or Kazhdan–Lusztig theory; see Remark 5.3 and Proposition 5.4.

The rest of the paper is organized as follows. In Section 2, we recall the
relevant background on Coxeter systems, subregular $J$-rings, path algebras,
and quiver representations. In Section 3, we define uniform families of
polynomials $\\{f_{n}\\}$ and their associated evaluation ideals, then we
realize $J_{C}$ and the algebra $A_{K}$ as quotients of path algebras by
suitable evaluation ideals via Theorems 3.6 and 3.7. Section 4 deals with
quiver contractions and its main result is Theorem 4.5, which asserts that
$KQ/\mathcal{I}_{f}$ is Morita equivalent to $K\bar{Q}/\bar{\mathcal{I}}_{f}$
if the quiver $\bar{Q}$ is obtained from $Q$ via a sequence of contractions.
We define contractions in § 4.1, give detailed examples of contractions in §
4.2 and prove Theorem 4.5 in § 4.3, then we analyze and give examples of
representations of contracted quivers in § 4.4. Finally, we state and prove
the results on mod-$A_{K}$ in Section 5. Most of the examples from § 4.2 and §
4.4 will be used in the proofs.

### Acknowledgements

The first three named authors are supported by the National Sciences and
Engineering Research Council of Canada. The second and third named authors are
also supported by the Canadian Defence Academy Research Programme. We thank R.
M. Green for reading a draft of the paper and for his helpful comments.

##  2\. Background

###  2.1. Coxeter Systems

A _Coxeter system_ is a pair $(W,S)$ where $S$ is a finite set and $W$ is the
group given by the presentation

$W=\langle S\,|\,(ab)^{m(a,b)}=1\;\text{for all $a,b\in S$ with
$m(a,b)<\infty$}\rangle,$

where $m$ denotes a map $m:S\times S\rightarrow\mathbb{Z}_{\geq
1}\cup\\{\infty\\}$ such that for all $a,b\in S$, we have $m(a,b)=m(b,a)$, and
$m(a,b)=1$ if and only if $a=b$. These conditions imply that $a^{2}=1$ for all
$a\in S$ and that

(2) $aba\dots=bab\dots,$

where both sides contain $m(a,b)$ factors, for every two distinct generators
$a,b\in S$ with $m(a,b)<\infty$. We call each side of Equation (2) an
$\\{a,b\\}$-_braid_ and call the equation a _braid relation_.

Each Coxeter system $(W,S)$ can be encoded via its _Coxeter diagram_ , the
weighted, undirected graph $G$ whose vertex set is $S$, whose edge set is
$\\{\\{a,b\\}:m(a,b)\geq 3\\}$, and where the weight of an edge $\\{a,b\\}$ is
$m(a,b)$. An edge with weight $m$ in $G$ is _simple_ if $m=3$ and is _heavy_
otherwise. When drawing $G$, we label each edge with its weight except for
simple edges. A Coxeter system $(W,S)$ is said to be _irreducible_ if its
Coxeter diagram $G$ is connected and _reducible_ otherwise.

For the rest of the paper, we let $(W,S)$ be an irreducible Coxeter system and
let $G$ be its Coxeter diagram. The irreduciblity assumption is made to
simplify our statements, as the reducible case can be easily derived from the
irreducible case for all the relevant results; see Remark 3.8.

###  2.2. The Subregular J-ring

Let $S^{*}$ be the free monoid generated by $S$. For each element $w\in W$,
the words in $S^{*}$ that express $w$ and have minimal length are called the
_reduced words_ of $w$. The common length of these words, denoted $l(w)$, is
called the _length_ of $w$. By the well-known Matsumoto–Tits theorem, every
two reduced words of $w$ can be obtained from each other via a finite sequence
of braid relations.

An element in $W$ is called _rigid_ if it has a unique reduced word. In this
paper we are particularly interested in the set

$C=\\{w\in W:w\neq 1,w\text{\; is rigid}\\}.$

The set $C$ is known to be a _two-sided Kazhdan–Lusztig cell_ of $W$, and is
called the _subregular cell_ or _submaximal cell_ of $W$ (see [Xu19] and
[Bon17, Chapter 12]).

######  Remark 2.1.

(a) By the Matsumoto–Tits theorem, a word $w\in S^{*}$ expresses an element in
$C$ if and only if $w$ is nonempty and does not contain as a contiguous
subword a word of the form $aa$ for any $a\in S$ or an $\\{a,b\\}$-braid for
any distinct elements $a,b\in S$.

(b) Henceforth we will identify each element $w\in C$ with its unique reduced
word. In particular, we will also use $w$ to denote the reduced word of the
element (as in Propositions 2.2 and 3.9, for example).

To define the subregular $J$-ring, we first recall the construction of the
_$J$ -ring_, or the _asymptotic Hecke algebra_ , of $(W,S)$. The construction
is due to Lusztig, who defined the $J$-ring as the free abelian group
$J:=\oplus_{w\in W}\mathbb{Z}t_{w}$ and defined multiplication in $J$ by the
formula

$t_{x}t_{y}=\sum_{z\in W}\gamma_{x,y,z^{-1}}t_{z}$

where each coefficient $\gamma_{x,y,z^{-1}}$ is a certain nonnegative integer
extracted from the structure constants for the _Kazhdan–Lusztig basis_ of the
_Iwahori–Hecke algebra_ of $(W,S)$; see [Lus87] and [Lus14b, Section 18.3].
Lusztig showed that for each two-sided cell $E$ of $W$, the subgroup
$J_{E}:=\oplus_{w\in E}\mathbb{Z}t_{w}$ is in fact a subring of $J$. We define
the _subregular $J$-ring_ to be the subring $J_{C}$ of $J$ arising from the
subregular cell $C$ of $W$.

While the definition of $J$ relies heavily on Kazhdan–Lusztig theory, it is
shown in [Xu19] that we can describe products in the subregular $J$-ring via
simple manipulations of reduced words. To do so, for each pair of distinct
generators $a,b\in S$, let us call an element $w\in C$ an _$\\{a,b\\}$
-element_ if $w$ lies in the subgroup of $W$ generated by $a$ and $b$. For two
words $x=\dots a_{2}a_{1},y=b_{1}b_{2}\dots\in S^{*}$ with $a_{1}=b_{1}$, let
$x*y$ be the word $\dots a_{2}b_{1}b_{2}\dots$, the result of concatenating
$x$ and $y$ and deleting one duplicate copy of the letter $a_{1}=b_{1}$. Then
products in $J_{C}$ behave as follows:

######  Proposition 2.2 ([Xu19, Corollary 4.2, Propositions 4.4 & 4.5]).

Let $x,y$ be elements of $C$ with reduced words $x=\dots a_{2}a_{1}$ and
$y=b_{1}b_{2}\dots$, where we take $a_{2}$ and $b_{2}$ to be nonexistant when
$l(x)=1$ and $l(y)=1$, respectively. Then the following holds.

  1. (a)

If $a_{1}\neq b_{1}$, then $t_{x}t_{y}=0$.

  2. (b)

If $a_{1}=b_{1}$ and $a_{2}\neq b_{2}$ (including the vacuous cases where
$a_{2}$ or $b_{2}$ do not exist), then $t_{x}t_{y}=t_{x*y}$.

  3. (c)

If $a_{1}=b_{1}$ and $x,y$ are both $\\{a,b\\}$-elements for some $a,b\in S$,
then $t_{x}t_{y}$ is a linear combination of the form $\sum_{z\in Z}t_{z}$
where $Z$ is a certain set of $\\{a,b\\}$-elements.

Note that the first two parts of the proposition imply that $J_{C}$ has a
unit, namely, the element $\sum_{a\in S}t_{a}$. In the last part, the set $Z$
can be obtained via a _truncated Clebsch–Gordan rule_ , but the exact
description of $Z$ is not essential to this paper, so we omit it. Instead, we
describe below the product $t_{x}t_{y}$ from Proposition 2.2.(c) in the
special case where $l(x)=2$. The special case is in fact equivalent to the
general case because one can deduce the latter from the former by induction.

######  Proposition 2.3 ([Xu19, Corollary 4.2]).

Let $a,b\in S$ and let $m=m(a,b)$. Suppose that $m\geq 3$. For all $1<i<m$,
let $w_{a,i}$ be the $\\{a,b\\}$ element $aba\dots$ of length $i$ and let
$t_{a,i}=t_{w_{a,i}}$, then define $w_{b,i}=bab\dots$ and $t_{b,i}$ similarly.
Then for all $1<i<m$, we have

(3) $t_{ab}t_{b,i}=\begin{cases}t_{a,i-1}+t_{a,i+1}&\text{if $i<m-1$};\\\
t_{a,i-1}&\text{if $i=m-1$}.\end{cases}$

The following example illustrates how Proposition 2.2 can be used to compute
the product $t_{x}t_{y}$ for all $x,y\in C$: Suppose $(W,S)$ is a Coxeter
system where $S=\\{a,b,c\\}$ and $m(a,b)=3,m(a,c)=4,m(b,c)=5$. Let
$x=abcb,y=bcbcac$. Then $x,y\in C$ by Remark 2.1.(a). The first two parts of
Proposition 2.2 imply that $t_{y}t_{x}=0$ and

$t_{x}t_{y}=(t_{ab}t_{bcb})(t_{bcbc}t_{cac})=t_{ab}(t_{bcb}t_{bcbc})t_{cac}.$

The product $t_{bcb}t_{bcbc}$ can be computed using Part (c) and turns out to
equal $t_{bc}$. Applying Part (b) again completes the computation:

$t_{x}t_{y}=t_{ab}t_{bc}t_{cac}=t_{abcac}.$

Intuitively, as the example shows, the reductions allowed by the first two
parts of Proposition 2.2 mean that the most interesting multiplication in
$J_{C}$ happen “locally”, for elements within subgroups of $W$ generated by
two elements. This fact is a key reason why Theorem 3.6 holds.

###  2.3. Path Algebras

In this and the next subsection, we recall the background on quivers, path
algebras and quiver representations that is relevant to the paper. Our main
reference is [Sch14].

A _quiver_ is a directed graph $Q=(Q_{0},Q_{1})$ where $Q_{0}$ is the set of
vertices and $Q_{1}$ is the set of directed edges, or _arrows_. The sets
$Q_{0}$ and $Q_{1}$ will be finite for all quivers in this paper. For each
arrow $\alpha:a\rightarrow b$, we call $a$ and $b$ the _source_ and the
_target_ of $\alpha$ and denote them by
$\operatorname{\mathrm{source}}(\alpha)$ and
$\operatorname{\mathrm{target}}(\alpha)$, respectively. An arrow $\alpha$ is
called a _loop at $a$_ if
$\operatorname{\mathrm{source}}(\alpha)=\operatorname{\mathrm{target}}(\alpha)=a$.

A _path_ on $Q$ is an element of the form
$p=\alpha_{1}\alpha_{2}\dots\alpha_{n}$ where
$\operatorname{\mathrm{target}}(\alpha_{i})=\operatorname{\mathrm{source}}(\alpha_{i+1})$
for all $1\leq i\leq n-1$; we define the _source_ of $p$ to be
$\operatorname{\mathrm{source}}(p):=\operatorname{\mathrm{source}}(\alpha_{1})$
and the _target_ of $p$ to be
$\operatorname{\mathrm{target}}(p):=\operatorname{\mathrm{target}}(\alpha_{n})$.
To each vertex $a\in Q_{0}$, we associate a special path $e_{a}$ called the
_stationary path at $a$_; we consider it as a path that “stays at $a$”, so in
particular we have
$\operatorname{\mathrm{source}}(e_{a})=\operatorname{\mathrm{target}}(e_{a})=a$.
The _length_ of the path $p$, denoted by $\mathrm{length}(p)$, is defined to
be the number of arrows it traverses. In other words, each arrow has length 1,
each stationary path has length 0, and we have
$\operatorname{\mathrm{length}}(p)=\sum_{i=1}^{n}\operatorname{\mathrm{length}}(\alpha_{i})$
for each path $p=\alpha_{1}\dots\alpha_{n}$.

Let $\mathcal{P}$ be the set of all paths on $Q$, and let $R$ be a commutative
ring. The _path algebra of $Q$ over $R$_, denoted by $RQ$, is the $R$-algebra
with $\mathcal{P}$ as an $R$-basis and with multiplication induced by path
concatenation: for paths
$p=\alpha_{1}\dots\alpha_{m},q=\beta_{1}\dots\beta_{n}\in\mathcal{P}$, we
define $pq$ to be the path $\alpha_{1}\dots\alpha_{m}\beta_{1}\dots\beta_{n}$
if $\operatorname{\mathrm{target}}(p)=\operatorname{\mathrm{source}}(q)$ and
to be 0 otherwise. In particular, for any path $p$ with source $a$ and target
$b$, we have $e_{a}p=p=pe_{b}$ in $RQ$. Consequently, $RQ$ contains the unit
$1=\sum_{a\in Q_{0}}e_{a}$, and we can describe $RQ$ as the algebra generated
by the arrows and stationary paths in $Q$ subject only to the relations
$e_{a}e_{b}=\delta_{a,b}e_{a}$ for all $a,b\in Q_{0}$ and
$e_{a}\alpha=\alpha=\alpha e_{b}$ for each arrow $\alpha:a\rightarrow b$ in
$Q_{1}$.

Among the elements of $RQ$, we will be especially interested in elements of
the form $r=\sum_{p}c_{p}p\in RQ$ where the sum is taken over a finite set of
paths on $Q$ which share the same source and the same target. Following
[Sch14, Definition 3.1], we call such an element $r$ a _uniform relation_ or
simply a _relation_ on $Q$. We define the source and target of $r$ to be the
common source and common target of the paths appearings in it, respectively.
Our first main theorem, Theorem 3.6, asserts that
$J_{C}\cong\mathbb{Z}Q/\mathcal{I}_{f}$ for a suitable quiver $Q$ and a
suitable ideal $\mathcal{I}_{u}^{\mathbb{Z}}$ generated by a set of relations
of the form $\mathcal{R}=\\{r_{u}(\alpha):\alpha\in Q_{1}\\}$, where each
relation corresponds to an arrow in $Q$.

###  2.4. Quiver Representations

Let $Q$ be a quiver and let $K$ be an arbitrary field. We recall below some
basic facts about the representation theory of the path algebra $KQ$ and its
quotients. All representations and modules we mention in this paper will be
finite dimensional.

Let mod-$KQ$ be the category of finite dimensional right $KQ$-modules. It is
well-known that mod-$KQ$ is naturally equivalent to the category
$\mathrm{rep}_{K}Q$ of finite dimensional representations of $Q$ over $K$.
Here, a _representation_ of a quiver $Q$ over $K$ is an assignment

$M=(M_{a},M_{\alpha})_{a\in Q_{0},\alpha\in Q_{1}}$

of a $K$-vector space $M_{a}$ to each vertex $a$ of $Q$ and a linear map
$M_{\alpha}:M_{a}\rightarrow M_{b}$ for each arrow $\alpha:a\rightarrow b$ in
$Q$; the _dimension_ of $M$ is defined by $\dim(M):=\sum_{a\in
Q_{0}}\dim(M_{a})$. A morphism $\varphi:M\rightarrow N$ between two
representations $M,N$ of $Q$ consists of the data $\varphi=(\varphi_{a})_{a\in
Q_{0}}$ of linear maps $\varphi_{a}:M_{a}\rightarrow N_{a}$ for $a\in Q_{0}$
such that $\varphi_{b}\circ M_{\alpha}=N_{\alpha}\circ\varphi_{a}$ for every
arrow $\alpha:a\rightarrow b$ in $Q$. The equivalence between the two
categories can be established by two naturally defined quasi-inverse functors
$\mathcal{F}:\textrm{mod-}KQ\rightarrow\mathrm{rep}_{K}Q$ and
$\mathcal{G}:\mathrm{rep}_{K}Q\rightarrow\textrm{mod-}KQ$; see [Sch14, Chapter
5].

We can modify the equivalence between mod-$KQ$ and $\mathrm{rep}_{K}Q$ to
account for relations on $Q$. To do so, for each representation $M$ of $Q$, we
set $M_{e_{a}}=\operatorname{\mathrm{id}}_{M_{a}}$ for all $a\in Q_{0}$ and
associate to each path $p=\alpha_{1}\dots\alpha_{n}$ on $Q$ the map

$M_{p}:=M_{\alpha_{n}}\circ\dots\circ M_{\alpha_{1}},$

and we say that $M$ _satisfies_ a relation $r=\sum c_{p}p$ if $\sum
c_{p}M_{p}=0$. For an ideal $\mathcal{I}$ of $KQ$ generated by a set of
relations $\mathcal{R}$, define a representation of $Q$ to be a
_representation of $(Q,\mathcal{I})$_ if it satisfies all relations in
$\mathcal{R}$. Finally, let $\mathrm{rep}_{K}(Q,\mathcal{I})$ be the full
subcategory of $\mathrm{rep}_{K}Q$ whose objects are the representations of
$(Q,\mathcal{I})$. Then it is well-known that
$\mathrm{rep}_{K}(Q,\mathcal{I})$ is equivalent to mod-$KQ/\mathcal{I}$, the
category of finite dimensional right modules of the quotient $KQ/\mathcal{I}$.

######  Remark 2.4.

We introduce two types of shorthand notation to be used for the rest of the
paper. First, for a category $\mathbf{C}$, we will write $M\in\mathbf{C}$ to
mean that $M$ is an object in $\mathbf{C}$. Second, given a two-sided ideal
$I$ in a ring $R$ and an element $r\in R$, we will denote the coset $r+I$
simply by $r$.

Familiar notions from mod-$KQ$ have obvious counterparts in
$\mathrm{rep}_{K}Q$: The _zero representation_ in $\mathrm{rep}_{K}Q$ is the
representation $M$ with $M_{a}=0$ for all $a\in Q_{0}$. A _subrepresentation_
of a representation $M$ is an assignment $N=(N_{a},N_{\alpha})_{a\in
Q_{0},\alpha\in Q_{1}}$ such that for every arrow $\alpha:a\rightarrow b$ in
$Q$, we have $N_{a}\subseteq M_{a}$, $M_{\alpha}(N_{a})\subseteq N_{b}$, and
$N_{\alpha}$ equals the restriction of $M_{\alpha}$ to $N_{a}$. A
representation is _simple_ if it does not contain any proper, nonzero
subrepresentation. The _direct sum_ of two representations $M,N$ is the
reprentation $M\oplus N$ where $(M\oplus N)_{a}=M_{a}\oplus N_{a}$ and
$(M\oplus N)_{\alpha}((m,n))=(M_{\alpha}(m),N_{\alpha}(n))$ for every arrow
$\alpha:a\rightarrow b$ and every element $(m,n)\in M_{a}\oplus N_{a}$.
Finally, a representation is _semisimple_ if it is a direct sum of simple
representations, and each of $\mathrm{rep}_{K}Q$ and
$\mathrm{rep}_{K}(Q,\mathcal{I})$ is _semisimple_ if all representations in it
are semisimple. These notions agree with their counterparts in mod-$KQ$ under
the equivalences $\mathcal{F}$ and $\mathcal{G}$. For example, a
representation $M\in\mathrm{rep}_{K}Q$ is simple if and only if the module
$\mathcal{G}(M)\in\text{mod-}KQ$ is simple, and
$\mathrm{rep}_{K}(Q,\mathcal{I})$ is semisimple if and only if
mod-$KQ/\mathcal{I}$ is semisimple. Indeed, the agreement of the notions can
be attributed to the facts that mod-$KQ$ and $\mathrm{rep}_{K}Q$ are abelian
categories, that the definitions in $\mathrm{rep}_{K}Q$ and mod-$KQ$ are
specializations of the corresponding categorical notions, and that
$\mathcal{F},\mathcal{G}$ are equivalences of abelian categories.

##  3\. Quiver Realizations

Henceforth, let $K$ be an algebraically closed field of characteristic zero,
let $(W,S)$ be a Coxeter system, and let $G,C$ and $J_{C}$ be the Coxeter
diagram, subregular cell and subregular $J$-ring of $(W,S)$, respectively. Let
$A=A_{K}:=K\otimes_{\mathbb{Z}}J_{C}$. In this section, we associate a quiver
$Q$ to $(W,S)$ and then show that
$J_{C}\cong\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}$ and $A\cong
KQ/\mathcal{I}_{f}$ for suitable ideals
$\mathcal{I}_{u}^{\mathbb{Z}}\subseteq\mathbb{Z}Q$ and
$\mathcal{I}_{f}\subseteq KQ$. By § 2.4, the latter isomorphism will allow us
to study the category $\mathrm{rep}_{K}A$ via the equivalent category
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$.

###  3.1. Statement of Results

Let $Q=(Q_{0},Q_{1})$ be the quiver with $Q_{0}=S$ and $Q_{1}=\\{(a,b):a,b\in
S,m({a,b})\geq 3\\}$. Each edge $a-b$ in the Coxeter diagram $G$ gives rise to
a pair of arrows $a\rightarrow b$ and $b\rightarrow a$ in $Q$, and all arrows
of $Q$ arise this way. For an arrow $\alpha:a\rightarrow b$ in $Q$, we call
the arrow $b\rightarrow a$ arising from the same edge in $G$ the _dual arrow
of $\alpha$_ and denote it by $\bar{\alpha}$; we define the _weight_ of
$\alpha$ to be $m(a,b)$ and denote it by $m_{\alpha}$. We call the quiver $Q$
the _double quiver of $(W,S)$_ or the _double quiver of $G$_.

The ideal $\mathcal{I}_{u}^{\mathbb{Z}}$ for which
$J_{C}\cong\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}$ is generated by a set of
(uniform) relations obtained via _arrow evaluations_ of polynomials from
suitable polynomial families. We first define arrow evaluations:

######  Definition 3.1.

For each arrow $\alpha$ in $Q$, let $\mathrm{Eval}_{\alpha}:K[x]\rightarrow
KQ$ be the unique $K$-linear map such that $\mathrm{Eval}_{\alpha}(1)=e_{a}$
where $a=\operatorname{\mathrm{source}}(\alpha)$ and

$\mathrm{Eval}_{\alpha}(x^{n})=\alpha\bar{\alpha}\alpha\dots,$

the product with $n$ factors that start with $\alpha$ and alternate in
$\alpha$ and $\bar{\alpha}$, for all $n>0$. For each polynomial $f\in K[x]$,
we write

$f(\alpha,\bar{\alpha}):=\mathrm{Eval}_{\alpha}(f)$

and call $f(\alpha,\bar{\alpha})$ the _$\alpha$ -evaluation of $f$_.

By a “polynomial family” we mean a countable collection
$\\{f_{n}:n\in\mathbb{Z}_{\geq 2}\\}$ of polynomials in $K[x]$. Note that for
$f(\alpha,\bar{\alpha})$ to yield a uniform relation on $Q$, the polynomial
$f$ needs to be either even or odd, therefore we will consider only polynomial
families $\\{f_{n}\\}$ where each $f_{n}$ is either an even or an odd
polynomial. To describe further conditions we would like to impose on
$\\{f_{n}\\}$, we need more notation:

######  Definition 3.2.

For each even polynomial $f=\sum c_{i}x^{2i}\in K[x]$, let

$\tilde{f}=\sum{c_{i}}x^{i};$

for each odd polynomial $f=\sum c_{i}x^{2i+1}\in K[x]$, let

$\tilde{f}=\sum{c_{i}}x^{i}.$

Note that when $f$ is an even or odd polynomial of degree $n$, the polynomial
$\tilde{f}$ has degree $\lfloor{n/2}\rfloor$ where $\lfloor{-}\rfloor$ denotes
the floor function; moreover, we have

(4)
$f(\alpha,\bar{\alpha})=\begin{cases}\tilde{f}(\alpha\bar{\alpha})&\text{if
$f$ is even};\\\
\tilde{f}(\alpha\bar{\alpha})\cdot\alpha=\alpha\cdot\tilde{f}(\bar{\alpha}\alpha)&\text{if
$f$ is odd},\end{cases}$

where we evaluate a constant term $c$ in $\tilde{f}$ to $ce_{a}$ for
$a=\operatorname{\mathrm{source}}(\alpha)$. For example, if $f=x^{3}-2x$ then
$\tilde{f}=x-2$ and $f(\alpha,\bar{\alpha})=\alpha\bar{\alpha}\alpha-2\alpha$,
and if $f=x^{4}-1$ then $\tilde{f}=x^{2}-1$ and
$f(\alpha,\bar{\alpha})=\alpha\bar{\alpha}\alpha\bar{\alpha}-e_{a}$ where
$a=\operatorname{\mathrm{source}}(\alpha)$.

We are ready to define the polynomial families we need.

######  Definition 3.3.

A _uniform_ family of polynomials (over $K$) is a set

$\\{f_{n}\in K[x]:n\in\mathbb{Z}_{\geq 2}\\}$

such that for all $n\in\mathbb{Z}_{\geq 2}$, we have

  1. (a)

$f_{n}$ has degree $n$, is even when $n$ is even, and is odd when $n$ is odd.

  2. (b)

zero is not a root of $\tilde{f}_{n}$, and no root of $\tilde{f}_{n}$ is
repeated.

Given a uniform polynomial family, we assign one relation to each arrow and
define an ideal $\mathcal{I}_{f}$ of $KQ$ as follows:

######  Definition 3.4.

Let $\\{f_{n}:n\geq 2\\}$ be a uniform family of polynomials.

  1. (a)

For each arrow $\alpha$ in $Q$, we set $m=m_{\alpha}$ and define

(5) $r_{f}(\alpha)=\begin{cases}0&\text{if $m=\infty$};\\\
f_{m-1}(\alpha,\bar{\alpha})&\text{if $m<\infty$}.\end{cases}$

  2. (b)

We define the _evaluation ideal of $\\{f_{n}\\}$_ to be the two-sided ideal

$\mathcal{I}_{f}:=\langle r_{f}(\alpha):\alpha\in Q_{1}\rangle$

of $KQ$ generated by the relations of the form $r_{f}(\alpha)$. More
generally, if $f_{n}\in R[x]$ for all $n\geq 2$ for some subring $R$ of $K$,
we define $\mathcal{I}_{f}^{R}$ to be the two-sided ideal of $RQ$ given by

$\mathcal{I}_{f}^{R}:=\langle r_{f}(\alpha):\alpha\in Q_{1}\rangle\subseteq
RQ.$

######  Example 3.5.

Suppose $K=\mathbb{C}$, and consider the polynomials $u_{n}$ for $n\geq 0$
where

(6) $u_{0}=1,\quad u_{1}=x,\quad\text{and\;
}u_{n}=xu_{n-1}-u_{n-2}\;\text{\;for all $n\geq 2$}.$

These polynomials are normalizations of the _Chebyshev polynomials of the
second kind_. It is easy to see by induction that for each $n\geq 2$, the
polynomial $u_{n}$ has degree $n$, is even when $n$ is even, and is odd when
$n$ is odd. Moreover, it is known that $u_{n}$ has $n$ distinct nonzero real
roots $z_{1},\dots,z_{n}$ where

$z_{i}=2\cos\left(\dfrac{i\pi}{n+1}\right)$

for each $i$. The definition of the polynomial $\tilde{u}_{n}$ implies that
$\tilde{u}_{n}$ has $\lfloor\frac{n}{2}\rfloor$ distinct nonzero roots,
namely, the numbers $z_{i}^{2}$ where $1\leq i\leq\lfloor\frac{n}{2}\rfloor$.
It follows that $\\{u_{n}:n\geq 2\\}$ forms a uniform family of polynomials
over $\mathbb{C}$. Note that $u_{n}\in\mathbb{Z}[x]$ for all $n\geq 2$, so
$\mathcal{I}_{u}^{\mathbb{Z}}$ makes sense as an ideal of $\mathbb{Z}Q$.

We state our first two results below.

######  Theorem 3.6.

Let $\\{u_{n}:n\in\mathbb{Z}_{\geq 2}\\}$ be as in Example 3.5. Then
$J_{C}\cong\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}$ as unital rings.

######  Theorem 3.7.

Let $K$ be an algebraically closed field of characteristic zero and let
$\\{f_{n}\\},\\{g_{n}\\}$ be two uniform families of polynomials over $K$.
Then $KQ/\mathcal{I}_{f}\cong KQ/\mathcal{I}_{g}$ as $K$-algebras.

######  Remark 3.8.

We can now explain why it suffices to deal with only irreducible Coxeter
systems in this paper. Recall that if $(W,S)$ is reducible, then the connected
components of its Coxeter diagram are the diagrams of Coxeter systems
$(W_{i},S_{i})$ for $1\leq i\leq k$ for some $k\geq 2$, and we have
$S=\sqcup_{i}S_{i},W=\Pi_{i}W_{i}$, where the symbols $\sqcup$ and $\Pi$
denote disjoint union and direct product, respectively. Now let $C(i),Q(i)$ be
the subregular cell and the double quiver of $(W_{i},S_{i})$ for each $i$.
Then $C=\sqcup_{i}C(i)$ by definition and $J_{C}=\Pi_{i}J_{C(i)}$ by Part (a)
of Proposition 2.2. On the other hand, for any uniform polynomial family
$\\{f_{n}\\}$ over $K$ where $f_{n}\in\mathbb{Z}[x]$ for all $n$ (such as
$\\{u_{n}\\}$) it is easy to see that
$\mathbb{Z}Q/\mathcal{I}_{f}^{\mathbb{Z}}=\Pi_{i}\mathbb{Z}Q(i)/\mathcal{I}_{f}^{\mathbb{Z}}(i)$
and $KQ/\mathcal{I}_{f}\cong\Pi_{i}KQ(i)/\mathcal{I}_{f}(i)$, where for each
$i$ the ideals $\mathcal{I}_{f}^{\mathbb{Z}}(i)$ and $\mathcal{I}_{f}(i)$ are
the evaluation ideals of $\\{f_{n}\\}$ in $\mathbb{Z}Q(i)$ and $KQ(i)$,
respectively. It follows that we can deduce Theorem 3.6 and Theorem 3.7 for
reducible Coxeter systems from the irreducible cases by taking suitable direct
products.

###  3.2. Proof of Theorem 3.6

In this section we prove Theorem 3.6 by constructing an explicit isomorphism
$\bar{\varphi}:\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}\rightarrow J_{c}$. To
connect the two sides of the isomorphism, first observe that given any element
$w=s_{1}s_{2}\dots s_{k}\in C$, we must have $m(s_{i},s_{i+1})\geq 3$ for all
$1\leq i\leq k-1$: otherwise we can exchange $s_{i}$ and $s_{i+1}$ to obtain
another reduced word of $w$, contradicting the fact that $w$ is rigid. It
follows that the quiver $Q$ contains an arrow $\alpha_{i}:s_{i}\rightarrow
s_{i+1}$ for all $1\leq i\leq k$ as well as the path

$p_{w}:=\alpha_{1}\alpha_{2}\cdots\alpha_{k-1}.$

Recall the notation $\mathcal{P}$ for the set of all paths on $Q$, and
consider the map $\iota:C\rightarrow\mathcal{P}$ which sends $w$ to $p_{w}$
for all $w\in C$. For each arrow $\alpha:a\rightarrow b$ in $Q$ with
$m_{\alpha}<\infty$, let $p_{\alpha}:=\alpha\bar{\alpha}\alpha\dots$ be the
path of length $m_{\alpha}-1$ obtained by concatenating $\alpha$ and
$\bar{\alpha}$ repeatedly. Define a path $p\in\mathcal{P}$ to be _unbraided_
if it does not contain $p_{\alpha}$ as a subpath, i.e., if we cannot write
$p=p_{1}p_{\alpha}p_{2}$ for some paths $p_{1},p_{2}\in\mathcal{P}$, for all
$\alpha\in Q_{1}$ with $m_{\alpha}<\infty$. Let $\mathrm{Unbr}(Q)$ be the set
of unbraided paths in $\mathcal{P}$. Then by Remark 2.1.(a), the image of
$\iota$ is exactly $\mathrm{Unbr}(Q)$. Since $\iota$ is clearly injective, it
gives a bijection from $C$ to $\mathrm{Unbr}(Q)$. We will henceforth use
$\iota$ exclusively to denote this bijection. The definitions and notation of
this paragraph are inspired by those from [Bon17, Chapter 12], where the
bijection $\pi:\mathrm{Unbr}(Q)\rightarrow C$ is essentially the inverse of
$\iota$.

Having connected $C$ to $\mathbb{Z}Q$, let us next consider the effect of
quotienting $\mathbb{Z}Q$ by the ideal $\mathcal{I}_{u}^{\mathbb{Z}}$. Let
$\alpha\in Q_{1}$ and let $m=m_{\alpha}$. Since the polynomial $u_{m-1}$ has
degree $(m-1)$, the relation
$r_{f}(\alpha)=r_{m-1}(\alpha,\bar{\alpha})\in\mathcal{I}_{u}^{\mathbb{Z}}$
must be a linear combination of the alternating path
$q:=\alpha\bar{\alpha}\alpha\cdots$ of length $m-1$ and strictly shorter,
unbraided paths in $\mathbb{Z}Q$. Since $\alpha$ is arbitrary, it follows that
modulo $\mathcal{I}_{u}^{\mathbb{Z}}$ we can rewrite every path as a linear
combination of unbraided paths. In other words, every element in the quotient
$\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}$ can be represented in the form
$\sum_{p\in\mathrm{Unbr}(Q)}c_{p}p$ where the coefficients
$c_{p}\in\mathbb{Z}$ are zero for all but finitely many paths.

The final tool we need concerns a natural filtration of $J_{C}$. For each
$i\in\mathbb{Z}_{\geq 0}$, let $C^{(i)}=\\{w\in C:l(w)\leq i+1\\}$ and let
$J_{C}^{(i)}=\oplus_{w\in C^{(i)}}\mathbb{Z}t_{w}$. As the example at the end
of § 2.2 illustrates, Propositions 2.2 and 2.3 imply that given elements
$x,y\in C$ with length $l(x)=p+1$ and $l(y)=q+1$ for some $p,q\geq 0$, the
product $t_{x}t_{y}$ is always a linear combination of terms of the form
$t_{z}$ where $l(z)\leq p+q+1$. It follows that the filtration

(7) $0\subseteq J_{C}^{(0)}\subseteq J_{C}^{(1)}\subseteq\dots.$

equips $J_{C}$ with the structure of a filtered algebra. The same propositions
also imply the following result.

######  Proposition 3.9.

Let $w=s_{1}s_{2}\dots s_{k}\in C$. Then in $J_{C}$, we have

$t_{s_{1}s_{2}}t_{s_{2}s_{3}}\dots t_{s_{k-1}s_{k}}\in t_{w}+J_{C}^{(k-2)}$

where $t_{w}+J_{C}^{(k-2)}=\\{t_{w}+z:z\in J_{C}^{(k-2)}\\}$. In other words,
the product is the sum of $t_{w}$ and a linear combination of terms $t_{y}$
for which $l(y)<k$.

This proposition will be useful for proving that the map
$\bar{\varphi}:\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}\rightarrow J_{C}$ is
an isomorphism, because we will examine several outputs of the map
$\bar{\varphi}$ which have the form $t_{s_{1}s_{2}}t_{s_{2}s_{3}}\dots
t_{s_{k-1}s_{k}}$. Rather than giving a formal proof of it, however, let us
only sketch the main ideas needed with an example. The proposition follows
from repeated application of Proposition 2.3 in the special case that $w$ is
an $\\{a,b\\}$-element for some $a,b\in S$. The general case then reduces to
the special case in the way illustrated by the following example: suppose
$w=abacacb\in C$ for some Coxeter system and let
$T=t_{ab}t_{ba}t_{ac}t_{ca}t_{ac}t_{cb}$. Then $k=7$, and by the special case
we have

$T=(t_{ab}t_{ba})(t_{ac}t_{ca}t_{ac})(t_{cb})\in\left(t_{aba}+J_{C}^{(1)}\right)\left(t_{acac}+J_{C}^{(2)}\right)\left(t_{cb}+J_{C}^{(0)}\right)$

where the factors in parentheses correspond to the longest “dihedral” subwords
$aba,acac,cb$ of $w$. The filtration (7) implies that all terms $t_{w}$ with
$l(w)=k$ which appear in $T$ must come from the product
$t_{aba}t_{acac}t_{cb}$, where each factor is the “highest degree part” in a
pair of parentheses. This product is nothing but $t_{w}$ by Proposition
2.2.(b), therefore

$T\in t_{aba}t_{acac}t_{cb}+J_{C}^{(5)}=t_{w}+J_{C}^{(k-2)},$

as desired.

We are ready to prove Theorem 3.6. Roughly speaking, the isomorphism holds for
two main reasons: first, as we mentioned in § 2.2, all interesting
multiplications in $J_{C}$ happen “locally” along individual edges of the
Coxeter diagram, just as the relations generating
$\mathcal{I}_{u}^{\mathbb{R}}$ are defined in the same fashion; second, via
arrow evaluations, the recursion from Equation (3) which controls the local
multiplication in $J_{c}$ “agrees with” the recursive definition of
$\\{u_{n}\\}$ which controls the generators of $\mathcal{I}_{u}^{\mathbb{Z}}$.
We make these remarks more precise in the following proposition, where Theorem
3.6 appears as its last assertion.

######  Proposition 3.10.

Let $(W,S)$ be an irreducible Coxeter system and let $Q$ be its double quiver.

  1. (a)

There exists a unique algebra homomorphism $\varphi:\mathbb{Z}Q\rightarrow
J_{C}$ such that for every pair of dual arrows $\alpha:a\rightarrow b$ and
$\beta:b\rightarrow a$ in $Q$, we have

(8)
$\varphi(e_{a})=t_{a},\quad\varphi(e_{b})=t_{b},\quad\varphi(\alpha)=t_{ab},\quad\varphi(\beta)=t_{ba}.$

Moreover, for all $1\leq i\leq m:=m(a,b)$, we have

(9) $\varphi(u_{i-1}(\alpha,\beta))=\begin{cases}t_{a,i}&\text{if $i<m$};\\\
0&\text{if $i=m<\infty$}\end{cases}$

and similarly

(10) $\varphi(u_{i-1}(\beta,\alpha))=\begin{cases}t_{b,i}&\text{if $i<m$};\\\
0&\text{if $i=m<\infty$},\end{cases}$

where $w_{a,i},w_{b,i},t_{a,i},t_{b,i}$ are as in Proposition 2.3.

  2. (b)

The map $\varphi$ factors through the ideal $\mathcal{I}_{u}^{\mathbb{Z}}$ and
induces a homomorphism
$\bar{\varphi}:\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}\rightarrow J_{C}$
given by $\bar{\varphi}(p)=\varphi(p)$ for all $p\in\mathrm{Unbr}(Q)$.

  3. (c)

The map $\bar{\varphi}$ is a unital algebra isomorphism, therefore
$J_{C}\cong\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}$.

###### Proof.

(a) Recall from Section 2.3 that $\mathbb{Z}Q$ is generated by the arrows and
stationary paths of $Q$ subject only to the relations
$e_{u}e_{v}=\delta_{u,v}e_{u}$ for all $u,v\in Q_{0}$ and
$e_{a}\alpha=\alpha=\alpha e_{b}$ for every arrow $\alpha:a\rightarrow b$ in
$Q_{1}$. On the other hand, in $J_{c}$ we have $t_{u}t_{v}=\delta_{u,v}t_{u}$
for all $u,v\in Q_{0}$ and $t_{a}t_{ab}=t_{ab}=t_{ab}t_{b}$ for every arrow
$\alpha:a\rightarrow b$ in $Q_{1}$ by Proposition 2.2. Thus, the relations
satisfied by the generators of $\mathbb{Z}Q$ are respected in the assignment
$e_{a}\mapsto t_{a},\alpha\mapsto t_{ab}$ for all arrows $\alpha:a\rightarrow
b$ in $Q_{1}$. It follows that this assignment extends to a unique algebra
homomorphism $\varphi:\mathbb{Z}Q\rightarrow J_{C}$ which satisfies Equation
(8). The homomorphism is unital since

$\varphi(1)=\varphi\left(\sum_{a\in Q_{0}}e_{a}\right)=\sum_{a\in
Q_{0}}\varphi(e_{a})=\sum_{a\in Q_{0}}t_{a}=1.$

Note that for each $w=s_{1}s_{2}\dots s_{k}\in C$, Equation (8) and the fact
that $\varphi$ is a homomorphism imply that $\varphi(p_{w})$ is exactly the
element $t_{s_{1}s_{2}}t_{s_{2}s_{3}}\dots t_{s_{k-1}s_{k}}$. It follows from
Proposition 3.9 that

(11) $\varphi(p_{w})\in t_{w}+J_{C}^{(l(w)-2)}.\quad$

To prove Equations (9) and (10), we induct on $i$. For $i\leq 2$, the
equations follow from the definition of $\varphi$ and the fact that
$u_{0}=1,u_{1}=x$. For $i>2$, the recursion $u_{i}=xu_{i-1}-u_{i-2}$ implies
that

$u_{i}(\alpha,\beta)=\alpha u_{i-1}(\beta,\alpha)-u_{i-2}(\alpha,\beta),$

therefore

$\displaystyle\varphi(u_{i}(\alpha,\beta))$ $\displaystyle=$
$\displaystyle\varphi(\alpha)\varphi(u_{i-1}(\beta,\alpha))-\varphi(u_{i-2}(\alpha,\beta))$
$\displaystyle=$ $\displaystyle t_{ab}t_{b,i}-t_{a,i-1}$ $\displaystyle=$
$\displaystyle\begin{cases}t_{a,i+1}&\text{if $i<m$};\\\ 0&\text{if
$i=m<\infty$},\end{cases}$

where the second equality holds by induction and the third equality follows
from Proposition 2.3. This proves Equation (9); the proof of Equation (10) is
similar.

(b) By construction, the ideal $\mathcal{I}_{u}^{\mathbb{Z}}$ is generated by
elements of the form $r_{u}(\alpha)=u_{m_{\alpha}-1}(\alpha,\bar{\alpha})$
where $\alpha$ is an arrow in $Q$ with finite weight. Such elements vanish via
$\varphi$ by Equation (9), therefore $\varphi$ factors through
$\mathcal{I}_{u}^{\mathbb{Z}}$ and descends to the map $\bar{\varphi}$ as
claimed.

(c) The map $\bar{\varphi}$ is unital since $\varphi$ is unital. To show that
$\bar{\varphi}$ is surjective, we prove that
$t_{w}\in\operatorname{\mathrm{im}}\bar{\varphi}$ for all $w\in C$ by
induction on $l(w)$. In the base case where $l(w)=1$, we must have $w=a$ for
some $a\in S$ and hence
$t_{w}=\bar{\varphi}(e_{a})\in\operatorname{\mathrm{im}}\bar{\varphi}$. When
$l(w)>1$, we have

$\bar{\varphi}(p_{w})=\varphi(p_{w})\in t_{w}+J_{C}^{(l(w)-2)}$

by (11) and $J_{C}^{(l(w)-2)}\subseteq\operatorname{\mathrm{im}}\bar{\varphi}$
by induction, therefore $t_{w}\in\operatorname{\mathrm{im}}\bar{\varphi}$.

It remains to prove that $\bar{\varphi}$ is injective. Let
$x=\sum_{p\in\mathrm{Unbr}(Q)}c_{p}p$ be a nonzero element in
$\mathbb{Z}Q/\mathcal{I}_{u}^{\mathbb{Z}}$. We need to show that
$\bar{\varphi}(x)\neq 0$. To do so, let $k$ be the maximal number such that
$c_{p}\neq 0$ for some unbraided path $p$ of length $k$, and let
$\\{p_{1},\dots,p_{n}\\}$ be the set of paths of length $k$ appearing with
nonzero coefficients in $x$. Let $w_{i}=\iota^{-1}(p_{i})$ and write
$c_{i}:=c_{p_{i}}$ for all $1\leq i\leq n$. Then $l(w_{i})=k+1$ for all $i$
and we have

$\bar{\varphi}(x)=\sum_{i=1}^{n}c_{i}\varphi(p_{i})\in\sum_{i=1}^{n}c_{i}t_{w_{i}}+J_{C}^{(k-1)}$

by (11). It follows that $\bar{\varphi}(x)\neq 0$, and the proof is complete.
∎

###  3.3. Proof of Theorem 3.7: Dihedral Case

Let $\\{f_{n}\\}$ be a uniform family of polynomials over $K$. As the
generators of the ideal $\mathcal{I}_{f}$ correspond to individual pairs of
dual arrows in $Q$, we first prove Theorem 3.7 in the _dihedral_ case, the
case where $\lvert S\rvert=2$. Let $S=\\{a,b\\}$, let $m=m(a,b)\geq 3$, and
denote the arrows $a\rightarrow b$ and $b\rightarrow a$ in $Q$ by $\alpha$ and
$\beta$, respectively. If $m=\infty$, then $\mathcal{I}_{f}=0$ and the theorem
clearly holds, so until Corollary 3.14 we assume that $m$ is finite. Under
this assumption, we show that $KQ/\mathcal{I}_{f}$ is semisimple and find its
Artin–Wedderburn decomposition. We start with the category
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ in light of the equivalence between
mod-$KQ/\mathcal{I}_{f}$ and $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ (see §
2.4). The simple modules of $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ turn out to
have the following forms:

######  Lemma 3.11.

  1. (a)

For each root $\lambda$ of the polynomial $\tilde{f}_{m-1}$, the assignment

$M(\lambda):=(M_{a},M_{b},M_{\alpha},M_{\beta})=(K,K,\operatorname{\mathrm{id}},\lambda\cdot\operatorname{\mathrm{id}})$

defines a simple representation in $\mathrm{rep}(Q,\mathcal{I}_{f})$.
Moreover, if $\lambda$ and $\lambda^{\prime}$ are distinct roots of
$\tilde{f}_{m-1}$, then $M(\lambda)\not\cong M({\lambda^{\prime}})$.

  2. (b)

If $m$ is even, then the assignments

$S(a):=(K,0,0,0),\quad S(b):=(0,K,0,0)$

define two non-isomorphic simple representations in
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$.

###### Proof.

(a) Recall that $\mathcal{I}_{f}$ is generated by the relations

(12)
$r_{f}(\alpha)=f_{m-1}(\alpha,\beta)=\begin{cases}\tilde{f}_{m-1}(\alpha\beta)&\text{if
$m$ is odd};\\\ \tilde{f}_{m-1}(\alpha\beta)\alpha&\text{if $m$ is
even},\end{cases}$

and

(13)
$r_{f}(\beta)=f_{m-1}(\beta,\alpha)=\begin{cases}\tilde{f}_{m-1}(\beta\alpha)&\text{if
$m$ is odd};\\\ \tilde{f}_{m-1}(\beta\alpha)\beta&\text{if $m$ is
even}.\end{cases}$

The maps $M_{\alpha}M_{\beta}$ and $M_{\beta}M_{\alpha}$ both equal
$\lambda\cdot\operatorname{\mathrm{id}}$ as maps from $K$ to $K$. Since
$\lambda$ is a root of $\tilde{f}_{m-1}$, it follows that
$\tilde{f}_{m-1}(M_{\alpha}M_{\beta})=\tilde{f}_{m-1}(M_{\beta}M_{\alpha})=0$,
hence $M(\lambda)$ satisfies the relations $r_{f}(\alpha)$ and $r_{f}(\beta)$
and forms a representation in $\mathrm{rep}(Q,\mathcal{I}_{f})$. Note that
$M(\lambda)$ is simple by basic linear algebra.

To check that $M(\lambda)\not\cong M({\lambda^{\prime}})$ for distinct roots
$\lambda,\lambda^{\prime}$ of $\tilde{f}_{m-1}$, let
$M({\lambda^{\prime}})=(M^{\prime}_{a},M^{\prime}_{b},M^{\prime}_{\alpha},M^{\prime}_{\beta})$.
Then an isomorphism $\phi:M_{\lambda}\rightarrow M_{\mu}$ must consist of two
linear isomorphisms $\phi_{a}:M_{a}\rightarrow M_{a},\phi_{b}:M_{b}\rightarrow
M_{b}$ such that

$\phi_{b}M_{\alpha}=M^{\prime}_{\alpha}\phi_{a},\quad\phi_{a}M_{\beta}=M^{\prime}_{\beta}\phi_{b}.$

The isomorphisms $\phi_{a},\phi_{b}$ must be multiplication by nonzero scalars
$x,y$, respectively, whence the above equations become $y=x$ and $\lambda
y=\lambda^{\prime}x$. This cannot happen, therefore $M(\lambda)\not\cong
M(\lambda^{\prime})$.

(b) When $m$ is even the assignments $M_{\alpha}=M_{\beta}=0$ clearly satisfy
the relations $r_{f}(\alpha)$ and $r_{f}(\beta)$, so $S(a)$ and $S(b)$ define
representations in $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$. Moreover, the
representations are simple and non-isomorphic by dimension considerations. ∎

To prove $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ is semisimple, we will use the
following linear algebra facts to decompose every representation in
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ into a direct sum of simple modules.

######  Lemma 3.12.

Let $h\in K[X]$ be a polynomial with degree $k\geq 1$ and with $k$ distinct
nonzero roots $z_{1},z_{2},\ldots,z_{k}$ in $K$. Let $U$ and $V$ be finite
dimensional vector spaces, and let $A:U\to V$ and $B:V\to U$ be linear maps
such that

(14) $h(BA)=0_{U}\quad\quad\quad{\text{and}}\quad\quad\quad h(AB)=0_{V}$

or

(15) $h(AB)A=0_{U}\quad\quad\quad{\text{and}}\quad\quad\quad h(BA)B=0_{V}.$

Then the following results hold.

  1. (a)

Both $AB$ and $BA$ are diagonalizable; their eigenvalues lie in the set
$\\{z_{1},z_{2},\ldots,z_{k}\\}$ if (14) holds and in the set
$\\{0,z_{1},z_{2},\ldots,z_{k}\\}$ if (15) holds. In particular, we have
eigenspace decompositions

$U=U_{z_{1}}\oplus U_{z_{2}}\oplus\ldots\oplus U_{z_{k}},\quad
V=V_{z_{1}}\oplus V_{z_{2}}\oplus\ldots\oplus V_{z_{k}}$

if (14) holds and

$U=U_{0}\oplus U_{z_{1}}\oplus U_{z_{2}}\oplus\ldots\oplus U_{z_{k}},\quad
V=V_{0}\oplus V_{z_{1}}\oplus V_{z_{2}}\oplus\ldots\oplus V_{z_{k}}$

if (15) holds, where $U_{\lambda}$ and $V_{\lambda}$ denotes the
$\lambda$-eigenspace of $BA$ and $AB$ for each scalar $\lambda$, respectively.

  2. (b)

For all $1\leq i\leq k$, the restrictions of $A$ to $U_{z_{i}}$ and of
${z_{i}}^{-1}\cdot B$ to $V_{z_{i}}$ form mutually inverse isomorphisms. When
(15) holds, the restrictions of $A$ to $U_{0}$ and of $B$ to $V_{0}$ are both
zero maps.

###### Proof.

(a) The equations in (14) and in (15) imply that the minimal polynomials of
both $AB$ and $BA$ divide $h$ and the polynomial $g:=x\cdot h\in K[x]$,
respectively. The result follows since the polynomials $h$ and $g$ have
distinct roots in the sets $\\{z_{1},\dots,z_{k}\\}$ and
$\\{0,z_{1},\dots,z_{k}\\}$, respectively.

(b) Let $1\leq i\leq k$. Set $U_{i}=U_{z_{i}},V_{i}=V_{z_{i}}$ and
$B^{\prime}=z_{i}^{-1}\cdot B$. Use $|$ to denote restriction of maps so that,
for example, $A|_{U_{i}}$ stands for the restriction of $A$ to $U_{i}$. By
direct computation, we have $Au\in V_{i}$ for all $u\in U_{i}$,
$B^{\prime}v\in U_{i}$ for all $v\in V_{i}$, and
$B^{\prime}A|_{U_{i}}=\operatorname{\mathrm{id}}_{U_{i}},AB^{\prime}|_{V_{i}}=\operatorname{\mathrm{id}}_{V_{i}}$.
This proves the first claim. To prove the second claim, assume the equations
in (15) hold and write $h=x\cdot\bar{h}+c$ where $c$ is the constant term of
$h$. Then $c\neq 0$ since $0$ is not a root of $h$, and we have

$h(AB)A=Ah(BA)=A(\bar{h}(BA)\cdot BA+c)=A\bar{h}(BA)\circ BA+cA.$

Let $u\in U_{0}$. Then $BA(u)=0$, therefore

$0=[h(AB)A](u)=[A\bar{(}BA)](BA(u))+cA(u)=cA(u).$

where the first equality holds since $h(AB)A=0_{U}$. It follows that $A(u)=0$,
so $A|_{U_{0}}$ is the zero map. The proof that $B|_{V_{0}}$ is the zero map
is similar. ∎

######  Theorem 3.13.

Let $(W,S)$ be an irreducible Coxeter system where $S=\\{a,b\\}$ and $3\leq
m:=m(a,b)<\infty$.

  1. (a)

Suppose $m$ is odd. Then the category $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ is
semisimple and has exactly $(m-1)/2$ non-isomorphic simple representations,
all of dimension 2. The algebra $KQ/\mathcal{I}_{f}$ is semisimple, and is
isomorphic to the direct product of $(m-1)/2$ copies of the matrix algebra
$M_{2\times 2}(K)$.

  2. (b)

Suppose $m$ is even. Then the category $\mathrm{rep}(Q,\mathcal{I}_{f})$ is
semisimple and has exactly $(m-2)/2+2$ non-isomorphic simple representations;
two of these representations have dimension 1 and the other representations
have dimension 2. The algebra $KQ/\mathcal{I}_{f}$ is semisimple, and is
isomorphic to the direct product of two copies of $K$ and $(m-2)/2$ copies of
$M_{2\times 2}(K)$.

###### Proof.

Let $M=(M_{a},M_{b},M_{\alpha},M_{\beta})$ be a representation in
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ where $\alpha$ and $\beta$ are the
arrows $a\rightarrow b$ and $b\rightarrow a$ in $Q$, respectively. Set
$h=\tilde{f}_{m-1},U=M_{a},V=M_{b},A=M_{\alpha}$ and $B=M_{\beta}$. If $m$ is
odd, then the equations in (14) hold by Equations (12) and (13). Using Lemma
3.12, we may then decompose $M$ into a direct sum where each summand is of the
form $N(\lambda):=(U_{\lambda},V_{\lambda},A|_{U_{\lambda}},B|_{V_{\lambda}})$
where $\lambda$ is one of the $(m-1)/2$ roots of $h=\tilde{f}_{m-1}$ and
$B|_{V_{\lambda}}A|_{\lambda}=\lambda\cdot\operatorname{\mathrm{id}}_{U_{\lambda}},A|_{U_{\lambda}}B|_{V_{\lambda}}=\lambda\cdot\operatorname{\mathrm{id}}_{V_{\lambda}}$.
It is easy to verify that $N(\lambda)$ is isomorphic to the representation
$M(\lambda)$ from Lemma 3.11. The claims in Part (a) now follow from the
Artin–Wedderburn theorem and the equivalence between
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ and mod-$KQ/\mathcal{I}_{f}$. Similarly,
if $m$ is even, then the equations in (15) hold and we may use Lemma 3.12 to
decompose $M$ into a direct sum of the representations

$\ker\alpha:=(U_{0},0,0,0),\quad\ker\beta:=(0,V_{0},0,0)$

and simple representations isomorphic to $M(\lambda)$ where $\lambda$ is one
of the $(m-2)/2$ roots of $\tilde{f}_{m-1}$. The represenations $\ker\alpha$
and $\ker\beta$ further decompose into $\dim(U_{0})$ and $\dim(V_{0})$ copies
of the modules $S(a)$ and $S(b)$ from Lemma 3.11, respectively. Part (b)
follows. ∎

We are ready to prove that in the case $\lvert S\rvert=2$, the isomorphism
type of the algebra $KQ/\mathcal{I}_{f}$ does not depend on the choice of the
uniform family of polynomials $\\{f_{n}\\}$. Note that we no longer assume
that $m$ is finite in the result below.

######  Corollary 3.14.

Let $(W,S)$ be an irreducible Coxeter system where $S=\\{a,b\\}$ and $3\leq
m:=m(a,b)\leq\infty$. Let $\\{f_{n}\\},\\{g_{n}\\}$ be two uniform families of
polynomials over $K$. Then there is an algebra isomorphism
$\Phi:KQ/\mathcal{I}_{f}\rightarrow KQ/\mathcal{I}_{g}$ such that
$\Phi(e_{i})=e_{i}$ for $i\in\\{a,b\\}$.

###### Proof.

When $m=m(a,b)=\infty$, we have $\mathcal{I}_{f}=\mathcal{I}_{g}=0$, so we may
take $\phi$ to be the identity map. Now assume $m$ is finite. We first treat
the case where $m$ is even. Consider the direct product $B=B_{1}\times
B_{2}\times B_{3}\times\dots\times B_{r}$ where $r=(m-2)/2+2$,
$B_{1}=B_{2}=K$, and $B_{i}$ equals the matrix algebra $M_{2\times 2}(K)$ for
all $3\leq i\leq r$. By Theorem 3.13, there exist algebra isomorphisms
$\phi:KQ/\mathcal{I}_{f}\rightarrow B$ and $\psi:KQ/\mathcal{I}_{g}\rightarrow
B$.

Let $x=\phi(e_{1}),y=\phi(e_{2})$ and write
$x=(x_{1},\dots,x_{r}),y=(y_{1},\dots,y_{r})$. Then we have:

  1. (a)

Since $e_{1},e_{2}$ are idempotents, $x,y$ must be idempotents, therefore
$x_{i},y_{i}$ are idempotents in $B_{i}$ for all $1\leq i\leq r$. This implies
that $x_{1},x_{2},y_{1},y_{2}\in\\{0,1\\}$ and that for all $3\leq i\leq r$,
$x_{i},y_{i}$ must be each conjugate to the zero matrix, the identity matrix,
or the matrix $E_{11}:=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}$.

  2. (b)

Since $1=e_{1}+e_{2}$ in $KQ/\mathcal{I}_{f}$, we must have $x+y=1$ and hence
$x_{i}+y_{i}=1$ for all $1\leq i\leq r$.

  3. (c)

Since $\phi$ is an isomorphism, we have

$\dim(xBx)=\dim(\phi(e_{1})B\phi(e_{1}))=\dim(e_{1}(KQ/\mathcal{I}_{f})e_{1}).$

Here, we have $\dim(xBx)=\sum_{i=1}^{r}\dim(x_{i}B_{i}x_{i})$ in the direct
product $B$. We also have $\dim(e_{1}(KQ/\mathcal{I}_{f})e_{1})=r-1$ because
it is easy to see that the classes of the elements
$e_{1},\alpha\beta,(\alpha\beta)^{2},\dots,(\alpha\beta)^{r-2}$ form a basis
of $e_{1}(KQ/\mathcal{I}_{f})e_{1}$. It follows that
$\sum_{i=1}^{r}\dim(x_{i}B_{i}x_{i})=r-1$. Similarly, we must have
$\sum_{i=1}^{r}\dim(y_{i}B_{i}y_{i})=r-1$. Notice that for all $1\leq i\leq
r$, the dimensions of $x_{i}B_{i}x_{i}$ and $y_{i}B_{i}y_{i}$ depend only on
the conjugacy classes of the idempotents $x_{i},y_{i}$, respectively.

By straightforward dimension considerations, the above three facts force that
$x_{i},y_{i}$ are conjugate to $E_{11}$ for all $3\leq i\leq r$ and that we
either have $x_{1}=0,x_{2}=1,y_{1}=1,y_{2}=0$ or have
$x_{1}=1,x_{2}=0,y_{1}=0,y_{2}=1$. Similarly, the same conclusions apply to
the coordinates $x^{\prime}_{i},y^{\prime}_{i}$ of the elements
$x^{\prime}=(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},\dots,x^{\prime}_{r})=\psi(e_{1})$
and
$y^{\prime}=(y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3},\dots,y^{\prime}_{r})=\psi(e_{2})$.
Thus, we either have $x\sim x^{\prime},y\sim y^{\prime}$ or have $x\sim
y^{\prime},y\sim x^{\prime}$ where $\sim$ means two elements are conjugate. In
both cases, it is easy to find an automorphism $\eta$ of $B$ such that the map
$\Phi:=\psi^{-1}\eta\phi:KQ/\mathcal{I}_{f}\rightarrow KQ/\mathcal{I}_{g}$ is
an isomorphism sending $e_{i}$ to $e_{i}$ for $i\in\\{1,2\\}$. This proves the
corollary in the case where $m$ is even.

The proof for the case where $m$ is odd is similar but simpler: let
$B=B_{1}\times\dots\times B_{r}$ where $r=(m-1)/2$ and $B_{i}=M_{2\times
2}(K)$ for all $1\leq i\leq r$, then consider the isomorphisms $\phi,\psi$
guaranteed by Theorem 3.13 as before. This time, facts similar to (a), (b),
(c) will force all coordinates of
$\phi(e_{1}),\phi(e_{2}),\psi(e_{1}),\psi(e_{2})$ to be conjugate to $E_{11}$,
allowing us to form an isomorphism $\Phi$ with the desired properties as
before. ∎

###  3.4. Proof of Theorem 3.7: General Case

We now prove Theorem 3.7 for a general Coxeter system $(W,S)$. The rough idea
is to notice that each edge in the Coxeter diagram corresponds to a dihedral
system, so we can take the “local” isomorphisms provided by Corollary 3.14 and
then assemble them to a “global” isomorphism between the quotients of $KQ$.

###### Proof of Theorem 3.7.

Let $E$ be the set of edges of the Coxeter diagram of $(W,S)$. For each $e\in
E$ of the form $a-b$, let $\alpha:a\rightarrow b$ and $\beta:b\rightarrow a$
be the dual arrows arising from $e$ in $Q$ and consider the subquiver
$Q_{e}=(\\{a,b\\},\\{\alpha,\beta\\})$ of $Q$. Let
$\mathcal{I}_{f}(e),\mathcal{I}_{g}(e)$ be the evaluation ideals of
$\\{f_{n}\\}$ and $\\{g_{n}\\}$ in $Q_{e}$, respectively. Fix an isomorphism
$\Phi_{e}:KQ_{e}/\mathcal{I}_{f}(e)\rightarrow KQ_{e}/\mathcal{I}_{g}(e)$ such
that $\Phi_{e}(e_{a})=e_{a},\Phi_{e}(e_{b})=e_{b}$ for $e$; such an
isomorphism exists by Corollary 3.14. Note that $KQ_{e}/\mathcal{I}_{g}(e)$
naturally embeds into $KQ/\mathcal{I}_{g}$, so we can naturally view an
element of $KQ_{e}/\mathcal{I}_{g}(e)$ as an element in $KQ/\mathcal{I}_{g}$.
We will do so without further comment.

Let $Q^{\leq 1}=\\{e_{a}:a\in Q_{0}\\}\cup Q_{1}$ be the set of stationary
paths and arrows of $Q$. Consider the function $\phi:Q^{\leq 1}\rightarrow
KQ/\mathcal{I}_{g}$ such that for every edge $e=\\{a,b\\}$ in $G$ and the
arrows $\alpha:a\rightarrow b,\beta:b\rightarrow a$ in $Q$, we have

$\phi(e_{a})=\Phi_{e}(e_{a}),\;\phi(e_{b})=\Phi_{e}(e_{b}),\;\phi(\alpha)=\Phi_{e}(\alpha),\;\phi(\beta)=\Phi_{e}(\beta).$

This function is well-defined because even if a vertex $a$ in $G$ is incident
to two distinct edges $e,e^{\prime}$ in $G$, the maps $\Phi_{e}$ and
$\Phi_{e^{\prime}}$ both send $e_{a}$ to $e_{a}$, causing no ambiguity for the
value of $\phi(e_{a})$. Next, recall again that the path algebra $KQ$ is
generated by $Q^{\leq 1}$ subject only to the relations that
$e_{u}e_{v}=\delta_{u,v}e_{u}$ for $u,v\in Q_{0}$ and the relations
$e_{a}\alpha=\alpha=\alpha e_{b}$ for each arrow $\alpha:a\rightarrow b$ in
$Q_{1}$, and note that the map $\phi$ respects these relations: we have
$\phi(e_{u})\phi(e_{v})=e_{u}e_{v}=e_{u}e_{v}=\delta_{u,v}e_{u}=\delta_{u,v}\phi(e_{u}),$
$\phi(e_{a})\phi(\alpha)=\Phi_{e}(e_{a})\Phi_{e}(\alpha)=\Phi_{e}(e_{a}\alpha)=\Phi(\alpha)=\phi(\alpha)$,
and similarly $\phi(\alpha)\phi(e_{b})=\phi(\alpha)$. It follows that $\phi$
extends to a unique homomorphism $\Phi:KQ\rightarrow KQ/\mathcal{I}_{g}$ with
$\Phi(x)=\phi(x)$ for all $Q^{\leq 1}$. Finally, for each edge $e:a-b$ in $Q$
and the corresponding arrows $\alpha:a\rightarrow b,\beta:b\rightarrow a$, the
restriction of $\Phi$ to $KQ_{e}$ agrees with $\Phi_{e}$, therefore $\Phi$
sends both $r_{f}(\alpha)$ and $r_{f}(\beta)$ to zero because $\Phi_{e}$ does.
It follows that $\Phi$ factors through $\mathcal{I}_{f}$ to induce a
homomorphism $\bar{\Phi}:KQ/\mathcal{I}_{f}\rightarrow KQ/\mathcal{I}_{g}$.
Starting from the collection $\\{\Psi_{e}:e\in E\\}$ where
$\Psi_{e}=\Phi_{e}^{-1}$ for all $e\in E$, we may obtain in the same way a
homomorphism $\bar{\Psi}:KQ/\mathcal{I}_{g}\rightarrow KQ/\mathcal{I}_{f}$,
and it is clear that $\bar{\Psi}$ and $\bar{\Phi}$ are mutual inverses,
therefore $KQ/\mathcal{I}_{f}\cong KQ/\mathcal{I}_{g}$. ∎

##  4\. Quiver Contractions

Let $J_{C}$ be the subregular $J$-ring of an arbitrary Coxeter system $(W,S)$,
let $K$ be an algebraically closed field of characteristic zero, and let
$A=A_{K}=K\otimes_{\mathbb{Z}}J_{C}$. Let mod-$A$ be the category of finite
dimensional right $A$-modules. The rest of the paper is dedicated to the study
of mod-$A$.

In this section we introduce a procedure to modify a quiver $Q$ to a new
quiver $\bar{Q}$ such that the algebra $K\bar{Q}/\bar{\mathcal{I}}_{f}$ is
Morita equivalent to $KQ/\mathcal{I}_{f}$ for any uniform family of
polynomials $\\{f_{n}\\}$ over $K$, where $\bar{\mathcal{I}}_{f}$ is the
evaluation ideal of $\\{f_{n}\\}$ in $K\bar{Q}$. We call the procedure a
_quiver contraction_. In applications, we will often iterate contractions to
obtain sequences of the form

(16) $Q^{(0)}:=Q\rightarrow Q^{(1)}\rightarrow\dots\rightarrow Q^{(n)}$

where $Q$ is the double quiver of $(W,S)$. Denote the evaluation ideal of
$\\{f_{n}\\}$ in $KQ^{(i)}$ by $\mathcal{I}_{f}^{(i)}$ for each $i$. Then
$A\cong KQ/\mathcal{I}_{f}$ by Theorems 3.6 and 3.7, therefore mod-$A$ is
equivalent to $\mathrm{rep}_{K}(Q^{(i)},\mathcal{I}^{(i)}_{f})$ for all $0\leq
i\leq n$. For this reason, we shall develop tools for studying the last type
of category in this section to prepare for the study of mod-$A$ in Section §
5.

###  4.1. Definition of Quiver Contractions

Consider the following generalization of double quivers of Coxeter systems:

######  Definition 4.1.

A _generalized double quiver_ is a triple $(Q,d,m)$ consisting of a quiver
$Q=(Q_{0},Q_{1})$, a map $d:Q_{1}\rightarrow Q_{1}$, and a map
$m:Q_{1}\rightarrow\mathbb{Z}_{\geq 1}\cup\\{\infty\\}$ such that

  1. (a)

$d(Q_{ab})=Q_{ba}$ for all $a,b\in Q_{0}$, where $Q_{c,d}$ denotes the set of
all arrows in $Q$ from $c$ to $d$ for all $c,d\in Q_{0}$.

  2. (b)

$d^{2}(\alpha)=\alpha$ for all $\alpha\in Q_{1}$;

  3. (c)

$m(\alpha)=m({d(\alpha)})$ for all $\alpha\in Q_{1}$.

Given such a triple, we also call $Q$ a _generalized double quiver_. We say
that two arrows $\alpha,\beta\in Q_{1}$ are _dual_ to each other if
$\beta=d(\alpha)$, and call $m(\alpha)$ the _weight_ of $\alpha$ for all
$\alpha\in Q_{1}$.

Note that we may (and will) naturally view the double quiver $Q$ of a Coxeter
system as a generalized double quiver by setting $m(\alpha)=m_{\alpha}$ and
$d(\alpha)=\bar{\alpha}$ for all $\alpha\in Q_{1}$.

We now define quiver contractions as operations on generalized double quivers
$(Q,d,m)$. Roughly speaking, we will define a contraction along a suitable
pair of arrows $\alpha:a\rightarrow b$ and $\beta:b\rightarrow a$ where $a,b$
are distinct vertices in $Q$. The contraction will identify $a,b$ by
collapsing them into a new vertex $v_{ab}$, replace $\alpha,\beta$ by a loop
at $v_{ab}$, and reroute all other arrows incident to $a$ or $b$ in $Q$ to new
arrows incident to $v_{ab}$. The assignments of duals and weights of arrows in
the new quiver will be naturally inherited from $d$ and $m$.

######  Definition 4.2.

Let $(Q,d,m)$ be a generalized double quiver. A pair of arrows
$\\{\alpha,\beta\\}$ is called _contractible_ if they are of the form
$\alpha:a\rightarrow b,\beta:b\rightarrow a$ where $a,b$ are distinct vertices
in $Q$, $\beta=d(\alpha)$, and $m(\alpha)=m(\beta)$ is an odd integer that is
at least 3.

For a contractible pair of arrows $\\{\alpha:a\rightarrow b,\beta:b\rightarrow
a\\}$, the _contraction of $(Q,d,m)$ along $\\{\alpha,\beta\\}$_ is the
generalized double quiver $(\bar{Q},\bar{d},\bar{m})$ where

  1. (a)

The vertex set of the quiver $\bar{Q}$ is

$\bar{Q}_{0}:=Q_{0}\setminus\\{a,b\\}\sqcup\\{v_{ab}\\},$

where $v_{ab}\notin Q_{0}$ is a newly introduced vertex. The arrow set
$\bar{Q}_{1}$ of $\bar{Q}$ is defined as follows: write

$c^{\prime}:=\begin{cases}v_{ab}&\text{ if $c\in\\{a,b\\}$};\\\ c&\text{
otherwise}\end{cases}$

for all $c\in Q_{0}$ and define $\gamma^{\prime}$ to be the arrow
$u^{\prime}\rightarrow v^{\prime}$ for each arrow $\gamma:u\rightarrow v$ in
$Q_{1}$, then let

$\bar{Q}_{1}:=\\{\gamma^{\prime}:\gamma\in
Q_{1}\setminus\\{\alpha,\beta\\}\\}\sqcup\\{\varepsilon_{ab}\\},$

where $\varepsilon_{ab}\notin Q_{1}$ is a newly introduced loop at $v_{ab}$.

  2. (b)

$\bar{d}$ is defined by $\bar{d}(\varepsilon_{ab})=\varepsilon_{ab}$ and
$\bar{d}(\gamma^{\prime})=d(\gamma)^{\prime}$ for all $\gamma\in
Q_{1}\setminus\\{\alpha,\beta\\}$.

  3. (c)

$\bar{m}$ is defined by $\bar{m}(\varepsilon_{ab})=m(\alpha)$ and
$\bar{m}(\gamma^{\prime})=m(\gamma)$ for all $\gamma\in
Q_{1}\setminus\\{\alpha,\beta\\}$.

Note that quiver contractions introduce loops and may lead to multiple pairs
of arrows between two distinct vertices in the resulting quiver (see the
quiver $Q^{(2)}$ in Example 4.10), features that cannot be present in double
quivers of Coxeter diagrams. This is the reason why we do not forbid these
features in Definition 4.3. On the other hand, the generalization from double
quivers to generalized ones is mild enough that we can extend the definition
of evaluation ideals easily, at least for the cases we are interested in:

######  Definition 4.3.

Let $\\{f_{n}:n\geq 2\\}$ be a uniform family of polynomials over $K$, and let
$(Q,d,m)$ be a generalized double quiver. We define the _evaluation ideal_ of
$\\{f_{n}\\}$ in $KQ$ to be the two-sided ideal $\mathcal{I}_{f}\subseteq KQ$
given by

$\mathcal{I}_{f}:=\langle r_{f}(\alpha):\alpha\in Q_{1}\rangle$

where

$r_{f}(\alpha)=\begin{cases}0&\text{if $m=\infty$};\\\
f_{m-1}(\alpha,d(\alpha))&\text{if $m<\infty$ and $d(\alpha)\neq\alpha$};\\\
\tilde{f}_{m-1}(\alpha)&\text{if $m<\infty$ and
$d(\alpha)=\alpha$},\end{cases}$

where $m=m(\alpha)$. Here, as in Equation (4), the evaluation of $\alpha$
through a constant term $c$ in $\tilde{f}_{m-1}(\varepsilon)$ returns $ce_{a}$
for $a=\operatorname{\mathrm{source}}(\alpha)$. For example, if $\alpha$ is a
self-dual loop $\varepsilon:a\rightarrow a$ with $m=5$, then
$r_{f}(\alpha)=\varepsilon^{2}-e_{a}$ if $f_{4}=x^{4}-1$ and
$r_{f}(\alpha)=2\varepsilon^{2}-\varepsilon-3e_{a}$ if $f_{4}=2x^{4}-x^{2}-3$.

######  Remark 4.4.

In this paper we are only interested in generalized double quivers $\bar{Q}$
obtained from the double quiver of a Coxeter diagram via iterated
contractions. In this case, every self-dual arrow $\alpha$ in $\bar{Q}$ must
be either a loop of the form $\varepsilon=\varepsilon_{ab}$ at a vertex
$v=v_{ab}$ introduced during a contraction of a quiver $Q$ along a dual pair
of arrows $\gamma:a\rightarrow b,\delta:b\rightarrow a$ or a reroute of such a
loop. In particular, $m$ must be a finite, odd integer, so the third case in
the definition of $r_{f}(\alpha)$ applies and gives
$r_{f}(\alpha)=\tilde{f}_{m-1}(\varepsilon)$. The relation
$r_{f}(\alpha)=\tilde{f}_{m-1}(\varepsilon)$ in $K\bar{Q}$ mirrors the
relation $r_{f}(\gamma)=\tilde{f}_{m-1}(\gamma\delta)$ in the evaluation ideal
$\mathcal{I}_{f}$ of $KQ$ via the replacements
$\gamma\delta\mapsto\varepsilon$ and $a\mapsto v$. For example, if $m=5$ and
$f_{4}=x^{4}-1$, then the relation
$r_{f}(\gamma)=\tilde{f}_{4}(\gamma\delta)=(\gamma\delta)^{2}-e_{a}\in\mathcal{I}_{f}$
is mirrored by the relation
$r_{f}(\varepsilon)=\tilde{f}_{4}(\varepsilon)=\varepsilon^{2}-e_{v}$.

Our main result on contractions is the following theorem.

######  Theorem 4.5.

Let $(Q,d,m)$ be a generalized double quiver and let
$(\bar{Q},\bar{d},\bar{m})$ be a contraction of $(Q,d,m)$ along a contractible
pair of arrows $\\{\alpha,\beta\\}$. Let $\\{f_{n}:n\geq 2\\}$ be a uniform
family of polynomials over $K$, and let $\mathcal{I}_{f}$ and
$\bar{\mathcal{I}}_{f}$ be the evaluation ideal of $\\{f_{n}\\}$ in $KQ$ and
$K\bar{Q}$, respectively. Then the algebras $KQ/\mathcal{I}_{f}$ and
$K\bar{Q}/\bar{\mathcal{I}}_{f}$ are Morita equivalent.

We postpone the proof of the theorem to § 4.3. Before the proof, we discuss
several detailed examples of quiver contractions and some consequences of the
theorem in the next subsection.

###  4.2. Examples of Quiver Contractions

Throughout this subsection, $G$ denotes the Coxeter diagram of a Coxeter
system $(W,S)$ and $Q$ stands for the double quiver of $G$. When drawing
generalized double quivers, we label each pair of dual arrows
$\\{\alpha,d(\alpha)\\}$ with their common weight $m(\alpha)$ except when
$m(\alpha)=3$, including for the case where $\alpha$ is a self-dual loop of
the form $\varepsilon_{ab}$ introduced by a contraction. For convenience, we
consider only the polynomials $\\{f_{n}:n\geq 2\\}$ where

(17) $f_{n}=\begin{cases}x^{n}-1&\text{if $n$ is even};\\\ x^{n}-x&\text{if
$n$ is odd}.\end{cases}$

for all $n\geq 2$. Note that $\\{f_{n}\\}$ is a uniform family over $K$ since
$K$ is algebraically closed (see Remark 4.17).

######  Example 4.6.

Suppose that $G$ and $Q$ are as shown at the top of Figure 1. The arrows
$\alpha$ and $\beta$ are dual to each other in $Q$ and have weight $3$,
therefore they form a contractible pair. The contraction along
$\\{\alpha,\beta\\}$ results in the quiver $\bar{Q}$ shown in the bottom right
corner of the figure, where $v=v_{ab}$ and $\varepsilon=\varepsilon_{ab}$.

$G:$$a$$b$$c$$d$54$\longrightarrow$$Q:$$a$$b$$c$$d$45$\alpha$$\beta$$\gamma$$\delta$$\zeta$$\eta$$\kappa$$\lambda$$\downarrow$$\downarrow$$\bar{G}:$$v$$c$$d$54$\longrightarrow$$\bar{Q}:$$v$$c$$d$45$\varepsilon$$\gamma^{\prime}$$\delta^{\prime}$$\zeta$$\eta$$\kappa^{\prime}$$\lambda^{\prime}$
Figure 1.

Let us examine the effect of the contraction. The three pairs of arrows
$\\{\gamma,\delta\\},\\{\eta,\zeta\\}$ and $\\{\kappa,\lambda\\}$ in $Q$ have
weights $3,5,4$ and give rise to the elements rl r_1=γδ-e_b, & r_2=δγ-e_c
r_3=(ζη)^2-e_c, r_4=(ηζ)^2-e_d,
r_5=κλκ-κ, r_6=λκλ-λ of the evaluation ideal $\mathcal{I}_{f}$. The
contraction reroutes the arrows $\gamma,\delta,\kappa,\lambda$ since they are
incident to $a$ or $b$, but the rerouting preserves weights by definition, so
the rerouted arrows give rise to “duplicates” of the relations
$r_{1},r_{2},r_{5},r_{6}$ in the ideal $\bar{\mathcal{I}}_{f}$, namely, the
relations

$r_{1}^{\prime}=\gamma^{\prime}\delta^{\prime}-e_{v},\quad
r_{2}^{\prime}=\delta^{\prime}\gamma^{\prime}-e_{c},\quad
r_{5}^{\prime}=\kappa^{\prime}\lambda^{\prime}\kappa^{\prime}-\kappa^{\prime},\quad
r_{6}^{\prime}=\lambda^{\prime}\kappa^{\prime}\lambda^{\prime}-\lambda^{\prime}.$

The arrows $\zeta$ and $\eta$ and their weights remain unchanged in the
contraction since they are not incident to $a$ or $b$. Consequently, they
contribute the same relations $r_{3}$ and $r_{4}$ to $\bar{\mathcal{I}}_{f}$
just as they do to $\mathcal{I}_{f}$. Finally, the arrows $\alpha,\beta$ are
replaced by a single loop at $v$. Since $m:=m(\alpha)=m(\beta)$ is odd,
$\alpha$ and $\beta$ contribute the relations

$r_{f}(\alpha)=(\alpha\beta)^{k}-e_{a},\quad
r_{f}(\beta)=(\beta\alpha)^{k}-e_{b}$

to $\mathcal{I}_{f}$, and their replacement $\varepsilon$ contributes a single
relation

$r_{f}(\varepsilon)=r_{f}(d(\varepsilon))=\varepsilon^{k}-e_{v}$

to $\bar{\mathcal{I}}_{f}$; here, we have $m=3$ and $k=(m-1)/2=1$.

By Theorem 4.5, the algebra $A$ is Morita equivalent to the quotient
$K\bar{Q}/\bar{\mathcal{I}}_{f}$ where $\bar{\mathcal{I}}_{f}=\langle
r_{1}^{\prime},r_{2}^{\prime},r_{3}^{\prime}=r_{3},r_{4}^{\prime}=r_{4},r_{5}^{\prime},r_{6}^{\prime},r_{f}(\varepsilon)\rangle$.
The relation $r_{f}(\varepsilon)=\varepsilon-e_{v}$ implies that
$\varepsilon=e_{v}$ in the quotient, therefore the quotient is isomorphic to
the quotient $K\hat{Q}/\hat{\mathcal{I}}_{f}$ where $\hat{Q}$ is obtained from
$\bar{Q}$ by removing the loop $\varepsilon$ and
$\hat{\mathcal{I}}_{f}=\langle r_{i}:1\leq i\leq 6\rangle\subseteq K\hat{Q}$.
More generally, we define a quiver contraction with respect to a pair of
arrows $\\{\alpha,\beta\\}$ to be _simple_ if $m(\alpha)=m(\beta)=3$. By the
above discussion, Theorem 4.5 remains true for a simple contraction if we omit
the loop $\varepsilon$ in the construction of $\bar{Q}$. We shall do so from
now on.

For a double quiver $Q$ of a Coxeter diagram $G$, a contraction of $Q$ along a
pair of arrows $\\{\alpha:a\rightarrow b,\beta:b\rightarrow a\\}$ is simple if
and only if the corresponding edge $a-b$ in $G$ is simple. When this is the
case, we may define a simple contraction of $G$ by “contracting” the edge
$a-b$ until $a,b$ are identified as a new vertex $v:=v_{ab}$, thus effectively
rerouting all edges incident to $a$ or $b$ to $v$. More precisely, we may
define a weighted graph $\bar{G}$ whose vertex set is
$S\setminus\\{a,b\\}\sqcup\\{v\\}$ where $v\notin S$ and whose edge set is
$\\{e^{\prime}:e\text{ is an edge in $G$ other than $a-b$}\\}$, where
$e^{\prime}=e$ if $e$ is not incident to $a$ or $b$ and $e^{\prime}$ is the
edge $v-c$ whenever $e$ is of the form $a-c$ or $b-c$ for a vertex $c\in
S\setminus\\{a,b\\}$; the weight $m(e^{\prime})$ of $e^{\prime}$ is defined to
be the same as that of $e$. We call $\bar{G}$ the _simple contraction of $G$
along $a-b$_. For the above example, the contraction of $G$ along $a-b$
results in the graph $\bar{G}$ shown in the lower left corner of Figure 1.
Note that if $a,b$ share no neighbor in $G$, which is the case for our example
and must be the case if $G$ has no cycles, then the graph $\bar{G}$ can again
be viewed as the Coxeter diagram of a Coxeter system $(\bar{W},\bar{S})$. In
this case, the double quiver of $\bar{G}$ makes sense, and it is clear that
the double quiver of the simple contraction $\bar{G}$ of $G$ coincides with
the simple contraction $\bar{Q}$ of the double quiver $Q$ of $G$ (once we
ignore the loop $\varepsilon=\varepsilon_{ab}$). This phenomenon is manifest
in our example: the diagram in Figure 1 commutes once we ignore the loop
$\varepsilon$.

######  Remark 4.7.

Maintain the assumptions and notation of the previous paragraph. In
particular, assume that $a-b$ is a simple edge in $G$ where $a,b$ have no
common neighbor. Then Theorem 4.5 implies that the algebra
$A=K\otimes_{\mathbb{Z}}J_{C}$ associated to the Coxeter system $(W,S)$ is
Morita equivalent to the algebra $\bar{A}:=K\otimes_{\mathbb{Z}}\bar{J}_{C}$
associated to the system $(\bar{W},\bar{S})$. Note that we may state the
equivalence purely in terms of contractions of Coxeter diagrams, with no
reference to quivers. Also note that the equivalence implies that when we
study the category mod-$A$, we may assume $G$ has no simple edges whenever $G$
is a tree. The reason is that, since no two vertices can share a neighbor in a
tree, we may repeatedly remove all simple edges in $G$ by simple contractions.

The following example illustrates the reduction allowed by Remark 4.7. After
the example, we record an application of the reduction for future use.

######  Example 4.8.

Suppose that $G$ is the tree shown on the left in Figure 2. By Remark 4.7, the
algebra $A$ associated to $(W,S)$ is Morita equivalent to the algebra
$A^{\prime}:=K\otimes J^{\prime}_{C}$ where $J^{\prime}_{C}$ is the subregular
$J$-ring of the Coxeter system whose diagram is the graph $G^{\prime}$
obtained by contracting all simple edges in $G$; the graph $G^{\prime}$ is
shown on the right on Figure 2.

$G:$$a$$b$$c$$f$$g$$d$$e$$\rightarrow$$G^{\prime}:$$x$$y$$z$$w$574547 Figure
2.

######  Proposition 4.9.

Let $(W,S)$ be a Coxeter system whose Coxeter graph $G$ is a tree, has no edge
with infinite weight, and has at most one heavy edge. Then the category
mod-$A$ associated to $(W,S)$ is semisimple.

###### Proof.

Let $e$ be an edge of maximal weight in $G$. In other words, let $e$ be the
unique heavy edge if $G$ has one, and let $e$ be any simple edge in $G$
otherwise. By contracting all edges different from $e$ in $G$ if necessary, we
may assume that $e$ is the only edge in $G$ and hence $\lvert S\rvert=2$.
Theorem 3.13 then implies that mod-$A$ is semisimple. ∎

The next two examples involve iterated quiver contractions.

######  Example 4.10.

Suppose that $G=C_{n}(m)$ is a cycle with at most one heavy edge, where $n$ is
the number of vertices in $G$ and $m$ is the maximal edge weight. In other
words, suppose that $G$ has $n$ vertices $v_{1},v_{2},\dots,v_{n}$ for some
$n\geq 3$ and has exactly $n$ edges
$v_{1}-v_{2},v_{2}-v_{3},\dots,v_{n-1}-v_{n},v_{n}-v_{1}$, then assume,
without loss of generality, that $G$ has edge weights
$m(v_{2},v_{3})=m(v_{3},v_{4})=\dots=m(v_{n-1},v_{n})=m(v_{n},v_{1})=3$ and
$m(v_{1},v_{2})=m\geq 3$. Note that when $m=3$, the Coxeter group arising from
the Coxeter diagram $C_{n}(m)=C_{n}(3)$ is exactly the affine Weyl group of
type $\tilde{A}_{n-1}$ for all $n\geq 3$.

By repeated simple contractions of Coxeter diagrams, we may reduce $G$ to a
triangle whose double quiver is the quiver $Q^{(1)}$ shown on the left of
Figure 3. Contracting $Q^{(1)}$ along the arrows $\alpha_{3},\beta_{3}$
results in the quiver $Q^{(2)}$ in the same figure, where $a=v_{v_{1}v_{3}}$
and the loop $\varepsilon_{v}$ is omitted as usual. Furthermore, since
$m(\alpha_{2}^{\prime})=m(\alpha_{2})=3$, we may further contract $Q^{(2)}$
along the arrows $\alpha_{2}^{\prime},\beta_{2}^{\prime}$ to obtain the quiver
$Q^{(3)}$, where $b=v_{av_{2}}$, the loop $\varepsilon_{b}$ is omitted, and
the arrows $\alpha_{1}^{\prime\prime},\beta_{1}^{\prime\prime}$ are dual to
each other and have weight $m$. Note that as the last contraction
demonstrates, given a contractible pair of arrows $\alpha:a\rightarrow
b,\beta:b\rightarrow a$ in a generalized double quiver $Q$, every pair of dual
arrows of the form $\gamma:a\rightarrow b,\delta:b\rightarrow a$ where
$\gamma\neq\alpha$ is rerouted to a pair of distinct loops
$\gamma^{\prime},\delta^{\prime}$ at $v_{ab}$ which are dual to each other and
have the same weight as $\gamma$ and $\delta$.

$Q^{(1)}:$$v_{1}$$v_{2}$$v_{3}$$\rightarrow$$Q^{(2)}:$$a$$v_{2}$$\rightarrow$$Q^{(3)}:$$b$$m$$m$$m$$m$$\alpha_{1}$$\beta_{1}$$\alpha_{2}$$\beta_{2}$$\alpha_{3}$$\beta_{3}$$\beta_{2}^{\prime}$$\alpha_{2}^{\prime}$$\alpha_{1}^{\prime}$$\beta_{1}^{\prime}$$\alpha_{1}^{\prime\prime}$$\beta_{1}^{\prime\prime}$
Figure 3.

By Theorem 4.5, the algebra $A$ associated to the Coxeter system whose Coxeter
diagram is the cycle $G$ is Morita equivalent to the algebra
$\bar{A}:=KQ^{(3)}/\mathcal{I}^{(3)}_{f}$ where $\mathcal{I}^{(3)}_{f}$ is the
evaluation ideal of $\\{f_{n}\\}$ in $KQ^{(3)}$. The path algebra $KQ^{(3)}$
is isomorphic to the free unital associative algebra $K\langle x,y\rangle$ on
two variables via the identification $e_{b}\mapsto
1,\alpha_{1}^{\prime\prime}\mapsto x,\beta_{1}^{\prime\prime}\mapsto y$, and
$\mathcal{I}^{(3)}_{f}$ is given by

$\mathcal{I}^{(3)}_{f}:=\langle(\alpha_{1}^{\prime\prime}\beta_{1}^{\prime\prime})^{k}-e_{b},(\beta_{1}^{\prime\prime}\alpha_{1}^{\prime\prime})^{k}-e_{b}\rangle$

where $k=(m-1)/2$, so $\bar{A}$ is isomorphic to the algebra

(18) $T_{k}:=K\langle x,y\rangle/\langle(xy)^{k}=(yx)^{k}=1\rangle.$

In particular, if $m=3$, then $k=1$ and $A$ is isomorphic to the Laurent
polynomial algebra $K[t,t^{-1}]$ via the identification $e_{b}\mapsto
1,\alpha_{1}^{\prime\prime}\mapsto t,\beta_{1}^{\prime\prime}\mapsto t^{-1}$.
To summarize, we have just proved the following result.

######  Proposition 4.11.

Let $n\geq 3$. Let $m\geq 3$ be an odd integer, let $k=(m-1)/2$, and let
$(W,S)$ be the Coxeter system with Coxeter diagram $G=C_{n}(m)$. Then the
algebra $A$ associated to $(W,S)$ is Morita equivalent to the algebra $T_{k}$
defined by Equation (18). In particular, the Morita equivalence class of $A$
does not depend on the value of $n$, and if $m=3$, i.e., if $(W,S)$ is of type
$\tilde{A}_{n-1}$, then $A$ is Morita equivalent to $K[t,t^{-1}]$.

######  Example 4.12.

Apart from the algebras of the form $T_{k}$ from Equation (18), group algebras
of free products of finite cyclic groups can also be realized as the algebras
of the form $A$ associated to Coxeter systems up to Morita equivalence. To see
this, let $A_{k}$ be the $K$-algebra given by the presentation

$A_{k}:=\langle x:x^{k}=1\rangle$

for each integer $k>1$, and let

$A_{\mathbf{k}}=\langle x_{j}:1\leq j\leq n,x_{j}^{k_{j}}=1\rangle$

for each tuple $\mathbf{k}=(k_{1},\dots,k_{n})$ where $n\geq 1$ and
$k_{i}\in\mathbb{Z}_{\geq 1}$ for all $1\leq i\leq n$. Then $A_{k}$ is
isomorphic to the group algebra of the cyclic group $C_{k}$ of order $k$, and
$A_{\mathbf{k}}$ is isomorphic to the group algebra of the free product
$C_{\mathbf{k}}:=C_{k_{1}}*\dots*C_{k_{n}}$ of $C_{k_{1}},\dots,C_{k_{n}}$.
For each tuple $\mathbf{k}=(k_{1},\dots,k_{n})$, let $(W,S)$ be the Coxeter
system where $S=\\{0,1,2,\dots,n\\}$, $m_{j}:=m(0,j)=2k_{j}+1$ for each $1\leq
j\leq n$, and $m(i,j)=2$ for all $1\leq i<j\leq n$. Let $Q$ be the double
quiver of $(W,S)$, and for each $1\leq j\leq n$ denote the arrows
$0\rightarrow j$ and $j\rightarrow 0$ in $Q$ by $\alpha_{j}$ and $\beta_{j}$,
respectively. It is easy to see that by starting from the quiver $Q$ and
performing successive contractions, first along $\\{\alpha_{1},\beta_{1}\\}$,
then along (the reroutes of) $\\{\alpha_{2},\beta_{2}\\}$, then along (the
reroutes of) $\\{\alpha_{3},\beta_{3}\\}$, and so on, we can transform $Q$ to
a generalized double quiver $\bar{Q}$ with a single vertex $v$ and $n$ self-
dual loops $\varepsilon_{1},\dots,\varepsilon_{n}$ at $v$ of weight
$m_{1},m_{2},\dots,m_{n}$, respectively. Figure 4 demonstrates the
construction of $\bar{Q}$ from the Coxeter diagram $G$ of $(W,S)$ when $n=4$.

By Theorem 4.5, the algebra $A$ is Morita equivalent to the quotient
$K\bar{Q}/\bar{\mathcal{I}}_{f}$ where $\bar{\mathcal{I}}_{f}$ is the
evaluation ideal of $\\{f_{n}\\}$ in $K\bar{Q}$. The following proposition is
now immediate.

######  Proposition 4.13.

Let $(W,S)$ be as described in the above paragraph. Then the algebra $A$
associated to $(W,S)$ is Morita equivalent to the algebra $A_{\mathbf{k}}$.

###### Proof.

Note that $K\bar{Q}$ is the free unital associative algebra generated by the
loops $\varepsilon_{j}$ where $1\leq j\leq n$ and that
$\bar{\mathcal{I}}_{f}=\langle\varepsilon_{j}^{k_{j}}-e_{v}:1\leq j\leq
n\rangle$. It follows that we may induce an algebra isomorphism
$\varphi:A_{\mathbf{k}}\rightarrow K\bar{Q}/\bar{\mathcal{I}}_{f}$ from the
assignment $\varphi(x_{j})=\varepsilon_{j}$ for all $1\leq j\leq n$. ∎

$G:$$1$$0$$2$$3$$4$$\rightarrow$$Q:$$1$$2$$3$$4$$0$$\rightarrow$$\bar{Q}:$$v$$m_{1}$$m_{4}$$m_{2}$$m_{3}$$m_{1}$$m_{2}$$m_{3}$$m_{4}$$m_{1}$$m_{2}$$m_{3}$$m_{4}$$\alpha_{1}$$\beta_{1}$$\alpha_{2}$$\beta_{2}$$\alpha_{3}$$\beta_{3}$$\alpha_{4}$$\beta_{4}$$\varepsilon_{1}$$\varepsilon_{2}$$\varepsilon_{3}$$\varepsilon_{4}$
Figure 4.

######  Remark 4.14.

A pleasant feature of iterated quiver contractions is that for every sequence
of the form (16), since we reduce the number of vertices in the quiver with
every contraction, representations in the category
$\mathrm{rep}_{K}(Q^{(n)},\mathcal{I}^{(n)}_{f})$ are often relatively easy to
describe. For instance, in Example 4.12 a representation in
$\mathrm{rep}_{K}(Q^{(n)},\mathcal{I}^{(n)}_{f})$ is simply the data of a
vector space $M_{v}$ and endormophisms $\phi_{1},\dots\phi_{n}$ of $M_{v}$
where $\phi_{j}^{k_{j}}=\operatorname{\mathrm{id}}$ for all $1\leq j\leq n$.
In Example 4.10, to define a representation in
$\mathrm{rep}_{K}(Q^{(3)},\mathcal{I}^{(3)}_{f})$ it suffices to specify a
space $M_{b}$ and two endormorphisms
$M_{\alpha_{1}^{\prime\prime}},M_{\beta_{1}^{\prime\prime}}$ of $M_{v_{1}}$
satisfying the relations
$f_{m-1}(\alpha_{1}^{\prime\prime},\beta_{1}^{\prime\prime})$ and
$f_{m-1}(\beta_{1}^{\prime\prime},\alpha_{1}^{\prime\prime})$. Finally, for
the Coxeter system $(W,S)$ from Example 4.8, by repeated contractions along
arrows corresponding to the edges of weight 5 and 7 in $G^{\prime}$, we may
transform the double quiver of $G^{\prime}$ to a quiver $\bar{Q}$ of the form
shown in Figure 5, where the loop $\varepsilon_{1}$ is self-dual and has
weight 7, the loop $\varepsilon_{2}$ is self-dual and has weight 5, and
$\alpha,\beta$ are dual to each other and have weight 4. It follows that
mod-$A$ is equivalent to the category
$\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ where

$\bar{\mathcal{I}}_{f}=\langle\varepsilon_{1}^{2}-e_{v},\varepsilon_{2}^{3}-e_{v},\alpha\beta\alpha-\alpha,\beta\alpha\beta-\beta\rangle.$

A representation in the latter category is then simply the data of two vector
spaces $M_{v},M_{z}$, two operators $M_{\varepsilon_{1}},M_{\varepsilon_{2}}$
on $M_{v}$ and two maps $M_{\alpha}:M_{v}\rightarrow
M_{z},M_{\beta}:M_{z}\rightarrow M_{v}$ which satisfy the four relations in
the above equation. In all these examples, the representations are much easier
to describe than those in the category $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$
attached to the original double quiver $Q$. Endomorphism like $\phi_{j}$,
$M_{\alpha}M_{\beta},M_{\beta}M_{\alpha}$ and
$M_{\varepsilon_{1}},M_{\varepsilon_{2}}$ will be key tools in our study of
mod-$A$; we will elaborate on their use in § 4.4.

$\bar{Q}:$$v$$z$754$\alpha$$\beta$$\varepsilon_{1}$$\varepsilon_{2}$ Figure 5.

###  4.3. Proof of Theorem 4.5

Let $(Q,d,m)$ be a generalized double quiver and let
$(\bar{Q},\bar{d},\bar{m})$ be its contraction along a contractible pair of
arrows $\\{\alpha:a\rightarrow b,\beta:b\rightarrow a\\}$. We prove Theorem
4.5 in this subsection. To begin, we introduce an intermediate generalized
double quiver $(Q^{\prime},d^{\prime},m^{\prime})$ where

  1. (a)

$Q^{\prime}_{0}=Q_{0}$, and $Q_{1}^{\prime}$ is defined as follows: write

$c^{\prime}=\begin{cases}a&\text{if }c=b;\\\ c&\text{otherwise}\\\
\end{cases}$

for all $c\in Q_{0}$, let

$\gamma^{\prime}=\begin{cases}\gamma&\text{if }\gamma\in\\{\alpha,\beta\\};\\\
(c^{\prime}\rightarrow d^{\prime})&\text{otherwise, if $\gamma$ is of the form
$c\rightarrow d$}\\\ \end{cases}$

for each arrow $\gamma\in Q_{1}$, then set

$Q_{1}^{\prime}=\\{\gamma^{\prime}:\gamma\in Q_{1}\\}.$

  2. (b)

$d^{\prime}$ is defined by $d^{\prime}(\gamma^{\prime})=[d(\gamma)]^{\prime}$
for all $\gamma\in Q^{\prime}_{1}$;

  3. (c)

$m^{\prime}$ is defined by $m^{\prime}(\gamma^{\prime})=m(\gamma)$ for all
$\gamma\in Q_{1}$.

Intuitively, we consider $Q^{\prime}$ an intermediate rerouted version of $Q$
similar to $\bar{Q}$: in $\bar{Q}$, we identify $a,b$ with $v=v_{ab}$ and
“transfer” all data relevant to $a$ or $b$ in $Q$ to $v$ by rerouting all
arrows in $Q$ incident to $a$ or $b$ to $v$; in $Q^{\prime}$, however, we
transfer almost all data relevant to $b$ to $a$ except for the arrows $\alpha$
and $\beta$. We may thus think of $a\in Q^{\prime}_{0}$ as a partial copy of
$a,b\in Q_{0}$ and $v\in\bar{Q}_{0}$ as a complete copy of $a,b$. To obtain
$\bar{Q}$ from $Q^{\prime}$, it remains to rename $a$ as $v_{ab}$, rename each
arrow $\gamma\in Q^{\prime}_{1}\setminus\\{\alpha,\beta\\}$ incident to $a$ as
$\gamma^{\prime}$, replace $\alpha^{\prime},\beta^{\prime}$ with
$\varepsilon_{{ab}}$, and remove $b$. For the quiver $Q$ from Example 4.6,
this procedure, along with the construction of $Q^{\prime}$ from $Q$, is
illustrated in Figure 6, where $v=v_{ab}$ and $\varepsilon=\varepsilon_{ab}$.

$Q:$$a$$d$$c$$b$45$\beta$$\alpha$$\gamma$$\delta$$\zeta$$\eta$$\kappa$$\lambda$$\rightarrow$$Q^{\prime}:$$a$$d$$c$$b$45$\beta$$\alpha$$\gamma^{\prime}$$\delta^{\prime}$$\zeta$$\eta$$\kappa^{\prime}$$\lambda^{\prime}$$\rightarrow$$\bar{Q}:$$v$$c$$d$45$\varepsilon$$\gamma^{\prime}$$\delta^{\prime}$$\zeta$$\eta$$\kappa^{\prime}$$\lambda^{\prime}$
Figure 6.

Maintain the notation of Theorem 4.5, and let $\mathcal{I}^{\prime}_{f}$ be
its evaluation ideal of $KQ^{\prime}$ associated to the polynomials
$\\{f_{n}\\}$. We show below that the algebra
$KQ^{\prime}/\mathcal{I}^{\prime}_{f}$ is both isomorphic to
$KQ/\mathcal{I}_{f}$ and Morita equivalent to
$K\bar{Q}/\bar{\mathcal{I}}_{f}$; Theorem 4.5 immediately follows. Note that
by the last paragraph, we “favored” the vertex $a$ in the construction of
$Q^{\prime}$ by choosing it as the partial copy of $a$ and $b$ before renaming
it $v_{ab}$ in $\bar{Q}$, and we could equally have chosen to favor $b$ in a
similar way. This choice is insignificant in that if we favored $b$ when
constructing $Q^{\prime}$ then the promised isomorphism and Morita equivalence
still hold. In particular, we emphasize that in the direct construction of
$\bar{Q}$ described by Definition 4.2, the vertices $a$ and $b$ clearly play
equal roles, so a quiver contraction is insensitive to the choice of the
favored endpoint of the contractible pair of arrows.

######  Proposition 4.15.

Maintain the setting of Theorem 4.5. Then the algebra
$KQ^{\prime}/\mathcal{I}^{\prime}_{f}$ is isomorphic to $KQ/\mathcal{I}_{f}$.

###### Proof.

We will construct mutually inverse homomorphisms
$\Phi:KQ/\mathcal{I}_{f}\rightarrow KQ^{\prime}/\mathcal{I}^{\prime}_{f}$ and
$\Psi:KQ^{\prime}/\mathcal{I}^{\prime}_{f}\rightarrow KQ/\mathcal{I}_{f}$ to
prove the proposition. To do so, we use presentations of the algebras as
usual: since $KQ$ is the algebra generated by the set $Q^{\leq
1}=\\{e_{u}:u\in Q_{0}\\}\cup Q_{1}$ subject only to the relations that
$e_{u}e_{v}=\delta_{u,v}e_{u}$ for all $u,v\in Q_{0}$ and the relations
$e_{u}\gamma=\gamma=\gamma e_{v}$ for each arrow $\gamma:u\rightarrow v$ in
$Q_{1}$, the algebra $KQ/\mathcal{I}_{f}$ is generated by the same set
$Q^{\leq 1}$ subject to the above relations and the relations $r_{f}(\alpha)$
for all $\alpha\in Q_{1}$. We may therefore construct $\Phi$ by inducing it
from a function $\varphi:Q^{\leq 1}\rightarrow
KQ^{\prime}/\mathcal{I}_{f}^{\prime}$ which respects all the necessary
relations. Similarly, we may construct the homomorphism $\Psi$ from a function
$\psi:Q^{\prime\leq 1}=\\{e_{s}:s\in Q^{\prime}_{0}\\}\cup
Q^{\prime}_{1}\rightarrow KQ/\mathcal{I}_{f}$ which respects the necessary
conditions.

Let $m=m(a,b)$. Then $m\geq 3$ and $m$ is odd by Definition 4.2. By scaling if
necessary, we may assume the polynomial $f_{m-1}$ has constant term $-1$, in
which case the relations $r_{f}(\alpha),r_{f}(\beta)\in\mathcal{I}_{f}$ must
be of the form

$f_{m-1}(\alpha,\beta)=g(\alpha\beta)\alpha\beta-e_{a},\quad
f_{m-1}(\beta,\alpha)=g(\beta\alpha)\beta\alpha-e_{b}=\beta
g(\alpha\beta)\alpha-e_{b}$

for some polynomial $g\in K[x]$, respectively. Let

$\sigma_{1}=g(\alpha\beta)\alpha,\quad\sigma_{2}=\beta.$

Then $\sigma_{1},\sigma_{2}$ make sense in both $KQ$ and $KQ^{\prime}$, and we
have
$r_{f}(\alpha)=\sigma_{1}\sigma_{2}-e_{a},r_{f}(\beta)=\sigma_{2}\sigma_{1}-e_{b}$,
so that

(19) $\sigma_{1}\sigma_{2}=e_{a},\quad\sigma_{2}\sigma_{1}=e_{b}$

in both $KQ/\mathcal{I}_{f}$ and $KQ^{\prime}/\mathcal{I}_{f}^{\prime}$. To
define the functions $\varphi:Q^{\leq 1}\rightarrow
KQ^{\prime}/\mathcal{I}^{\prime}_{f}$ and $\psi:Q^{\prime\leq 1}\rightarrow
KQ/\mathcal{I}_{f}$, first let

$X_{+}=\\{\gamma\in
Q_{1}:\operatorname{\mathrm{source}}(\gamma)=b\\}\setminus\\{\beta\\},$

and

$X_{-}=\\{\gamma\in
Q_{1}:\operatorname{\mathrm{target}}(\gamma)=b\\}\setminus\\{\alpha\\}.$

Note that the set $X_{+}\cap X_{-}$ consists of all loops at $b$ in $Q_{1}$,
and each loop in it is rerouted to a loop at $a$ in $Q^{\prime}$. Next, let
$\varphi(e_{u})=e_{u}$ for all $u\in Q_{0}=Q^{\prime}_{0}$. Finally, recall
that $Q^{\prime}_{1}=\\{\gamma^{\prime}:\gamma\in Q_{1}\\}$ and define
$\varphi$ and $\psi$ on $Q_{1}$ and $Q_{1}^{\prime}$ by letting

$\varphi(\gamma)=\begin{cases}\sigma_{2}\gamma^{\prime}\sigma_{1}&\text{if
$\gamma\in X_{+}\cap X_{-}$};\\\ \sigma_{2}\gamma^{\prime}&\text{if $\gamma\in
X_{+}\setminus X_{-}$};\\\ \gamma^{\prime}\sigma_{1}&\text{if $\gamma\in
X_{-}\setminus X_{+}$};\\\ \gamma&\text{otherwise},\\\
\end{cases}\quad\psi(\gamma^{\prime})=\begin{cases}\sigma_{1}\gamma\sigma_{2}&\text{if
$\gamma\in X_{+}\cap X_{-}$};\\\ \sigma_{1}\gamma&\text{if $\gamma\in
X_{+}\setminus X_{-}$};\\\ \gamma\sigma_{2}&\text{if $\gamma\in X_{-}\setminus
X_{+}$};\\\ \gamma&\text{otherwise}\\\ \end{cases}$

for each $\gamma\in Q_{1}$. Using the relations in (19), it is straightforward
to verify that $\varphi$ and $\psi$ respect all necessary relations mentioned
in the previous paragraph and induce mutually inverse algebra homomorphisms
$\Phi:KQ/\mathcal{I}_{f}\rightarrow KQ^{\prime}/\mathcal{I}^{\prime}_{f}$ and
$\Psi:KQ^{\prime}/\mathcal{I}^{\prime}_{f}\rightarrow KQ/\mathcal{I}_{f}$, as
desired. ∎

######  Proposition 4.16.

Maintain the setting of Theorem 4.5. Then the algebra
$K\bar{Q}/\mathcal{I}(\bar{\mathcal{R}})$ is Morita equivalent to
$KQ^{\prime}/\mathcal{I}(\mathcal{R}^{\prime})$.

###### Proof.

Let $\Lambda=KQ^{\prime}/\mathcal{I}^{\prime}_{f}$ and let
$\sigma_{1},\sigma_{2}$ be as in the proof of Proposition 4.15. Since
$\sigma_{1}\sigma_{2}=e_{a}$ and $\sigma_{2}\sigma_{1}=e_{b}$ in $\Lambda$,
the maps $\phi_{1}:e_{a}\Lambda\rightarrow e_{b}\Lambda,x\mapsto\sigma_{2}x$
and $\phi_{2}:e_{b}\Lambda\rightarrow e_{a}\Lambda,y\mapsto\sigma_{1}y$ give
mutually inverse isomorphisms between the projective modules $\Lambda$-modules
$e_{a}\Lambda$ and $e_{b}\Lambda$. Set $V_{b}=Q^{\prime}_{0}\setminus\\{b\\}$
and let

$e=1-e_{b}=\sum_{u\in V_{b}}e_{u}\in\Lambda.$

Since $e_{a}\Lambda\cong e_{b}\Lambda$, the submodule
$\Lambda^{\prime}:=e\Lambda=\oplus_{u\in V_{b}}(e_{u}\Lambda)$ of the regular
module $\Lambda$ is a progenerator in the category of $\Lambda$-modules,
therefore $\Lambda$ is Morita equivalent to the endomorphism algebra
$\mathrm{End}_{\Lambda}(\Lambda^{\prime})$. We have
$\mathrm{End}_{\Lambda}(\Lambda^{\prime})\cong e\Lambda e$ since $e$ is an
idempotent, so to prove the proposition it suffices to show that
$K\bar{Q}/\mathcal{I}(\bar{\mathcal{R}})$ is isomorphic to $e\Lambda e$. We
will do so by inducing a homomorphism
$\Phi:K\bar{Q}/\mathcal{I}(\bar{\mathcal{R}})\rightarrow e\Lambda e$ from a
function $\phi:\bar{Q}^{\leq
1}:=\\{e_{u}:u\in\bar{Q}_{0}\\}\cup\bar{Q}_{1}\rightarrow e\Lambda e$ and then
showing that $\Phi$ is bijective.

Let $v=v_{ab}$ and $\varepsilon=\varepsilon_{ab}$. To define the function
$\phi:\\{e_{u}:u\in\bar{Q}_{0}\\}\cup\bar{Q}_{1}\rightarrow e\Lambda e$, let
$\varphi(e_{v})=e_{a}$ and let $\phi(e_{u})=e_{u}$ for all
$u\in\bar{Q}_{0}\setminus\\{v\\}$, then let $\phi(\varepsilon)=\alpha\beta$
and let $\phi(\gamma)=\gamma$ for all
$\gamma\in\bar{Q}_{1}\setminus\\{\varepsilon\\}$. Viewing
$K\bar{Q}/\bar{\mathcal{I}}_{f}$ as the algebra generated by $\bar{Q}^{\leq
1}$ subject to the suitable relations as usual, we can again check that $\phi$
respects all these relations: indeed, by the definitions of $\bar{Q}$ and
$Q^{\prime}$, it suffices to check only the relations involving the loop
$\varepsilon\in\bar{Q}_{1}$, i.e., the relation
$e_{v}\varepsilon=\varepsilon=\varepsilon e_{v}$ and the relation
$r_{f}(\varepsilon)=\tilde{f}_{m(\alpha)-1}(\varepsilon)$. These relations are
respected by $\phi$ since $\phi(e_{v})=e_{a},\phi(\varepsilon)=\alpha\beta$,
$e_{a}\alpha\beta=\alpha\beta=\alpha\beta e_{a}$, and
$\tilde{f}_{m(\alpha)-1}(\alpha\beta)=r_{f}(\alpha)\in\mathcal{I}_{f}$ (see
Remark 4.4). It follows that $\Phi$ induces a unique algebra homomorphism
$\Phi:K\bar{Q}/\bar{\mathcal{I}}_{f}\rightarrow e\Lambda e$.

To prove that $\Phi$ is bijective, we keep the notation from Definition 4.2
and from the definition of $Q^{\prime}$. Let
$X=Q_{1}\setminus\\{\alpha,\beta\\}$. Then

$\bar{Q}_{1}=\\{\gamma^{\prime}:\gamma\in X\\}\sqcup\\{\varepsilon\\},\quad
Q^{\prime}_{1}=\\{\gamma^{\prime}:\gamma\in X\\}\sqcup\\{\alpha,\beta\\},$

and $\Phi(\gamma^{\prime})=\gamma^{\prime}$ for all $\gamma\in X$ (where the
$\gamma^{\prime}$s stand for their respective images in
$K\bar{Q}/\bar{\mathcal{I}}_{I}$ and $e\Lambda e$). Let $\mathcal{P}_{b}$ be
the set of all paths on $Q^{\prime}$ which both start and end at a vertex in
$V_{b}$. Then $e\Lambda e$ is spanned by the classes of paths in
$\mathcal{P}_{b}$. Now, since $\alpha,\beta$ are the only arrows in
$Q^{\prime}$ with $b$ as its target and source, respectively, if a path
$p\in\mathcal{P}_{b}$ passes $b$ at any point then it must have traveled to
$b$ from $a$ via $\alpha$ and then immediately traveled back to $a$ via
$\beta$. Consequently, $p$ is a product of $\alpha\beta=\Phi(\varepsilon)$ and
arrows from the set $X=\Phi(X)\subseteq\operatorname{\mathrm{im}}\Phi$. It
follows that $p\in\operatorname{\mathrm{im}}\Phi$, so $\Phi$ is surjective.

It remains to prove that $\Phi:K\bar{Q}/\bar{\mathcal{I}}_{f}\rightarrow
e\Lambda e$ is injective. Since
$\Lambda=KQ^{\prime}/\mathcal{I}^{\prime}_{f}$, it suffices to show that
$e\mathcal{I}^{\prime}_{f}e\subseteq\Phi(\bar{\mathcal{I}}_{f})$. Since
$Q^{\prime}_{1}=\\{\alpha,\beta\\}\sqcup\\{\gamma^{\prime}:\gamma\in X\\}$,
the set $e\mathcal{I}_{f}^{\prime}e$ is spanned by nonzero elements of the
form

(20)
$y_{1}=p_{1}[r_{f}(\alpha)]q_{1}=p_{1}[\Phi(f(\varepsilon,\varepsilon))]q_{1},$
(21) $y_{2}=p_{2}[r_{f}(\beta)]q_{2},$

and

(22)
$y_{3}=p_{3}[r_{f}(\gamma^{\prime})]q_{3}=p_{3}[\Phi(r_{f}(\gamma))]q_{3}$

where $\gamma\in X$ and $p_{i},q_{i}$ are paths in $KQ^{\prime}$ for all
$i\in\\{1,2,3\\}$. We need to show that
$y_{1},y_{2},y_{3}\in\Phi(\bar{\mathcal{I}}_{f})$. Note that the following
holds for all $i\in\\{1,2,3\\}$:

  1. (a)

Since $e=\sum_{u\in V_{b}}e_{u}$ and $y_{i}\neq 0$, we have
$\operatorname{\mathrm{source}}(p_{i})$,
$\operatorname{\mathrm{target}}(q_{i})\in V_{b}$.

  2. (b)

Let $r_{i}$ be the bracketed relation in $y_{i}$ in Equations (20)-(22). Then
$\operatorname{\mathrm{target}}(p_{i})=\operatorname{\mathrm{source}}(r_{i}),\operatorname{\mathrm{target}}(r_{i})=\operatorname{\mathrm{source}}(q_{i})$
since $y_{i}\neq 0$. In particular, we have
$\operatorname{\mathrm{target}}(p_{1})=\operatorname{\mathrm{source}}(q_{1})=a\in
V_{b}$ and $\operatorname{\mathrm{target}}(p_{3})$,
$\operatorname{\mathrm{source}}(q_{3})\in V_{b}$ since the rerouted arrow
$\gamma^{\prime}$ cannot be incident to $b$.

  3. (c)

By (a) and (b),
$p_{3},q_{3}\in\mathcal{P}_{b}\subseteq\operatorname{\mathrm{im}}\Phi$ where
the last containment holds by the last paragraph. Furthermore, we have
$r_{1}=r_{f}(\alpha)=\Phi(r_{f}(\varepsilon))$ and
$r_{3}=r_{f}(\gamma^{\prime})=\Phi(r_{f}(\gamma))$ by the definition of
$\Phi$. It follows that $y_{1},y_{3}\in\Phi(\bar{\mathcal{I}}_{f})$, as
desired.

  4. (d)

By (b) we have
$\operatorname{\mathrm{target}}(p_{2})=b=\operatorname{\mathrm{source}}(q_{2})$,
but since $\alpha,\beta$ are the only arrows in $Q^{\prime}$ with $b$ as its
target and source, respectively, we must have $p_{2}=p^{\prime}_{2}\alpha$ and
$q_{2}=\beta q^{\prime\prime}_{2}$ for some paths
$p^{\prime}_{2},q^{\prime\prime}_{2}\in\mathcal{P}_{b}$. It follows that

(23) $y_{2}=p^{\prime\prime}_{2}\alpha[r_{f}(\beta)]\beta
q^{\prime\prime}_{2}=p^{\prime}_{2}[r_{f}(\alpha)]\alpha\beta
q^{\prime\prime}_{2}=p^{\prime}_{2}[\Phi(r_{f}(\varepsilon_{ab}))]q^{\prime}_{2}$

where $q^{\prime}_{2}=\alpha\beta q^{\prime\prime}_{2}$. Note that
$p_{2}^{\prime},q_{2}^{\prime}\in\mathcal{P}_{b}\subseteq\operatorname{\mathrm{im}}\Phi$,
therefore $y_{2}\in\Phi(\bar{\mathcal{I}}_{f})$.

The proof is now complete. ∎

######  Remark 4.17.

We have assumed that the field $K$ is algebraically closed in Theorem 3.7,
Theorem 3.13, Theorem 4.5, Proposition 4.15 and Proposition 4.16. However, it
is worth noting these results also hold, by the exact same proofs, if we
assume instead that $K$ is an arbitrary field of characteristic zero and that
$\\{f_{n}\\}$ is a family of polynomials which all split over $K$ and satisfy
Conditions (a) and (b) of Definition 3.3. The purpose of the assumption that
$K$ is algebraically closed is to guarantee that the polynomials $\\{f_{n}\\}$
defined by Equation 17 split and hence form a uniform family over $K$; the
simple forms of these polynomials will greatly simplify the study of
representations in categories of the form
$\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ appearing in § 4.2 and the
remaining parts of the paper.

###  4.4. Representations of Contracted Quivers

Let $(\bar{Q},\bar{d},\bar{m})$ be a generalized double quiver obtained from
the double quiver $Q$ of a Coxeter system $(W,S)$ via a sequence of
contractions. Let $\bar{\mathcal{I}}_{f}$ be the evaluation ideal of a uniform
family $\\{f_{n}\\}$ of polynomials over $K$. Then the category mod-$A$ is
equivalent to the category $\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$.
We develop tools for constructing and analyzing representations in
$\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ in this subsection.

Let $M=(M_{a},M_{\alpha})_{a\in\bar{Q}_{0},\alpha\in\bar{Q}_{1}}$ be a
representation in $\mathrm{rep}_{K}\bar{Q}$. The definition of
$\bar{\mathcal{I}}_{f}$ implies that $M$ is a representation in
$\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ if and only if for every
arrow of the form $\alpha:a\rightarrow b$ in $\bar{Q}$, the set of assignments

$M_{\\{\alpha,\beta\\}}:=\\{M_{a},M_{b},M_{\alpha},M_{\beta}\\}$

where $\beta=\bar{d}(\alpha)$ satisfies the relations $r_{f}(\alpha)$ and
$r_{f}(\beta)$ in the sense that the maps

(24)
$f_{m-1}(M_{\alpha},M_{\beta}):=\begin{cases}\tilde{f}_{m-1}(M_{\alpha}M_{\beta})&\text{if
$m$ is odd};\\\ \tilde{f}_{m-1}(M_{\alpha}M_{\beta})M_{\alpha}&\text{if $m$ is
even}\end{cases}$

and

(25)
$f_{m-1}(M_{\beta},M_{\alpha}):=\begin{cases}\tilde{f}_{m-1}(M_{\beta}M_{\alpha})&\text{if
$m$ is odd};\\\ \tilde{f}_{m-1}(M_{\alpha}M_{\beta})M_{\alpha}&\text{if $m$ is
even}\end{cases}$

where $m=m(\alpha)$ both equal 0. Call a set of the form
$M_{\\{\alpha,\beta\\}}$ a _local representation for $\\{\alpha,\beta\\}$_ if
it satisfies the equations (24) and (25). Then to construct a representation
$M\in\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ it suffices to
_assemble_ a collection of local representations

$\mathcal{M}:=\\{M_{\\{\alpha,\beta\\}}\,|\,\alpha\in
Q_{1},\beta=\bar{\alpha}\\}$

that is _consistent_ in the sense that for every vertex $a\in\bar{Q}_{0}$,
there is a common vector space $V_{a}$ such that $M_{a}=V_{a}$ for every local
representation $M_{\alpha,\beta\\}}\in\mathcal{M}$ where $\alpha$ is incident
to $a$. Here, we _assemble_ a consistent collection $\mathcal{M}$ into a
representation $M\in\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ as
follows: first, for each vertex $a\in\bar{Q}_{0}$, pick any arrow
$\alpha\in\bar{Q}_{1}$ incident to $a$, denote $M_{\\{\alpha,\bar{\alpha}\\}}$
by $M^{\prime}$, then let $M_{a}=M^{\prime}_{a}$; second, for each arrow
$\gamma\in\bar{Q}_{1}$ and $\beta=\bar{d}(\alpha)$, denote
$M_{\\{\alpha,\beta\\}}$ by $M^{\prime\prime}$, then let
$M_{\alpha}=M^{\prime\prime}_{\alpha}$ and
$M_{\beta}=M^{\prime\prime}_{\beta}$. Note that the first step can be done
unambiguously, independent of the choice of the arrow $\alpha$, if and only if
$\mathcal{M}$ is consistent. Also note that in the above discussion we allow
the possibility that $\alpha$ is self-dual, in which case $\beta=\alpha$ and
$a=b$.

Subsequently, we will often construct a representation in
$\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ by assembling a consistent
collection of local representations. The local representations can in turn be
studied, by comparison of the equations (12), (13) and (24), (25), similarly
to how we studied $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ in the dihedral case
in § 3.3:

######  Proposition 4.18.

Let $M\in\mathrm{rep}_{K}\bar{Q}$. Let $\alpha:a\rightarrow b$ be an arrow in
$\bar{Q}_{1}$, let $\beta=\bar{d}(\alpha)$, and let $m=\bar{m}(\alpha)$. Let
$M_{\\{\alpha,\beta\\}}=\\{M_{a},M_{b},M_{\alpha},M_{\beta}\\}$. Then the
following results hold.

  1. (a)

If $m=\infty$, then $M_{\\{\alpha,\beta\\}}$ is automatically a local
representation for $\\{\alpha,\beta\\}$. If $m<\infty$, then
$M_{\\{\alpha,\beta\\}}$ is a local representation whenever
$M_{\alpha}M_{\beta}$ and $M_{\beta}M_{\alpha}$ are diagonalizable maps whose
eigenvalues are roots of $\tilde{f}_{m-1}$.

  2. (b)

If $M_{\\{\alpha,\beta\\}}$ is a local representation for $\\{\alpha,\beta\\}$
and $m<\infty$, then $M_{\alpha}M_{\beta}$ and $M_{\beta}M_{\alpha}$ are
diagonalizable and their eigenvalues are either roots of $\tilde{f}_{m-1}$ or
zero.

###### Proof.

Part (a) follows from Equations (24) and (25). Part (b) follows from the same
equations and Lemma 3.12.(a). ∎

The following specializations of Proposition 4.18.(a) will be very useful:

######  Corollary 4.19.

Let $M,\alpha,\beta,a,b,m$ and $M_{\left\\{\alpha,\beta\right\\}}$ be as in
Proposition 4.18. Let $\\{f_{n}\\}$ be the polynomial family defined by (17).
Then the following holds for any positive integer $n$.

  1. (a)

If $M_{a}=M_{b}=K^{n}$ and $M_{\alpha}=M_{\beta}=\operatorname{\mathrm{id}}$,
then $M_{\\{\alpha,\beta\\}}$ defines a local representation for
$\\{\alpha,\beta\\}$.

  2. (b)

Suppose that $m\geq 5$ and $\beta\neq\alpha$. If $M_{a}=M_{b}=K^{n}$ and
$(M_{\alpha}M_{\beta})^{2}=(M_{\beta}M_{\alpha})^{2}=\operatorname{\mathrm{id}}$,
then $M_{\\{\alpha,\beta\\}}$ defines a local representation for
$\\{\alpha,\beta\\}$.

###### Proof.

The results follow from Proposition 4.18.(a), the fact that $1$ is a root for
$\tilde{f}_{m-1}$, and the fact that $x^{2}-1$ divides $\tilde{f}_{m-1}$
whenever $m\geq 5$. ∎

To prove results on mod-$A$ in Section § 5, we will often need to not only
construct a suitable representation $M$ in
$\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{R}})$ but also prove that $M$ is
simple or not semisimple. The proofs typically proceed in the following way.
Consider a specific vertex $a\in\bar{Q}_{0}$ and a number of paths
$p_{1},p_{2},\dots,p_{k}$ on $\bar{Q}$ that both start and end at $a$. Recall
from § 2.4 that these paths give rise to endormophisms
$\phi_{1}:=M_{p_{1}},\phi_{2}:=M_{p_{2}},\dots$ and $\phi_{k}:=M_{p_{k}}$ of
$M_{a}$, and that a subrepresentation $N$ of $M$ must assign to $a$ a vector
space $N_{a}\subseteq M_{a}$ that satisfies the invariance condition
$\phi_{i}(N_{a})\subseteq N_{a}$ for all $1\leq i\leq k$. Together, these
invariance conditions force $N_{a}$ to take certain forms, which in turn force
$M$ to satisfy certain properties such as being simple. We refer to the
analysis of what form $N_{a}$ can take as _subspace analysis at $a$_.

We now explain how we will construct representations to facilitate successful
subspace analysis via examples. All the examples will be used in the proofs of
Section 5, and $\\{f_{n}\\}$ stands for the uniform family of polynomials
defined by Equation (17) throughout the examples. Our first method starts with
an irreducible representation $\rho:G\rightarrow\mathrm{GL}(V)$ of a group
$G$:

######  Example 4.20.

We construct a simple representation $M$ in
$\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ for the generalized double
quiver $\bar{Q}$ in Figure 5 from Remark 4.14. To start, consider the
symmetric group $G=S_{q}$ where $q\geq 8$ and any irreducible representation
$\rho:G\rightarrow\mathrm{GL}(V)$ of $G$. By [Mil01], the group $S_{q}$ can be
generated by two elements $\sigma,\tau$ of orders 2 and 3, respectively. It
follows that
$\rho(\sigma)^{2}=\rho(\sigma^{2})=\rho(e)=\operatorname{\mathrm{id}}_{V}$ and
similarly $\rho(\tau)^{3}=\operatorname{\mathrm{id}}_{V}$.

To define $M$, first let $M_{v}=V,M_{\varepsilon_{1}}=\rho(\sigma)$ and
$M_{\varepsilon_{2}}=\rho(\tau)$. This defines local representations for the
sets $\\{\varepsilon_{1}\\}$ and $\\{\varepsilon_{2}\\}$ because
$M_{\varepsilon_{1}},M_{\varepsilon_{2}}$ satisfy the relations
$r_{f}(\varepsilon_{1})=\varepsilon_{1}^{2}-e_{v},r_{f}(\varepsilon_{2})=\varepsilon_{2}^{3}-e_{v}$,
respectively. To finish the definition of $M$, it remains to assign a local
representation for the dual arrows $\\{\alpha,\beta\\}$ that is consistent
with these two local representations. By Corollary 4.19, it suffices to define
$M_{z}=V$ and $M_{\alpha}=M_{\beta}=\operatorname{\mathrm{id}}_{V}$.

Let $N$ be a subrepresentation of $M$. Since the representation $V$ is
irreducible and $\sigma,\tau$ generate $G$, the only subspaces of $M_{v}$ that
are invariant under both $M_{\varepsilon_{1}}=\rho(\sigma)$ and
$M_{\varepsilon_{2}}=\rho(\tau)$ are $0$ and $M_{v}$ itself, therefore we have
$N_{v}=0$ or $N_{v}=M_{v}$. Since $M_{\alpha},M_{\beta}$ are isomorphisms, in
these two cases we must have $N=0$ or $N=M$, respectively, therefore $M$ is
simple.

The endomorphisms $\phi_{1},\dots,\phi_{k}$ of $M_{a}$ mentioned above are
often all diagonalizable in our examples. This makes the following well-known
fact from linear algebra very useful for subspace analysis: let $V$ be a
vector space and let $\phi$ be a diagonalizable endomorphism of $V$. Suppose
$\phi$ has $d$ distinct eigenvalues and $V=\oplus_{i=1}^{d}E_{i}$ is the
corresponding eigenspace decomposition. Then a subspace $W$ of $V$ is
invariant under $\phi$ if and only if $W$ is _compatible with the eigenspace
decomposition_ in the sense that $W=\oplus_{i=1}^{d}(W\cap E_{i})$. We use
this characterization in the following two examples.

######  Example 4.21.

Let $(W,S)$ be the Coxeter system whose Coxeter diagram $G$ is shown on the
left of Figure 7, where $m_{1},m_{2}\in\mathbb{Z}_{\geq 4}\cup\\{\infty\\}$.
The double quiver $Q$ of $G$ is shown on the right of the same figure. We
construct a representation $M$ in $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ and
apply subspace analysis at $b$ to show that $\mathrm{rep}_{K}(Q,\mathcal{R})$,
hence mod-$A$, is not semisimple.

$G:$$a$$b$$c$$m_{1}$$m_{2}$$\rightarrow$$Q:$$a$$b$$c$$\alpha$$\beta$$\gamma$$\delta$
Figure 7.

We begin by constructing the local representations $M_{\\{\alpha,\beta\\}}$
and $M_{\\{\gamma,\delta\\}}$ based on the values of $m_{1}$ and $m_{2}$,
respectively. For $\\{\alpha,\beta\\}$, first let $M_{a}=M_{b}=K^{2}$. Let
$\lambda_{2}=1$. Take $\lambda_{1}=-1$ if $m_{1}=\infty$, $\lambda=0$ if
$m_{1}=4$, and $\lambda=z$, a root of $\tilde{f}_{m-1}$ different from
$\lambda_{2}$, if $4<m_{1}<\infty$. Next, set $M_{\beta}$ to be the map given
by the matrix $\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}$ if $m_{1}=4$ and to be
the identity map otherwise, then set
$M_{\alpha}=\begin{bmatrix}\lambda_{1}&1\\\ 0&\lambda_{2}\end{bmatrix}$. It is
straightforward to check that
$M_{\\{\alpha,\beta\\}}:=\\{M_{a},M_{b},M_{\alpha},M_{\beta}\\}$ forms a local
representation: if $m_{1}=\infty$, then the relations
$r_{f}(\alpha),r_{f}(\beta)$ are zero and there is nothing to check; if
$4<m_{1}<\infty$, then the relations $r_{f}(\alpha),r_{f}(\beta)$ are
satisfied by Proposition 4.18 because
$M_{\alpha}M_{\beta}=M_{\beta}M_{\alpha}=M_{\alpha}$, a diagonalizable map
whose eigenvalues are roots of $\tilde{f}_{m-1}$; finally, if $m_{1}=4$, then
the relations
$r_{f}(\alpha)=\alpha\beta\alpha-\alpha,r_{f}(\beta)=\beta\alpha\beta-\beta$
are satisfied because
$M_{\alpha}M_{\beta}M_{\alpha}=M_{\alpha},M_{\beta}M_{\alpha}M_{\beta}=M_{\beta}$
by direct computation. We may similarly define a local representation for
$\\{\gamma,\delta\\}$ by letting $M_{b}=M_{c}=K^{2}$, defining numbers
$\mu_{2},\mu_{1}$ and the map $M_{\gamma}$ based on $m_{2}$ in the same way we
defined $\lambda_{2},\lambda_{1}$ and $M_{\beta}$ based on $m_{1}$, and
defining the map $M_{\delta}$ to be given by the matrix
$\begin{bmatrix}\mu_{1}&0\\\ 0&\mu_{2}\end{bmatrix}$. The two local
representations are consistent because they assign the same vector space
$K^{2}$ to the vertex $b$, therefore they can be assembled to a representation
$M\in\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$.

Consider the operators $\phi_{1}:=M_{\alpha}\circ M_{\beta}$ and
$\phi_{2}:=M_{\delta}\circ M_{\gamma}$ on $M_{b}$. Then

$\phi_{1}=\begin{bmatrix}\lambda_{1}&1\\\
0&\lambda_{2}\end{bmatrix},\quad\phi_{1}=\begin{bmatrix}\mu_{1}&0\\\
0&\mu_{2}\end{bmatrix},$

so the eigenspace decompositions of $M_{b}$ with respect to
$\phi_{1},\phi_{2}$ are given by

(26) $M_{b}=\langle e_{1}\rangle\oplus\langle
e_{1}+(\lambda_{2}-\lambda_{1})e_{2}\rangle=\langle e_{1}\rangle\oplus\langle
e_{2}\rangle$

where $\langle v\rangle$ stands for the span of $v$ for each vector $v$ and
$e_{1},e_{2}$ denote the standard basis vectors $\begin{bmatrix}1\\\
0\end{bmatrix},\begin{bmatrix}0\\\ 1\end{bmatrix}$ of $K^{2}$, respectively.
Let $N$ be a nonzero subrepresentation of $M$. Then the vector space $N_{b}$
must be invariant under both $\phi_{1}$ and $\phi_{2}$. Consequently, $N_{b}$
must be compatible with the decompositions in Equation (26) in the sense that

$N_{b}=(N_{b}\cap\langle e_{1}\rangle)\oplus(N_{b}\cap\langle
e_{2}\rangle)=(N_{b}\cap\langle e_{1}\rangle)\oplus(N_{b}\cap\langle
e_{1}+(\lambda_{2}-\lambda_{1})e_{2}\rangle).$

But each intersection in the above equation is either trivial or of dimension
1, so $N_{b}$ must be $\langle e_{1}\rangle$. This implies that $N$ cannot
have a complement in $M$, so $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ is not
semisimple.

######  Example 4.22.

Let $Q$ and $M$ be as in the previous example. We modify $M$ to produce an
infinite family of simple representations in
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ to be used in Section § 5. To begin,
define a representation $M^{x}$ for each scalar $x\in
K\setminus\\{\lambda_{1},\lambda_{2}\\}$ as follows: let
$M^{x}_{a}=M^{x}_{b}=M^{x}_{c}=K^{2},M^{x}_{\gamma}=M_{\gamma}$ and
$M^{x}_{\delta}=M_{\delta}$, then let
$M^{x}_{\beta}=\operatorname{\mathrm{id}}$ and let $M^{x}_{\alpha}$ be the map
given by the matrix

$B_{x}=\begin{bmatrix}x&x(\lambda_{1}+\lambda_{2}-x)-\lambda_{1}\lambda_{2}\\\
1&\lambda_{1}+\lambda_{2}-x\end{bmatrix}.$

As before, this defines local representations
$M_{\\{\gamma,\delta\\}}:=\\{M^{x}_{b},M^{x}_{c},M^{x}_{\gamma},M^{x}_{\delta}\\}$
and
$M_{\\{\alpha,\beta\\}}:=\\{M^{x}_{a},M^{x}_{b},M^{x}_{\alpha},M^{x}_{\beta}\\}$
for $\\{\gamma,\delta\\}$ and $\\{\alpha,\beta\\}$, respectively. Moreover,
the two local representations are consistent and assemble to a representation
$M^{x}$. Let $\phi^{x}_{1}:=M_{\alpha}^{x}\circ M_{\beta}^{x}$ and
$\phi^{x}_{2}:=M^{x}_{\delta}\circ M^{x}_{\gamma}$. The matrix $B_{x}$
guarantees that the map $\phi_{1}^{x}$ has an eigenvector
$v_{1}:=(x-\lambda_{1})e_{1}+e_{2}$ with eigenvalue $\lambda_{1}$ as well as
an eigenvector $v_{2}:=(x-\lambda_{2})e_{1}+e_{2}$ with eigenvalue
$\lambda_{2}$, so the eigenspace decompositions of $M^{x}_{b}$ with respect to
$\phi_{1}^{x}$ and $\phi_{2}^{x}$ are given by

$M^{x}_{b}=\langle v_{1}\rangle\oplus\langle v_{2}\rangle=\langle
e_{1}\rangle\oplus\langle e_{2}\rangle.$

It follows that in any subrepresentation $N$ of $M^{x}$ we must have

$N_{b}=N_{b}\cap\langle v_{1}\rangle\oplus N_{b}\cap\langle
v_{2}\rangle=N_{b}\cap\langle e_{1}\rangle\oplus N_{b}\cap\langle
e_{2}\rangle.$

The vectors $v_{1},v_{2},e_{1},e_{2}$ are pairwise distinct since
$\lambda_{1}\neq\lambda_{2},\mu_{1}\neq\mu_{2}$ and
$x\not\in\\{\lambda_{1},\lambda_{2}\\}$, therefore the above equation holds
only if $N_{b}=0$ or $N_{b}=K^{2}$. If $m_{2}\neq 4$, then $\mu_{1}\neq 0$ and
$M^{x}_{\gamma}$ is an isomorphism, therefore we must have $N=0$ or $N=M^{x}$,
which in turn implies that $M^{x}$ is simple. If $m_{2}=4$, it is easy to
check that $M^{x}$ contains a simple module $N^{x}$ such that
$N^{x}_{a}=N^{x}_{b}=K^{2}$ and $N^{x}_{c}=\langle e_{2}\rangle$. Set
$N^{x}=M^{x}$ when $m_{2}\neq 4$. Then $N^{x}$ is a simple representation in
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ for all $x\in
K\setminus\\{\lambda_{1},\lambda_{2}\\}$ regardless of the value of $m_{2}$.

Observe that if $x\neq y$ then $N^{x}\not\cong N^{y}$, for there cannot exist
linear isomorphisms $\phi_{a}:N^{x}_{a}\rightarrow
N^{y}_{a},\phi_{b}:N^{x}_{b}\rightarrow N^{y}_{b}$ such that
$\phi_{b}N^{x}_{\alpha}=N^{y}_{\alpha}\phi_{a}$ and
$\phi_{a}N^{x}_{\beta}=N^{y}_{\beta}\phi_{b}$ simultaneously: the latter
equation holds only if $\phi_{a}=\phi_{b}$ as maps from $K^{2}$ to $K^{2}$
because $N^{x}_{\beta}=N^{y}_{\beta}=\operatorname{\mathrm{id}}$, but then the
first equation implies that the matrices $B_{x},B_{y}$ are equal, which cannot
happen if $x\neq y$.

In the next example, we combine the ideas of the last three examples to
construct simple representations in a more flexible way.

######  Example 4.23.

Let $(W,S),G$, and $Q$ be as in Example 4.21. We construct a representation
$M\in\mathrm{rep}_{K}(Q,\mathcal{I})$ in the case that $m_{1}\geq 4,m_{2}\geq
6$ and prove that $M$ contains a simple subrepresentation $N$.

Consider the irreducible representation $\rho:S_{q}\rightarrow\mathrm{GL}(V)$
and the elements $\sigma,\tau\in S_{q}$ from Example 4.20. Using minimal
polynomials as we did in the proof of Lemma 3.12, we may deduce from the fact
$\rho(\sigma)^{2}=\operatorname{\mathrm{id}}_{V}$ that $\rho(\sigma)$ is a
diagonalizable map whose eigenvalues are from the set $\\{-1,1\\}$, so with
respect to $\rho(\sigma)$ we have an eigenspace decomposition $V=E_{1}\oplus
E_{2}$ where $E_{1},E_{2}$ are the eigenspaces for the eigenvalues $1$ and
$-1$, respectively. Similarly, since
$\rho(\tau)^{3}=\operatorname{\mathrm{id}}_{V}$, the map $\rho(\tau)$ is
diagonalizable and its eigenvalues lie in the set
$\\{\omega_{1},\omega_{2},\omega_{3}\\}$ containing the three third roots of
unity, therefore we have an eigenspace decomposition $V=F_{1}\oplus
F_{2}\oplus F_{3}$ of $V$ where each $F_{i}$ is the eigenspace for the
eigenvalue $\omega_{i}$. Note that 0 and $V$ are the only subspaces of $V$
compatible with both these decompositions, because they are the only subspaces
invariant under both $\rho(\sigma)$ and $\rho(\tau)$ by Example 4.20.

To define $M$, first set $M_{a}=M_{b}=M_{c}=V$. Next, assign the linear maps
$M_{\alpha},M_{\beta}$ based on the value of $m_{1}$: if $m_{1}>4$, then
$\tilde{f}_{m_{1}-1}$ contains at least two nonzero roots
$\lambda_{1},\lambda_{2}$, and we set

$M_{\alpha}=\operatorname{\mathrm{id}}_{E_{1}}\oplus\operatorname{\mathrm{id}}_{E_{2}},\quad
M_{\beta}=(\lambda_{1}\cdot\operatorname{\mathrm{id}}_{E_{2}})\oplus(\lambda_{2}\cdot\operatorname{\mathrm{id}}_{E_{2}})$

where the notation means, for example, that $M_{\beta}$ restricts to
$\lambda_{1}\cdot\operatorname{\mathrm{id}}$ on $E_{1}$ and to
$\lambda_{2}\cdot\operatorname{\mathrm{id}}$ on $E_{2}$; if $m_{1}=4$, we set

$M_{\alpha}=\operatorname{\mathrm{id}}_{E_{1}}\oplus 0_{E_{2}},\quad
M_{\beta}=(\lambda_{1}\cdot\operatorname{\mathrm{id}}_{E_{2}})\oplus
0_{E_{2}}$

where $\lambda_{1}$ is the unique nonzero root of $\tilde{f}_{3}$. Similarly,
if $m_{2}>6$ then $\tilde{f}_{m_{2}-1}$ has at least three nonzero roots
$\mu_{1},\mu_{2},\mu_{3}$ and we set

$M_{\gamma}=\operatorname{\mathrm{id}}_{F_{1}}\oplus\operatorname{\mathrm{id}}_{F_{2}}\oplus\operatorname{\mathrm{id}}_{F_{3}},\quad
M_{\delta}=(\mu_{1}\cdot\operatorname{\mathrm{id}}_{F_{1}})\oplus(\mu_{2}\cdot\operatorname{\mathrm{id}}_{F_{2}})\oplus(\mu_{3}\cdot\operatorname{\mathrm{id}}_{F_{3}}),$

while if $m_{2}=6$ then we set

$M_{\gamma}=\operatorname{\mathrm{id}}_{F_{1}}\oplus\operatorname{\mathrm{id}}_{F_{2}}\oplus
0_{F_{3}},\quad
M_{\delta}=(\mu_{1}\cdot\operatorname{\mathrm{id}}_{F_{1}})\oplus(\mu_{2}\cdot\operatorname{\mathrm{id}}_{F_{2}})\oplus
0_{F_{3}}.$

where $\mu_{1},\mu_{2}$ are the two nonzero roots of $\tilde{f}_{m_{2}-1}$. By
Parts (a) and (b) of Proposition 4.18, the assignments define a representation
in $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$. Moreover, the eigenspace
decompositions of $M_{b}$ with respect to the maps
$\phi_{1}=M_{\alpha}M_{\beta}$ and $\phi_{2}=M_{\delta}M_{\gamma}$ coincide
with the eigenspace decompositions of $V$ with respect to the maps
$\rho(\sigma)$ and $\rho(\tau)$, respectively, therefore a subrepresentation
of $M$ must assign the space $0$ or $V$ to the vertex $b$. It follows that $M$
contains a simple representation $N$ with $N_{b}=V$ and

$N_{a}=\begin{cases}M_{a}=E_{1}\oplus E_{2}&\text{if $m_{1}>4$};\\\
E_{1}&\text{if $m_{1}=4$},\\\
\end{cases}\;N_{c}=\begin{cases}M_{c}=F_{1}\oplus F_{2}\oplus F_{3}&\text{if
$m_{1}>6$};\\\ F_{1}\oplus F_{2}&\text{if $m_{1}=6$}.\\\ \end{cases}\quad$

######  Remark 4.24.

Given a vector space $V$ and operators $\phi_{1},\cdots,\phi_{k}$ on $V$, the
study of subspaces of $V$ that are simultaneously compatible with the
eigenspace decompositions of all the operators is closely related to
enumerative geometry and Schubert calculus. For example, if $K=\mathbb{C}$,
$V=K^{4}$, $n=2$ and the operators $\phi_{1},\phi_{2}$ yield eigenspace
decompositions $M_{b}=E_{1}\oplus E_{2}$ and $M_{b}=F_{1}\oplus F_{2}$ where
$E_{1},E_{2},F_{1},F_{2}$ all have dimension $2$, then “generically” there
exists a subspace $W\subseteq V$ with dimension 2 that is simultaneously
compatible with both $\phi_{1}$ and $\phi_{2}$, because a classical result in
Schubert calculus asserts that generically, given four lines in the projective
3-space $\mathbb{C}\mathbb{P}^{3}$, there are two lines that intersect all
these four lines. On the other hand, when the dimensions of the eigenspaces of
$V$ with respect to $\phi_{1},\cdots,\phi_{k}$ are known, we can often show
that no proper, nontrivial subspace of $V$ can be simultaneously compatible
with the corresponding eigenspace decompositions by certain codimension
computations involving products of Schubert classes. For instance, to
construct a simple representation in
$\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ in Example 4.20, it is
possible to specify for every positive integer $n$ a representation
$M\in\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{I}}_{f})$ in such a way that
$\dim(M_{b})=6n$, the maps $M_{\alpha}$ and $M_{\beta}$ are isomorphisms,
$\phi_{1}:=M_{\varepsilon_{1}}$ is diagonalizable with two eigenspaces
$E_{1},E_{2}$ of dimension $3n$, and $\phi_{2}:=M_{\varepsilon_{2}}$ is
diagonalizable with three eigenspaces $F_{1},F_{2},F_{3}$ of dimension $2n$. A
codimension computation using Schubert calculus guarantees that generically
$M_{b}$ has no subspace compatible with both $M_{\varepsilon_{1}}$ and
$M_{\varepsilon_{2}}$ other than 0 and $M_{b}$ itself, so $M$ is simple
(generically). This yields an alternative proof of the existence of certain
simple representations in $\mathrm{rep}_{K}(\bar{Q},\bar{\mathcal{R}})$. In
the interest of space, however, we omit details of the codimension computation
and the necessary background on Schubert calculus. In particular, we will not
make precise what the word “generically” means in this paragraph.

We end this subsection by proving a proposition to be used in § 5.2 under the
following setting: Let $Q$ be the double quiver of a Coxeter diagram $G$ and
let $\\{f_{n}\\}$ be as defined in Equation 17. Suppose that $G$ contains a
vertex $v$ which is adjacent to a unique vertex in $G$. Let $u$ be that unique
vertex, let $m=m(u,v)$, and let $\hat{G}$ be the graph obtained from $G$ by
removing the vertex $v$ and the edge $v-u$. Let $\hat{Q}$ be the double quiver
of $\hat{G}$, then let $\hat{\mathcal{I}}_{f}$ be the evaluation ideal of
$\\{f_{n}\\}$ in $K\hat{Q}$. The proposition allows us to “enlarge” certain
representations in $\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ to a simple
representation in $\mathrm{rep}_{K}(\hat{Q},\hat{\mathcal{I}}_{f})$:

######  Proposition 4.25.

Let $Q,\mathcal{R},\hat{Q},\hat{\mathcal{R}}$ and $u,v$ be as described above.
Suppose that $\\{S(1),S(2),\dots,S(k)\\}$ is a nonempty set of pairwise non-
isomorphic simple representations in
$\mathrm{rep}_{K}(\hat{Q},\hat{\mathcal{R}})$ and let $S=\oplus_{i=1}S(i)$. If
$m>3$, then there is a simple representation $M$ in
$\mathrm{rep}_{K}(Q,\mathcal{I}_{f})$ such that $M_{a}=S_{a}$ for all
$a\in\hat{Q}_{0}$.

We prove the proposition via subspace analysis at the vertex $u$, by using the
following lemma:

######  Lemma 4.26.

Let $\Lambda$ be an arbitrary ring. Let $\\{S(1),\ldots,S(k)\\}$ be a set of
pairwise non-isomorphic simple right $\Lambda$-modules, let
$S=\oplus_{i=1}^{k}S(i)$, and let $x=\sum_{1\leq i\leq k}x_{i}\in S$ where
$x_{i}\in S(i)$ for each $i$. If $x_{i}\neq 0$ for all $1\leq i\leq k$, then
the submodule generated by $x$ equals $S$.

###### Proof.