# On reflection representations of Coxeter groups
over non-commutative rings

Annette Pilkington  Department of Mathematics
University of Notre Dame
Room 255 Hurley Building
Notre Dame, Indiana, 46556<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider representations of Coxeter groups over a path
algebra, R, defined in Dyer [13]. We answer a question posed by Dyer about the
multiplicative properties of R, showing that it is “almost a domain”. We also
show that R cam be embedded in a matrix ring over a free product of extension
fields of the rational numbers and rings of Laurent polynomials.

###### Key words and phrases:

Path Algebras, Domain, Coxeter Groups

## Introduction

Coxeter groups admit representations as certain discrete reflection groups
associated to root systems in real vector spaces ([25], [3], [19]). In many
situations, these have an associated analogue of a Cartan matrix, describing
the pairings between simple roots in the reflection representation, and simple
coroots in the “dual” reflection representation. There are also reflection
representations and root systems of corresponding Iwahori-Hecke algebras
([4]). Reflection representations and root systems in modules over commutative
rings have been considered in [18], [9] and [11]-[12].

In [13], similar notions of reflection representation of Coxeter groups and
Hecke algebras are considered over non-commutative rings. They are associated
to a certain class of matrices (non-commutative Cartan matrices, NCMs) with
entries in non-commutative rings, defined by identities related to Chebyshev
polynomials on the $2\times 2$-principal submatrices. We call these “lax”
reflection representations in this paper. They have the property that if
$(W^{\prime},S^{\prime})$ is another Coxeter system and there is a group
homomorphism $f\colon W\to W^{\prime}$ which restricts to a bijection $S\to
S^{\prime}$, then lax reflection representations for $(W^{\prime},S^{\prime})$
pull back along $f$ to lax reflection representations of $(W,S)$.

An important role in the theory is played by a certain ring (the lax universal
coefficient ring) with specified NCM, denoted in this paper as
$\widetilde{R}=\widetilde{R}_{W,S}$, which gives (roughly speaking) an initial
object in a category of rings with a specified NCM for $(W,S)$. This ring,
which is of considerable interest111It is stated without proof in [13] that
$\widetilde{R}$ can be identified as based ring with the two-sided ideal
$J_{1}$ of Lusztig’s equal parameter asymptotic Hecke algebra $J$., may be
defined as quotient of the path algebra of the Coxeter digraph by certain rank
two relations. It contains a family $([e_{r}])_{r\in S}$ of non-zero
orthogonal idempotents (the images in the quotient of length $0$ paths) such
that $\widetilde{R}=\bigoplus_{r,s\in S}[r]\widetilde{R}[s]$, and is unital if
and only if $S$ is finite. The ring $\widetilde{R}$ is free as
$\mathbb{Z}$-module and admits commuting left and right lax reflection
$W$-actions, with the left (resp., right) action being by our conventions
right (resp., left) $\widetilde{R}$-linear.

This paper concerns a subclass of the lax reflection representations, called
here strict reflection representations, which is defined by strict NCMs,
satisfying stronger conditions than NCMs, and for which the above pullback
property no longer holds. There is again a universal ring with strict NCM,
with analogous properties to those of $\widetilde{R}$ mentioned above. We
denote this ring as $R=R_{W,S}$.

We state a result which partly motivates the definition of $R$ and clarifies
the relationship between lax and strict reflection representations, and
reflection representations in real vector spaces. For standard dual reflection
representations of $(W,S)$ on real vector spaces $V$ and ${V}^{\bigvee}$ with
bases of simple roots $\Pi$ and simple coroots ${\Pi}^{\bigvee}$, there is an
identification of ${V}^{\bigvee}\otimes_{\mathbb{R}}V$ with the ring $B$ of
$S\times S$ real matrices with only finitely many non-zero entries (so that
the matrix units are ${\alpha}^{\bigvee}\otimes\beta$ for
${\alpha}^{\bigvee}\in{\Pi}^{\bigvee}$ and $\beta\in\Pi$) and a ring
homomorphism $R\to B$ which is equivariant for the left and right $W$-actions
and which induces a bi-equivariant ring surjection
$R^{\prime}=R\otimes_{\mathbb{Z}}\mathbb{R}\to B$ Here, we regard $W$ as
acting on the left (resp., right) of ${V}^{\bigvee}$ (resp., $V$). This would
be true with $R$ replaced by $\widetilde{R}$, but that is of less interest
since $R$ is a quotient (as ring with two-sided $W$-actions) of
$\widetilde{R}$. For an irreducible finite Weyl group, $R^{\prime}\to B$ is an
isomorphism. In this paper, we do not discus more delicate results of a
similar nature to the last one for other classes of Coxeter systems, or
extensions of the above facts to Hecke algebras.

Although the class of strict reflection representations was not considered
explicitly in [13], both rings $R$ and $\widetilde{R}$ are constructed there.
The main result of this paper answers affirmatively a basic question, raised
in [13, 3.24], of whether the strict universal coefficient ring $R$ is nearly
a domain, in the sense that if $r,s,t\in S$ and $a\in[r]R[s]$ and
$b\in[s]R[t]$, then $ab=0$ implies $a=0$ or $b=0$. The analogous statement for
$\widetilde{R}$ fails whenever $\widetilde{R}\neq R$. One may hope that its
validity for $R$ will make the $W$-actions on $R$ better behaved in some
respects than those on $R$. We prove the main result by reducing to the case
of finite $S$ and then embedding $R$ as a subring of a ring
$\mathrm{Mat}_{S^{\prime}\times S^{\prime}}(A)$ of $S^{\prime}\times
S^{\prime}$-matrices, where $S^{\prime}:=S\cup\\{\bullet\\}$ and $A$ is a
$\mathbb{Q}$-algebra, depending on the Coxeter matrix, which is a free product
(in the sense of Cohn [5]), of certain real cyclotomic number fields, Laurent
polynomial rings and polynomial rings over $\mathbb{Q}$.

Since the techniques used come from diverse fields, some of which may be
unfamiliar to the reader, we include basic definitions and cite basic results
from the various fields. In section 1, we present basic definitions and
notation relating to rings and modules used throughout the paper. In section
2, we introduce Serre’s notation and terminology for graphs [23] and in
section 3, we give the basic definitions of path algebras and cite a number of
basic results on Gröbner bases of ideals in that context from Green [16]. In
section 4, we introduce the path algebras related to Coxeter systems and their
quotients, $\widetilde{R}_{W,S}$ and $R_{W,S}$, appearing in Dyer [13]. In
that section, we also prove some basic results about these algebras, which
allow us to restrict our attention to finite rank Coxeter groups and to the
ring $\hat{R}_{W,S}=R_{W,S}\otimes_{\mathbb{Z}}\mathbb{Q}$ when proving the
main result. In section 5, we provide for motivation some unpublished results
of Dyer extending material from [13] on the lax and strict reflection
representations associated to these rings and how they relate to reflection
representations on real vector spaces. We thank Matthew Dyer for his
permission to include these results. In section 6, we give the definition and
results on free products of rings from Cohn [5], [6], and [7] and we show that
certain rings related to the edges of the underlying path algebra are free
products in the sense of Cohn. In section 7, we show how the ring
$\hat{R}_{W,S}$ can be imbedded in a matrix rig over a free product of rings,
We conclude with a proof that the ring ${R}_{W,S}$ is nearly a domain, as
described above, in section 8.

##  1\. general Notation and Definitions

### 1.1.

The notation $X=\\{a,b,c\\}$ will be used to denote that $X$ is a set with
elements $a,b$ and $c$. The set $X\backslash Y$ will be the set of all
elements in $X$ but not in $Y$ and the set $X\bigcupdot Z$ will denote the
disjoint union of the sets $X$ and $Z$.

Unless otherwise stated, all rings considered have an identity (are unital).
If $R$ is a ring, we will denote the identity of $R$ by $1_{R}$ in situations
where the ring in question may not be clear from the context. If
$\phi:R_{1}\to R_{2}$ is a homomorphism of rings and $S\subset R_{1}$ is a
subring of $R_{1}$, we let $\phi|_{S}$ denote the restriction of $\phi$ to
$S$. If $\psi:R_{2}\to R_{3}$ is a homomorphism from $R_{2}$ to the ring
$R_{3}$, we let $\psi\circ\phi$ denote the composition map. We will use the
notation $R_{1}\cong R_{2}$ to indicate that the rings $R_{1}$ and $R_{2}$ are
isomorphic.

If $X$ is a subset of a ring $R$, the two sided ideal of $R$ generated by the
set $X$ will be denoted by $\langle X\rangle$. If $I$ is an ideal of a ring
$R$, we write $r_{1}\equiv r_{2}\mod I$ when $r_{1}-r_{2}\in I$. We denote the
ring of $n\times n$ matrices over a ring $R$ by $M_{n}(R)$. We will also use
the notation $\langle x\in X|r\in R\rangle$ to denote a group with generators
in the set $X$ and relations in the set $R$. In cases where the meaning of the
notation is not clear from the context, we will add clarification.

The symbol $\mathbb{N}$ will be used to denote the non-negative integers,
$\\{0,1,2,3,\dots\\}$, and the symbol $\mathbb{N}_{\geq k}$ will be used to
denote the subset of $\mathbb{N}$ consisting of numbers greater than or equal
to $k$. We let $[k]$ denote the set $\\{1,\dots,k\\}$. The integers will be
denoted by $\mathbb{Z}$, and the rational numbers by $\mathbb{Q}$. Polynomial
rings $\mathbb{Q}[\\{x_{i}\\}_{i\in I}]$ will have non-commuting variables
$\\{x_{i}\\}_{i\in I}$ unless stated otherwise.

######  Definition 1.2.

If $R$ is a ring and $a\in R,a\not=0$, we say that $a$ is a zero divisor in
$R$ if there exists $b\in R,b\not=0$ such that $ab=0$ or $ba=0$. We say that
$R$ is a domain if $R$ has no zero divisors.

Let $A$ be a commutative unital ring. An (associative, possibly non-unital)
$A$-algebra is a (possibly non-unital, associative) ring $B$ with an
additional structure of unital $A$-module for which the multiplication map
$B\times B\to B$ is $A$-bilinear (i.e.
$(\alpha_{1}b_{1}+\alpha_{2}b_{2})(\alpha_{3}b_{3}+\alpha_{4}b_{4})=\displaystyle\sum_{i=1}^{2}\sum_{j=1}^{2}\alpha_{i}\alpha_{j}b_{i}b_{j},$
for all $\alpha_{i}\in A,b_{j}\in B$). Morphisms of $A$-algebras are
$A$-linear ring homomorphisms. We call $A$ the coefficient ring of the
$A$-algebra $B$. We regard rings as $\mathbb{Z}$-algebras.

A module $M$ for the $A$-algebra $B$ is a module for the ring $B$ together
with a unital $A$-module structure on $M$ such that the multiplication map
$B\times M\to M$ is $A$-bilinear. Morphisms of modules for the algebra $B$ are
$B$-linear maps which are also $A$-linear. We write $\mathrm{End}_{B}(M)$ for
the unital $A$-algebra of endomorphisms of $M$ and $\mathrm{Aut}_{B}(M)$ for
its unit group. For a bimodule $M$ for a pair $(B,B^{\prime})$ of
$A$-algebras, it is required that the $A$-module structures from $B$ and
$B^{\prime}$ coincide.

##  2\. Graphs

We use the notation and terminology of Serre [23] for graphs and trees.

######  Definition 2.1.

A graph $\Gamma$ consists of a set of vertices $S=\mbox{Vert}\ \Gamma$, a set
of edges $Y=\mbox{Edge}\ \Gamma$ and two maps

$Y\to S\times S,\ \ y\to(o(y),t(y))$

and

$Y\to Y,\ \ y\to\bar{y},$

which satisfy the following condition: for each $y\in Y$, we have
$\bar{\bar{y}}=y,\bar{y}\not=y$ and $o(y)=t(\bar{y})$.

Each $s\in S$ is called a vertex of $\Gamma$ and each $y\in Y$ is called an
edge of $\Gamma$. If $y\in Y$, $o(y)$ is called the origin of $y$ and $t(y)$
is called the terminus of $y$, together $o(y)$ and $t(y)$ are called the
extremities  of $y$. We say that two vertices are adjacent if they are the
extremities of some edge.

######  Definition 2.2.

An orientation of a graph $\Gamma$ is a subset $Y_{+}$ of $Y=\mbox{Edge}\
\Gamma$ such that $Y$ is the disjoint union
$Y=Y_{+}\bigcupdot\overline{Y_{+}}$, where
$\overline{Y_{+}}=\left\\{\bar{y}|y\in Y_{+}\right\\}$.

###  2.3. Paths

A path of length $n\geq 1$ in a graph $\Gamma$ is a sequence of edges
$p=(y_{1},\dots,y_{n})$ such that $t(y_{i})=o(y_{i+1})$, $1\leq i\leq n-1$. We
will denote such a path by $p=y_{1}\dots y_{n}$. Paths of length zero are just
single vertices and will be denoted by $p=[s]$ for $s\in S$. The length of a
path $p$ will be denoted by $\ell(p)$. Let $\mathfrak{P}$ denote the set of
all paths in $\Gamma$, we can extend the maps $o$ and $t$ to $\mathfrak{P}$ by
letting $o(y_{1}\dots y_{n})=o(y_{1})$ and $t(y_{1}\dots y_{n})=t(y_{n})$ and
$o([s])=s=t([s])$. If $p=y_{1}\dots y_{n}$ with $s_{i}=t(y_{i})$ and
$s_{i-1}=o(y_{i})$, we say that $p$ is a path from $s_{0}$ to $s_{n}$ and that
$o(y_{1})=s_{0}$ and $t(y_{n})=s_{n}$ are the extremities of the path.

A graph is connected if any two vertices are the extremities of at least one
path. If $p=y_{1}\dots y_{n}$ and $y_{i+1}=\bar{y}_{i}$ for some $1\leq i\leq
n-1$, then the pair $(y_{i},y_{i+1})$ is called a backtracking. A path $p$ is
a circuit if it is a path without backtracking such that $o(p)=t(p)$ and
$\ell(p)\geq 1$. A circuit of length one is called a loop. A graph $\Gamma$ is
called combinatorial if it has no circuit of length less than or equal to two.
If a graph $\Gamma$ is combinatorial, a path $p=y_{1}\dots y_{n}$ is
determined by the extremities of its edges and can be characterized and
denoted by its vertices as

$p=y_{1}\dots y_{n}=[s_{0}s_{1}\dots s_{n}],\ \mbox{where}\
s_{i}=t(y_{i}),s_{i-1}=o(y_{i}).$

Since our graphs of interest in this paper will be combinatorial, we will make
frequent use of both notations for paths. In what follows, an edge may be
denoted by its label $y$ or it may be represented by its extremities as
$[o(y),t(y)]$.

A geometric edge of a combinatorial graph, $\Gamma$, is a set
$\\{s_{1},s_{2}\\}$ of extremities of an edge in $\Gamma$. Each such geometric
edge corresponds to a pair of edges $\\{y,\bar{y}\\}$ in the edge set of
$\Gamma$. A combinatorial graph is determined by its vertices and geometric
edges.

######  Definition 2.4.

let $\Gamma_{1}=(S_{1},Y_{1})$ and $\Gamma_{2}=(S_{2},Y_{2})$ be two graphs. A
graph homomorphism $\phi:\Gamma_{1}\to\Gamma_{2}$ is a pair of maps
$\phi_{S}:S_{1}\to S_{2}$ and $\phi_{Y}:Y_{1}\to Y_{2}$ such that
$o(\phi_{Y}(y))=\phi_{S}(o(y))$ and $t(\phi_{Y}(y))=\phi_{S}(t(y))$ for every
$y\in Y_{1}$.

######  Example 2.5.

Below we see a representation of the combinatorial graph $\Gamma$, with
vertices, $S=\\{r,s,t,u,v\\}$, and edges, $Y=Y_{+}\bigcupdot\overline{Y_{+}}$,
where

$Y_{+}=\\{y_{\alpha}=[rs],y_{\epsilon}=[st],y_{\beta}=[ut],y_{\zeta}=[tv],y_{\delta}=[uv],y_{\gamma}=[ru]\\}$

We show only the directed edges in $Y_{+}$ on the graph.

|
---|---
$\textstyle{s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\epsilon}}$$\textstyle{t\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\zeta}}$$\textstyle{v}$$\textstyle{{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\alpha}}$$\scriptstyle{y_{\gamma}}$$\textstyle{{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\beta}}$$\scriptstyle{y_{\delta}}$

The above example will serve as our running example throughout the paper.

##  3\. Path Algebras

In this section, we present the definition of a path algebra along with a
summary of some related results on Gröbner bases from Green [16], which will
prove useful in later sections.

### 3.1.

Let $\Gamma=\left(S,Y\right)$ be a combinatorial graph, with vertices
$S=\\{s_{i}\\}_{i\in I}$ and edges $Y=\\{y_{j}\\}_{j\in J}$. Let
$\mathfrak{P}$ denote the set of paths in $\Gamma$ of finite length. Given a
commutative ring $A$ with identity, the path algebra of $\Gamma$ over $A$,
denoted $A\Gamma$, is an associative $A$-algebra, with an $A$-basis of paths
in $\mathfrak{P}$. Multiplication is given by concatenation;

$[s_{0}s_{1}\dots s_{n}][s^{\prime}_{0}s^{\prime}_{1}\dots
s^{\prime}_{m}]=\begin{cases}0&\mbox{if}\ s_{n}\not=s^{\prime}_{0}\\\
[s_{0}s_{1}\dots s_{n}s^{\prime}_{1}\dots s^{\prime}_{m}]&\mbox{if}\
s_{n}=s^{\prime}_{0}\end{cases}$

Paths of length $0$ form a set of orthogonal idempotents with this
multiplication, and if $S$ is finite, $A\Gamma$ has an identity given by

$1=\sum_{s\in S}[s].$

###  3.2. Admissible Orders on $\mathfrak{P}$

For the remainder of this section, we will restrict our discussion to the case
where the base ring of our path algebra is a field, $K$, and the case where
the path algebra has finitely many vertices and edges, as in Green [16] and
Green et. al. in [17]. We will combine results from Green [16] presented below
with direct limits in subsequent sections to ensure that we can restrict our
attention to the finite case when proving our main theorem.

A well order on the basis $\mathfrak{P}$ of $K\Gamma$ is a total order on
$\mathfrak{P}$, with the property that every nonempty subset of $\mathfrak{P}$
has a minimal element, or equivalently, for every descending chain of elements
of $\mathfrak{P}$, $p_{1}\geq p_{2}\geq p_{3}\geq\dots$, there exists some
$N>0$ for which $p_{N}=p_{N+1}=p_{N+2}=\dots\ $.

Given a well order $>$ on $\mathfrak{P}$, and $w\not=0\in K\Gamma$, with
$\displaystyle w=\sum_{i=1}^{n}a_{i}p_{i},a_{i}\not=0,a_{i}\in
K,p_{i}\in\mathfrak{P}$, we say that $p_{i}$ is the tip of $w$, denoted
$T(w)$, if $p_{i}\geq p_{j}$ for $1\leq j\leq n$. If $X$ is a subset of
$K\Gamma$, the tip of $X$ is

$T(X)=\\{p\in\mathfrak{P}|p=T(x)\ \mbox{for some}\ x\not=0,x\in X\\}.$

We denote the set of non tips of $X$ by

$NT(X)=\mathfrak{P}\backslash T(X).$

Note both T(X) and NT(X) depend on the order chosen.

If $W$ is a subspace of $K\Gamma$. By Green [16], Theorem 2.1, we have

(3.2.1) $K\Gamma=W\oplus\mbox{Span}(NT(W)).$

We say that an order on $\mathfrak{P}$ is admissible if it is a well order and
satisfies the following conditions for $r,s,t,u\in\mathfrak{P}$:

  1. (1)

if $r<s$ then $rt<st$ if both $rt\not=0$ and $st\not=0$.

  2. (2)

if $r<s$ then $ur<us$ if both $ur\not=0$ and $us\not=0$.

  3. (3)

if $r=st\not=0$ then $r\geq s$ and $r\geq t$.

A number of admissible well orders can be imposed on the paths in a path
algebra, see Green [16]. We give one example of such an admissible ordering
below.

######  Example 3.3.

The (left) length lexicographic order on $\mathfrak{P}$:
Let $\Gamma=\left(S,Y\right)$ be a graph, with vertices
$S=\\{s_{i}\\}_{i\in[n]}$ and edges $Y=\\{y_{j}\\}_{j\in[m]}$. Let
$\mathfrak{P}$ denote the set of paths in $\Gamma$ of finite length. We choose
an ordering on paths of length $\leq 1$, in such a way that each element of
$S$ is less that each element of $Y$.

$[s_{1}]<[s_{2}]<\dots<[s_{n}]<y_{1}<y_{2}<\dots<y_{m}.$

If $p$ and $q$ are paths of length at least one, $p<q$ if $l(p)<l(q)$, or if
$l(p)=l(q)$ and $p=y_{1}y_{2}\dots y_{i}y_{i+1}\dots y_{n}$,
$q=y_{1}y_{2}\dots y_{i}y^{\prime}_{i+1}\dots y^{\prime}_{n}$ with
$y_{i+1}<y^{\prime}_{i+1}$. It is not difficult to check that this is in in
fact a well order and conditions (1), (2) and (3) in the definition of an
admissible well order hold.

###  3.4. Gröbner Basis of an Ideal

If $>$ is an admissible order on $\mathfrak{P}$, and $I$ an ideal in $K\Gamma$
($K$ a field), we say that a subset $\mathfrak{G}\subset I$ is a Gröbner basis
for the ideal $I$, with respect to $>$, if the two-sided ideal generated by
$T(\mathfrak{G})$ is equal to the two-sided ideal generated by $T(I)$.
Equivalently, $\mathfrak{G}\subset I$ is a Gröbner basis for $I$ if for every
$b\in T(I)$ there is some $g\in\mathfrak{G}$ such that $b=pT(g)q$ for some
$p,q\in\mathfrak{P}$.

Given a Gröbner basis, $\mathfrak{G}$, for an ideal $I$ of $K\Gamma$ with
respect to an admissible order $>$, we have, by Equation 3.2.1, that as a
vector space over $K$,

(3.4.1)
$K\Gamma=I\oplus\mbox{Span}(NT(I))=\langle\mathfrak{G}\rangle\oplus\mbox{Span}(NT(I)).$

Equation 3.4.1 allows us to use a Gröbner basis, $\mathfrak{G}$, to determine
ideal membership for $I$. Given a Gröbner basis, $\mathfrak{G}$, for an ideal
$I$ of $K\Gamma$, each $y\in K\Gamma$ has a unique normal form
$N(y)\in\mbox{Span}(NT(I))$ with respect to the order $>$, where
$y=y_{1}+N(y)$ and $y_{1}\in I$. Since $\left\langle
T(\mathfrak{G})\right\rangle=\left\langle T(I)\right\rangle$, we have that
$\mbox{Span}(NT(I))$ has a basis consisting of those paths in $\mathfrak{P}$
which are not divisible by $T(g)$ for some $g\in\mathfrak{G}$ (i.e. the paths
in $K\Gamma$ which are not of the form $pT(g)q$ for some $g\in\mathfrak{G}$,
$p,q\in\mathfrak{P}$). The following is special case of Proposition 2.9, Green
[16], which follows easily from Equation 3.4.1 and makes explicit the
relationship between $K\Gamma/I$ and $\mbox{Span}(NT(I))$:

######  Proposition 3.5.

(See Proposition 2.9, Green [16]) Let $\Gamma$ be a graph with finitely many
vertices and edges. Let $K\Gamma$ be the corresponding path algebra with basis
of paths, $\mathfrak{P}$, admitting an admissible order $>$. Let $I$ be an
ideal of $K\Gamma$ with Gröbner basis $\mathfrak{G}$ and let $x,y\in K\Gamma$
with equivalence classes $x+I$ and $y+I$ respectively in the quotient algebra
$K\Gamma/I$. Then

  1. (1)

$x+I=y+I$ if and only if $N(x)=N(y)$.

  2. (2)

$x+I=N(x)+I$.

  3. (3)

The map $\sigma:K\Gamma/I\to K\Gamma$ with $\sigma(x+I)=N(x)$ is a vector
space splitting to the canonical surjection $\pi:K\Gamma\to K\Gamma/I$.

  4. (4)

$\sigma$ induces a $K$-linear isomorphism between $K\Gamma/I$ and
$\mbox{Span}(NT(I))$.

  5. (5)

Identifying $K\Gamma/I$ with $\mbox{Span}(NT(I))$, then $NT(I)$ is a K-basis
of $K\Gamma/I$ contained in $\mathfrak{P}$.

###  3.6. Overlap Relations

Let $>$ be an admissible order on $\mathfrak{P}$. If $f,g\in K\Gamma$,
$f,g\not=0$, and $p,q\in\mathfrak{P}$ such that

  1. (1)

$T(f)q=pT(g)$.

  2. (2)

$T(f)$ does not divide $p$ and $T(g)$ does not divide $q$.

The overlap relation of $f$ and $g$ by $p,q$ is

$o(f,g,p,q)=(1/C(T(f)))fq-(1/C(T(g)))pg,$

where $C(T(h))$ denotes the coefficient of $T(h)$ in $h\in K\Gamma.$ Note that
$T(o(f,g,p,q))<T(f)q=pT(g)$.

We say the elements of a set $\mathfrak{G}$ is tip reduced if for distinct
elements $g_{1},g_{2}\in\mathfrak{G}$, $T(g_{1})$ does not divide $T(g_{2})$.
An element of the path algebra $K\Gamma$, $\sum_{i=1}^{n}a_{i}p_{i},a_{i}\in
A,a_{i}\not=0,p_{i}\in\mathfrak{P}$, is said to be left uniform if for each
$p\in\mathfrak{P}$, either $pp_{i}=0$ for all $i$ or $pp_{i}\not=0$ for all
$i$, $1\leq i\leq n$. Right uniform elements are defined similarly and an
element is uniform if it is both left and right uniform. An element of the
path algebra $K\Gamma$, $\sum_{i=1}^{n}a_{i}p_{i},a_{i}\in
K,a_{i}\not=0,p_{i}\in\mathfrak{P}$ is uniform if and only if there are
vertices $v,w$ such that for $1\leq i\leq n$, $o(p_{i})=v$ and $t(p_{i})=w$.
The following result, limited for simplicity to a special case of Theorem 2.3,
Green [16] which is sufficient for our needs, can be used to identify when a
set of polynomials form a Gröbner basis for an ideal in a path algebra:

######  Theorem 3.7.

(See Green [16], Theorem 2.3) Let $\Gamma$ be a graph with finitely many
vertices and edges. Suppose that $\mathfrak{G}$ is a set of uniform, tip
reduced elements of $K\Gamma$. If for every overlap relation,
$o(g_{1},g_{2},p,q)$, with $g_{1},g_{2}\in\mathfrak{G}$, $p,q\in\mathfrak{P}$,
we have;

$o(g_{1},g_{2},p,q)=0,$

then $\mathfrak{G}$ is a Gröbner basis for $\langle\mathfrak{G}\rangle$.

##  4\. Coxeter Systems and Associated Path Algebras

In this section, we introduce the Path Algebras related to Coxeter systems and
the rings $R=R_{W,S}$ and $\widetilde{R}=\widetilde{R}_{W,S}$ from Dyer [13],
discussed in the introduction. We also prove a number of lemmas which will be
used in later sections to simplify the proof of our main result.

###  4.1. Path Algebras associated to Coxeter Systems

A Coxeter matrix is a matrix $(m_{ij})_{i,j\in I}$, where $I$ is a set of
indices, with $m_{ii}=1$ for all $i\in I$ and
$m_{ij}=m_{ji}\in\mathbb{N}_{\geq 2}\cup\\{\infty\\}$ for all $i,j\in
I,i\not=j$. If $S=\\{s_{i}\\}_{i\in I}$ is a set indexed by $I$, the
corresponding Coxeter group is the group with presentation

$W=\langle s_{i}(i\in I)|(s_{k}s_{l})^{m_{kl}}=1\ \mbox{for}\ s_{k},s_{l}\in
S\ \mbox{with}\ m_{kl}\not=\infty\rangle.$

The pair $(W,S)$ is called a Coxeter system with Coxeter matrix
$(m_{ij})_{i,j\in I}$. We say that $(W,S)$ has finite rank if $S$ is finite.

We associate to a Coxeter system $(W,S)$ a graph $\Gamma_{W,S}$ with vertices
$s_{i}\in S$ and edges $Y_{W,S}=\left\\{y_{ij}=[s_{i}s_{j}]|\ i\not=j\
\mbox{and}\ m_{ij}\geq 3\right\\}$. Let $\mathfrak{P}_{W,S}$ denote the set of
paths in $\Gamma_{W,S}$. We let $P_{W,S}$ denote the path algebra
$\mathbb{Z}\Gamma_{W,S}$ and let $\hat{P}_{W,S}$ denote the path algebra
$\mathbb{Q}\Gamma_{W,S}$.

For $m\in\mathbb{Z},m\geq 3$, let $C_{m}(t)$ denote the minimum polynomial of
$\displaystyle 4\cos^{2}\frac{\pi}{m}$ in $\mathbb{Z}[t]$. We have

$C_{3}(t)=t-1,\ C_{4}(t)=t-2,\ C_{5}(t)=t^{2}-3t+1,\ C_{6}(t)=t-3,\dots.$

To each edge $y_{ij}=[s_{i}s_{j}]$ of $\Gamma_{W,S}$ where $3\leq
m_{ij}<\infty$, we associate an element $C_{y_{ij}}$ of the subring
$[s_{i}]P_{W,S}[s_{i}]\subset{P}_{W,S}$ (resp.
$[s_{i}]\hat{P}_{W,S}[s_{i}]\subset\hat{P}_{W,S}$), where the imbedding is
non-unital. Notice that the ring $[s_{i}]P_{W,S}[s_{i}]$ (resp.
$[s_{i}]\hat{P}_{W,S}[s_{i}]$) has identity $[s_{i}]$, making evaluation of
the above polynomials possible over the ring. We let

$C_{y_{ij}}=C_{m_{ij}}([s_{i}s_{j}s_{i}])=C_{m_{ij}}(y_{ij}\bar{y}_{ij}),$

with $C_{m_{ij}}$ viewed as a polynomial over the ring $[s_{i}]P_{W,S}[s_{i}]$
(resp. $[s_{i}]\hat{P}_{W,S}[s_{i}]$). Let $I_{W,S}$, (respectively
$\hat{I}_{W,S}$), be the ideal of $P_{W,S}$ (resp. $\hat{P}_{W,S}$) generated
by the set $\mathfrak{C}_{W,S}=\left\\{C_{y}|y\in Y_{W,S}\right\\}$, and let
$R_{W,S}=P_{W,S}/I_{W,S}$ (resp. $\hat{R}_{W,S}=\hat{P}_{W,S}/\hat{I}_{W,S}$)
denote the quotient algebras.

######  Lemma 4.2.

Let $\Gamma_{W,S}$ be a graph associated to a finite rank Coxeter System
$(W,S)$ with Coxeter matrix $(m_{ij})_{\\{i,j\in I\\}}$. Then
$\mathfrak{C}_{W,S}$ is a Gröbner basis for the ideal $\hat{I}_{W,S}$ in the
path algebra $\hat{P}_{W,S}$.

###### Proof.

Regardless of the admissible ordering used,
$T(C_{y_{ij}})=(y_{ij}\bar{y}_{ij})^{d_{ij}}$, where degree
$C_{m_{ij}}=d_{ij}$, for all $y_{ij}\in Y_{W,S}$ with $m_{ij}<\infty$. Clearly
the set $\mathfrak{C}_{W,S}$ is tip reduced and uniform, therefore, by Theorem
3.7, we need only verify that all overlap relations
$o(C_{y_{ij}},C_{y_{kl}},p,q)$, $y_{ij},y_{kl}\in
Y,m_{ij},m_{kl}<\infty,p,q\in\mathfrak{P}_{W,S}$ are zero. Consider such an
overlap relation, $T(C_{y_{ij}})q=pT(C_{y_{kl}})$, for $p,q\in\mathfrak{P}$,
where $T(C_{y_{kl}})$ does not divide $q$ and $T(C_{y_{ij}})$ does not divide
$p$. Since $T(C_{y_{ij}})=(y_{ij}\bar{y}_{ij})^{d_{ij}}$ and
$T(C_{y_{kl}})=(y_{kl}\bar{y}_{kl})^{d_{kl}}$, and $T(C_{y_{kl}})$ does not
divide $q$, we must have $T(C_{y_{kl}})=aq$ for some $a\in\mathfrak{P}_{W,S}$
where $\ell(a)\geq 1$. Hence

$T(C_{y_{ij}})q=(y_{ij}\bar{y}_{ij})^{d_{ij}}q=pT(C_{y_{kl}})=paq=p(y_{kl}\bar{y}_{kl})^{d_{kl}},$

and since $a$ divides both $(y_{ij}\bar{y}_{ij})^{d_{ij}}$ and
$(y_{kl}\bar{y}_{kl})^{d_{kl}}$, we must have that either $y_{kl}=y_{ij}$ or
$y_{kl}=\bar{y}_{ij}$. If $y_{kl}=y_{ij}$, then
$(y_{ij}\bar{y}_{ij})^{d_{ij}}q=p(y_{ij}\bar{y}_{ij})^{d_{ij}}$ and by
comparing lengths and edges, we see that $p=q=(y_{ij}\bar{y}_{ij})^{k}$ for
some $k$. Thus $C_{y_{ij}}p-qC_{y_{ij}}=0$ in this case. If
$y_{kl}=\bar{y}_{ij}$, then
$(y_{ij}\bar{y}_{ij})^{d_{ij}}q=p(\bar{y}_{ij}{y}_{ij})^{d_{ij}}$. In this
case, a comparison of lengths and edges shows that
$p=q=(y_{ij}\bar{y}_{ij})^{k}y_{ij}$ for some $k$ and it is clear that
$C_{y_{ij}}p-qC_{\bar{y}_{ij}}=0$. This completes our proof. ∎

### 4.3.

Let $J\subset I$ and $S_{J}=\\{s_{j}|j\in J\\}$. Let $(W_{J},S_{J})$ denote
the Coxeter system with Coxeter matrix $(m_{ij})_{i,j\in J}$. The group
$W_{J}$ has generators and relations

$W=\langle s_{j}(j\in J)|(s_{k}s_{l})^{m_{kl}}=1\ \mbox{for}\ s_{k},s_{l}\in
S_{J}\ \mbox{with}\ m_{kl}\not=\infty\rangle.$

and is called a standard parabolic subgroup of $W$. If $J_{1}\subseteq
J_{2}\subseteq I$ we have an inclusion map of graphs from the vertices
$S_{J_{1}}$ and edges $Y_{W_{J_{1}},S_{J_{1}}}$ in
$\Gamma_{W_{J_{1}},S_{J_{1}}}$ to the corresponding vertices and edges in
$\Gamma_{W_{J_{2}},S_{J_{2}}}$. This extends to an inclusion map
$i_{J_{1},J_{2}}:\mathfrak{P}_{W_{J_{1}},S_{J_{1}}}\to\mathfrak{P}_{W_{J_{2}},S_{J_{2}}}$,
which in turn gives us a non-unital monomorphism of $\mathbb{Q}$ algebras;
$\hat{i}_{J_{1},J_{2}}:\hat{P}_{W_{J_{1}},S_{J_{1}}}\to\hat{P}_{W_{J_{2}},S_{J_{2}}}$.
Since
$\hat{i}_{J_{1},J_{2}}(\mathfrak{C}_{W_{J_{1}},S_{J_{1}}})\subseteq\mathfrak{C}_{W_{J_{2}},S_{J_{2}}}$,
we get a well defined induced non-unital homomorphism of quotient algebras:

$\tilde{i}_{J_{1},J_{2}}:\hat{R}_{W_{J_{1}},S_{J_{1}}}\to\hat{R}_{W_{J_{2}},S_{J_{2}}}.$

######  Lemma 4.4.

Let $(W,S=\\{s_{i}\\}_{i\in I})$ be a Coxeter system. Let $J_{1},J_{2}$ be
finite subsets of $I$ with $J_{1}\subseteq J_{2}$, then the map
$\tilde{i}_{J_{1},J_{2}}:\hat{R}_{W_{J_{1}},S_{J_{1}}}\to\hat{R}_{W_{J_{2}},S_{J_{2}}}$
is a monomorphism of non-unital $\mathbb{Q}$-algebras.

###### Proof.

Let $>$ be an admissible ordering on $\mathfrak{P}_{W_{J_{2}},S_{J_{2}}}$. The
restriction of $>$ to $i_{J_{1},J_{2}}(\mathfrak{P}_{W_{J_{1}},S_{J_{1}}})$,
gives us an admissible ordering on $\mathfrak{P}_{W_{J_{1}},S_{J_{1}}}$ which,
for the sake of simplicity, we will also denote by $>$. By Proposition 3.5,
$\hat{R}_{W_{J_{i}},S_{J_{i}}}$ can be identified as a vector space over
$\mathbb{Q}$ with
$\mbox{Span}(NT(\hat{I}_{W_{J_{i}},S_{J_{i}}})),i\in\\{1,2\\}$ (with respect
to the ordering $>$). If $p\in NT(\hat{I}_{W_{J_{1}},S_{J_{1}}})$, then $p$ is
not divisible by $T(C_{y})$ for any
$C_{y}\in\mathfrak{C}_{W_{J_{1}},S_{J_{1}}}$. Since
$p\in\mathfrak{P}_{W_{J_{1}},S_{J_{1}}}$, $p$ cannot be divisible by
$T(C_{y})$ for any
$C_{y}\in\mathfrak{C}_{W_{J_{2}},S_{J_{2}}}\backslash\mathfrak{C}_{W_{J_{1}},S_{J_{1}}}$
and thus $i_{J_{1},J_{2}}(p)\in NT(\hat{I}_{W_{J_{2}},S_{J_{2}}})$. Thus, the
map
$i_{J_{1},J_{2}}:\mathfrak{P}_{W_{J_{1}},S_{J_{1}}}\to\mathfrak{P}_{W_{J_{2}},S_{J_{2}}}$
restricts to a one-to-one map
$i^{\prime}_{J_{1},J_{2}}:NT(\hat{I}_{W_{J_{1}},S_{J_{1}}})\to
NT(\hat{I}_{W_{J_{2}},S_{J_{2}}})$. Using the vector space identification
above, we see that
$\tilde{i}_{J_{1},J_{2}}:\hat{R}_{W_{J_{1}},S_{J_{1}}}\to\hat{R}_{W_{J_{2}},S_{J_{2}}}$
is a monomorphism as desired. ∎

###  4.5. Direct Systems

We shall prove some of our results involving rings attached to infinite rank
Coxeter groups by regarding them as direct limits of rings attached to the
finite rank standard parabolic subgroups.

######  Definition 4.6.

A directed poset is a poset $(P,\leq)$, $P\not=\emptyset$, such that for any
$p_{1},p_{2}\in P$, there is an element $p_{3}\in P$ such that $p_{1}\leq
p_{3}$ and $p_{2}\leq p_{3}$.

######  Definition 4.7.

A direct system $\left\\{P,\\{R_{p}\\},\\{i_{p_{1},p_{2}}\\}\right\\}$ in the
category of non-unital $\mathbb{Q}$-algebras consists of a directed poset,
$P$, a collection of non-unital $\mathbb{Q}$-algebras, $\\{R_{p}\\}_{p\in P}$,
and a set of morphisms of $\mathbb{Q}$ algebras,
$\\{i_{p_{1},p_{2}}:R_{p_{1}}\to R_{p_{2}}\\}_{p_{1},p_{2}\in P,p_{1}\leq
p_{2}}$ such that

  1. (1)

$i_{p,p}$ acts as the identity on $R_{p}$.

  2. (2)

$i_{p_{2},p_{3}}\circ i_{p_{1},p_{2}}=i_{p_{1},p_{3}}$ for
$p_{1},p_{2},p_{3}\in P,\ p_{1}\leq p_{2}\leq p_{3}$.

######  Note.

When considering direct systems in the category of non-unital
$\mathbb{Q}$-algebras throughout the remainder of this section, all associated
morphisms considered will be morphisms in that category.

######  Definition 4.8.

Let $\left\\{P,\\{R_{p}\\},\\{i_{p_{1},p_{2}}\\}\right\\}$ be a direct system
of non-unital $\mathbb{Q}$ algebras. A $\mathbb{Q}$ algebra $R$ is called the
direct limit of the direct system, denoted $\varinjlim{R_{p}}$, if there exist
homomorphisms, $i_{p}:R_{p}\to R,\ p\in P$ such that

  1. (1)

For any $p_{1},p_{2}\in P$, $p_{1}\leq p_{2}$, the following diagram commutes:

$\textstyle{R_{p_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{p_{1}}}$$\scriptstyle{i_{p_{1},p_{2}}}$$\textstyle{{R_{p_{2}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{p_{2}}}$$\textstyle{R}$

  2. (2)

Given a non-unital $\mathbb{Q}$ algebra, $Q$, and a set of homomorphisms
$\phi_{p}:R_{p}\to Q$, $p\in P$, such that the diagram

$\textstyle{R_{p_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{p_{1}}}$$\scriptstyle{i_{p_{1},p_{2}}}$$\textstyle{{R_{p_{2}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{p_{2}}}$$\textstyle{Q}$

commutes for all $p_{1},p_{2}\in P$, $p_{1}\leq p_{2}$, then there exists a
unique homomorphism $\phi:R\to Q$ such that the following diagram commutes for
all $p\in P$:

$\textstyle{R_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{p}}$$\scriptstyle{i_{p}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Q}$

######  Lemma 4.9.

Let $\left\\{P,\\{R_{p}\\},\\{i_{p_{1},p_{2}}\\}\right\\}$ be a direct system
of non-unital $\mathbb{Q}$ algebras. If $i_{p_{1},p_{2}}$ is a monomorphism
for all $p_{1},p_{2}\in P$, then the homomorphisms
$i_{p}:R_{p}\to\varinjlim{R_{p}}$ making the diagram below commute, are also
monomorphisms;

(4.9.1)
$\textstyle{R_{p_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{p_{1}}}$$\scriptstyle{i_{p_{1},p_{2}}}$$\textstyle{{R_{p_{2}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{p_{2}}}$$\textstyle{\varinjlim{R_{p}}}$

###### Proof.

One can construct the direct limit of non-unital $\mathbb{Q}$ algebras as
follows. Let

$R=\varinjlim{R_{p}}=\bigcupdot_{P\in P}R_{p}/\sim$

where $\sim$ is the equivalence relation on $\bigcupdot_{P\in P}R_{p}$ defined
by

$r_{1}\sim r_{2},r_{1}\in R_{p_{1}},r_{2}\in R_{p_{2}}\ \mbox{iff.}\
i_{p_{1},p_{3}}(r_{1})=i_{p_{2},p_{3}}(r_{2})\ \mbox{for some}\
p_{3}>p_{1},p_{2}.$

The ring and algebra operations on the equivalence classes of $R$, denoted by
square brackets $[]$, are defined as follows. For each $\alpha\in\mathbb{Q}$
and each $[r]\in R$, let $\alpha[r]=[\alpha r]$. Given $[r_{1}],[r_{2}]\in R$
such that $r_{1}\in R_{p_{1}}$ and $r_{2}\in R_{P_{2}}$, then for
$p_{3}>p_{1},p_{2}$, we define
$[r_{1}]+[r_{2}]=[i_{p_{1},p_{3}}(r_{1})+i_{p_{2},p_{3}}(r_{2})]$ and
$[r_{1}][r_{2}]=[i_{p_{1},p_{3}}(r_{1})i_{p_{2},p_{3}}(r_{2})]$. The
equivalence class $[0]$ gives a zero element for the ring $R$. It can be
checked easily that R along with the homomorphisms $i_{p}:R_{p}\to R,p\in P$
given by $i_{p}(r)=[r]$ for $r\in R_{p}$ form a non-unital $\mathbb{Q}$
algebra making the diagram 4.9.1 above commute. Given a non-unital
$\mathbb{Q}$ algebra $Q$ and a set of morphisms $\phi_{p}:R_{p}\to Q$, $p\in
P$, such that $\phi_{p_{2}}\circ i_{p_{1},p_{2}}=\phi_{p_{1}}$, we define
$\phi:R\to Q$ as $\phi([r_{p}])=\phi_{p}(r_{p})$ for $r_{p}\in R_{p}$. One can
easily check that $\phi$ is a well defined homomorphism with the property that
$\phi\circ i_{p}=\phi_{p}$ for all $p\in P$. It is unique since any
homomorphism $\psi:R\to Q$ with $\psi\circ i_{p}=\phi_{p}$ must agree with
$\phi$.

By the universal mapping property, the direct limit is unique up to
isomorphism, hence it is enough to show that the homomorphisms $i_{p}:R_{p}\to
R$ are monomorphism for all $p\in P$ for the non-unital $\mathbb{Q}$ algebra
$R$ constructed above. Now it is clear that if $i_{p}(r)=[0]$ for some $r\in
R_{p},p\in P$, then we must have $i_{p,p_{2}}(r)=0\in R_{p_{2}}$ for some
$p_{2}\in P$. Thus if all of the maps $i_{p_{1},p_{2}},p_{1},p_{2}\in P$ are
monomorphisms, we must have that each $i_{p},p\in P$ is also a monomorphism. ∎

### 4.10.

For the Coxeter system $(W,S=\\{s_{i}\\}_{i\in I})$, let $\mathscr{P}(I)$
denote the set of finite subsets of $I$. Then $\mathscr{P}(I)$ is a directed
poset with inclusion as a partial ordering. It is not hard to verify that the
quotient algebras associated to the family of parabolic subgroups,
$\\{\hat{R}_{W_{J},S_{J}}\\}_{J\in\mathscr{P}(I)}$, and the homomorphisms
$\tilde{i}_{J_{1},J_{2}}:\hat{R}_{W_{J_{1}},S_{J_{1}}}\to\hat{R}_{W_{J_{2}},S_{J_{2}}},$
$J_{1}\subseteq J_{2}$, form a direct system,
$\left\\{\mathscr{P}(I),\\{\hat{R}_{W_{J},S_{J}}\\},\\{\tilde{i}_{J_{1},J_{2}}\\}\right\\}$
in the category of non-unital algebras over $\mathbb{Q}$.

######  Lemma 4.11.

Let $(W,S=\\{s_{i}\\}_{i\in I})$, be a Coxeter system and
$\left\\{\mathscr{P}(I),\\{\hat{R}_{W_{J},S_{J}}\\},\\{\tilde{i}_{J_{1},J_{2}}\\}\right\\}$
the direct system of quotient algebras associated to the parabolic subgroups
of $W$ of finite rank. Then

  1. (1)

Given $x\in\hat{R}_{W,S}$, there exists $J\in\mathscr{P}(I)$ such that
$x=\tilde{i}_{J,I}(x^{\prime})$ for some $x^{\prime}\in\hat{R}_{W_{J},S_{J}}$.

  2. (2)

The homomorphisms $\tilde{i}_{J,I}:\hat{R}_{W_{J},S_{J}}\to\hat{R}_{W,S}$ have
the property that
$\tilde{i}_{J_{2},I}\circ\tilde{i}_{J_{1},J_{2}}=\tilde{i}_{J_{1},I}$ for all
$J_{1},J_{2}\in\mathscr{P}(I)$, $J_{1}\subseteq J_{2}$.

  3. (3)

Given a non-unital $\mathbb{Q}$ algebra $Q$ and a family of homomorphisms
$\\{\phi_{J}:\hat{R}_{W_{J},S_{J}}\to Q|J\in\mathscr{P}(I)\\}$ such that the
following diagram commutes for each $J_{1}\subseteq J_{2}\subset I$, $J_{1}$
and $J_{2}$ finite;

(4.11.1)
$\textstyle{\hat{R}_{W_{J_{1}},S_{J_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{J_{1}}}$$\scriptstyle{\tilde{i}_{J_{1},J_{2}}}$$\textstyle{{\hat{R}_{W_{J_{2}},S_{J_{2}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{J_{2}}}$$\textstyle{Q}$

then there exists a homomorphism $\phi$ such that the following diagram
commutes for all finite subsets $J$ of $I$:

(4.11.2)
$\textstyle{\hat{R}_{W_{J},S_{J}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{J}}$$\scriptstyle{\tilde{i}_{J,I}}$$\textstyle{\hat{R}_{W,S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Q}$

  4. (4)

$\hat{R}_{W,S}=\varinjlim{\hat{R}_{W_{J},S_{J}}}$ in the category of non-
unital $\mathbb{Q}$ algebras.

  5. (5)

The maps $\tilde{i}_{J,I}:\hat{R}_{W_{J},S_{J}}\to\hat{R}_{W,S}$ are
injective.

###### Proof.

Given an $x=p+\hat{I}_{W,S}\in\hat{R}_{W,S}$, where $p\in P_{W,S}$, then
$p=\sum_{1}^{n}\alpha_{i}p_{i}$ where $p_{i}\in\mathfrak{P}_{W,S}$. Since each
path $p_{i}$ is a product of only a finite number of edges, we have
$x=\tilde{i}_{J,I}(x^{\prime})$ for some $J\in\mathscr{P}(I)$ and
$x^{\prime}\in\hat{R}_{W_{J},S_{J}}$. This proves (1).

Since,
$(\tilde{i}_{J_{2},I}\circ\tilde{i}_{J_{1},J_{2}})(p+\hat{I}_{W_{J_{1}},S_{J_{1}}})=\tilde{i}_{J_{1},I}(p)$
for all $J_{1},J_{2}\in\mathscr{P}(I)$, $J_{1}\subseteq J_{2}$, and all
$p\in\mathfrak{P}_{W_{J_{1}},S_{J_{1}}}$, (2) follows.

Assume now that we are given a non-unital $\mathbb{Q}$ algebra $Q$ and a
family of homomorphisms $\\{\phi_{J}:\hat{R}_{W_{J},S_{J}}\to
Q|J\in\mathscr{P}(I)\\}$ making Diagram 4.11.1 commute. Let
$x=p+\hat{I}_{W,S}\in\hat{R}_{W,S}$. By (1), $x=\tilde{i}_{J,I}(x^{\prime})$
for some $J\in\mathscr{P}(I)$ and $x^{\prime}\in\hat{R}_{W_{J},S_{J}}$. We let
$\phi(x)=\phi_{J}(x^{\prime})$. By (2) and the commutativity of Diagram
4.11.1, $\phi(x)$ is independent of the choice of $J$. For each
$C_{y}\in\mathfrak{C}_{W,S}$,
$\phi(C_{y}+\hat{I}_{W,S})=\tilde{i}_{J,I}(C_{y}+\hat{I}_{W_{J},S_{J}})=\tilde{i}_{J,I}(0)=0$,
where $J=\\{i,j|o(y)=s_{i},t(y)=s_{j}\\}$. Thus $\phi$ is well defined. That
the diagram 4.11.2 commutes follows from the definition of $\phi$, and since
any homomorphism making 4.11.2 commute must agree with $\phi$, we see it is
the unique homomorphism with this property. This establishes (3).

From Definition 4.8, we see that (4) follows from (2) and (3). By Lemma 4.4,
the maps $\tilde{i}_{J_{1},J_{2}}$ in our direct system are monomorphisms.
Therefore (5) follows from (2), (3), (4) and Lemma 4.9. This completes the
proof. ∎

######  Lemma 4.12.

Let $(W,S)$ be a Coxeter system and $\Gamma_{W,S}$, the associated graph. Let
$\hat{\pi}$ denote the quotient map $\hat{P}_{W,S}$ to $\hat{R}_{W,S}$. Let
$\mathfrak{N}=\\{x\in\mathfrak{P}_{W,S}|x\not=ptq,t\in
T(\mathfrak{C}_{W,S}),p,q\in\mathfrak{P}_{W,S}\\}$. The set
$\hat{\pi}(\mathfrak{N})$ forms a basis for $\hat{R}_{W,S}$ as a vector space
over $\mathbb{Q}$.

###### Proof.

Let $x\in\hat{R}_{W,S}$, by Lemma 4.11 (1), we have $x=\tilde{i}_{J,I}(r)$ for
some finite $J\subseteq I$ and $r\in\hat{R}_{W_{J},S_{J}}$. Hence
$x\in\mbox{Span}(\tilde{i}_{J,I}(NT(\hat{I}_{W_{J},S_{J}})))\subseteq\mbox{Span}(\hat{\pi}(\mathfrak{N}))$.
let $x_{1},x_{2},\dots,x_{n}$ be a subset of $\hat{\pi}(\mathfrak{N})$ such
that $\sum_{i=1}^{n}\alpha_{i}x_{i}=0$ for some
$\alpha_{1},\alpha_{2},\dots,\alpha_{n}\in\mathbb{Q}$, then we have
$x_{i}=\tilde{i}_{J,I}(p_{i}+\hat{I}_{W_{J},S_{J}})$ for all $i$, for some
finite $J\subseteq I$ with each $p_{i}\in NT(\hat{I}_{W_{J},S_{J}})$. Thus,
since $\tilde{i}_{J,I}$ is a monomorphism by Lemma 4.11 (5), we have
$\sum_{i=1}^{n}\alpha_{i}p_{i}+\hat{I}_{W_{J},S_{J}}=0$ in
$\hat{R}_{W_{J},S_{J}}$ and since the cosets corresponding to
$NT(\hat{I}_{W_{J},S_{J}})$ form a vector space basis in
$\hat{R}_{W_{J},S_{J}}$, we see that $\alpha_{i}=0$ for $1\leq i\leq n$. Hence
the elements of $\hat{\pi}(\mathfrak{N})$ span $\hat{R}_{W,S}$ and are
linearly independent over $\mathbb{Q}$ and thus form a basis for
$\hat{R}_{W,S}$ as a vector space over $\mathbb{Q}$. ∎

######  Lemma 4.13.

Let $(W,S)$ be a Coxeter system and $\Gamma_{W,S}$, the associated graph. The
inclusion map $i:P_{W,S}\to\hat{P}_{W,S}$ which identifies the basis of paths
$\mathfrak{P}_{W,S}$ in both algebras, induces a monomorphism
$\tilde{i}:R_{W,S}\to\hat{R}_{W,S}$.

###### Proof.

It is clear that $\tilde{i}$ is a well defined homomorphism of rings, since
$\mathfrak{C}_{W,S}$ is a generating set for both $I$ and $\hat{I}$. We let
$\pi$ and $\hat{\pi}$ denote the quotient maps from ${P}_{W,S}$ to $R_{W,S}$
and $\hat{P}_{W,S}$ to $\hat{R}_{W,S}$ respectively. We let
$\mathfrak{N}=\\{x\in\mathfrak{P}_{W,S}|x\not=ptq,t\in
T(\mathfrak{C}_{W,S}),p,q\in\mathfrak{P}_{W,S}\\}$. By Lemma 4.12, we have
that $\hat{\pi}(\mathfrak{N})$ is a basis of $\hat{R}_{W,S}$ as a vector space
over $\mathbb{Q}$. Letting $M$ denote the $\mathbb{Z}$-submodule of
${P}_{W,S}$ generated by $\mathfrak{N}$, and $V=\mbox{Span}(\mathfrak{N})$ the
subspace of $\hat{P}_{W,S}$ as a $\mathbb{Q}$ vector space, we get a
commutative diagram;

(4.13.1) |
---|---
$\textstyle{{P_{W,S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{i}}$$\scriptstyle{\pi}$$\textstyle{{\hat{P}_{W,S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\textstyle{{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i|_{M}}$$\scriptstyle{\pi|_{M}}$$\textstyle{{V=\mbox{Span}(\mathfrak{N})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}|_{V}}$$\textstyle{{R_{W,S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{i}}$$\textstyle{\hat{R}_{W,S}}$

where ${\pi|_{M}}$ and $\hat{\pi}|_{V}$ denote the restrictions of the ring
homomorphisms $\pi$ and $\hat{\pi}$ as $\mathbb{Z}$ module and vector space
homomorphism respectively. By Lemma 4.12, $\hat{\pi}|_{V}$ is an isomorphism
of vector spaces. In particular $\hat{\pi}|_{V}$ is a monomorphism.

We claim that ${\pi|_{M}}$ maps $M$ onto $R_{W,S}$. We show that
$P_{W,S}=M+I_{W,S}$. First, we note that the paths of length zero,
$\\{[s_{i}]|s_{i}\in S\\}$ are in $\mathfrak{N}\subset M$. Next, we show that
given any $p\in\mathfrak{N}$, and any $y_{ij}\in Y$, with $j=o(p)$, then
$y_{ij}p=q+I_{W,S}$ for some $q\in M$. If $y_{ij}p\in\mathfrak{N}$, then
$q=y_{ij}p$ and we are done. Otherwise, $p\in\mathfrak{N}$ and
$y_{ij}p\not\in\mathfrak{N}$. Thus $p$ is not divisible by any element of
$T(\mathfrak{C}_{W,S})$ and $y_{ij}p$ is divisible by some element of
$T(\mathfrak{C}_{W,S})$. Comparing edges along the paths $p$ and $y_{ij}p$, we
see that $y_{ij}p=T(C_{y_{ij}})p_{1}$, for some $p_{1}\in\mathfrak{P}$. Thus,
since $C_{y_{ij}}$ is a monic polynomial over $\mathbb{Z}$ in
$y_{ij}\bar{y}_{ij}$, we have $y_{ij}p-C_{y_{ij}}p_{1}$ is a $\mathbb{Z}$\-
linear combination of paths which divide $p$. Since paths which divide $p$ are
already in normal form and are therefore in $\mathfrak{N}$, we have
$y_{ij}p-C_{y_{ij}}p_{1}\in M$ and $y_{ij}p\in M+I_{W,S}$ as asserted. To
prove our claim, we note that because $[s_{i}]\in\mathfrak{N}$ for all
$s_{i}\in S$, and $y_{ij}p\in M+I_{W,S}$ for all $y_{ij}\in Y$ and
$p\in\mathfrak{N}$ such that $o(p)=j$, we have $q+I_{W,S}\in M+I_{W,S}$ for
any $q\in\mathfrak{P}$. Therefore as a $\mathbb{Z}$ module,
$P_{W,S}=M+I_{W,S}$ and thus ${\pi|_{M}}$ maps $M$ onto $R_{W,S}$, proving our
claim.

Now since $\hat{P}_{W,S}\cong\mathbb{Q}\otimes_{\mathbb{Z}}P_{W,S}$ and
$P_{W,S}$ is a free $\mathbb{Z}$ module, we have that $i$ is a monomorphism.
Thus its restriction $i|_{M}$ is also a monomorphism. Because $\pi|_{M}$ is an
epimorphism and $i|_{M}$ and $\hat{\pi}|_{V}$ are both monomorphisms, we can
conclude from our diagram 4.13.1 that $\tilde{i}$ is a monomorphism as
desired. ∎

######  Corollary 4.14.

Let $(W,S)$ be a Coxeter system and $\Gamma_{W,S}$, the associated graph. Let
${\pi}$ denote the quotient map ${P}_{W,S}$ to ${R}_{W,S}$. Let
$\mathfrak{N}=\\{x\in\mathfrak{P}_{W,S}|x\not=ptq,t\in
T(\mathfrak{C}_{W,S}),p,q\in\mathfrak{P}_{W,S}\\}$. The elements of the set
${\pi}(\mathfrak{N})$ are pairwise distinct and form a basis for ${R}_{W,S}$
as a $\mathbb{Z}$-module.

###### Proof.

As in the proof of Lemma 4.13, let $M$ be the $\mathbb{Z}$-submodule of
${P}_{W,S}$ spanned by $\mathfrak{N}$. Since $\pi|_{M}$ is onto, we get that
$\pi(\mathfrak{N})$ spans ${R}_{W,S}$ as a $\mathbb{Z}$-module. From the proof
of Lemma 4.13, we have a commutative diagram:

(4.14.1) |
---|---
$\textstyle{{P_{W,S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{i}}$$\scriptstyle{\pi}$$\textstyle{{\hat{P}_{W,S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\textstyle{{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i|_{M}}$$\scriptstyle{\pi|_{M}}$$\textstyle{{V=\mbox{Span}(\mathfrak{N})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}|_{V}}$$\textstyle{{R_{W,S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{i}}$$\textstyle{\hat{R}_{W,S}}$

where $\hat{\pi}|_{V}\circ i|_{M}$ is a monomorphism. Thus
$\tilde{i}\circ\pi|_{M}$ is a monomorphism and since
$\tilde{i}\circ\pi|_{M}(\mathfrak{N})$ forms a basis over $\mathbb{Q}$ for
$\hat{R}_{W,S}$ by Lemma 4.12, we must have that ${\pi}(\mathfrak{N})$ are
linearly independent over $\mathbb{Z}$ in ${R}_{W,S}$. ∎

######  Example 4.15.

Let $(W,S)$ be the Coxeter system with Coxeter matrix

$\left(\begin{array}[]{ccccc}1&3&2&4&2\\\ 3&1&5&2&2\\\ 2&5&1&6&5\\\
4&2&6&1&\infty\\\ 2&2&5&\infty&1\end{array}\right),$

then $(W,S)$ be a Coxeter system with generators
$S=\\{s_{1}=r,s_{2}=s,s_{3}=t,s_{4}=u,s_{5}=v\\}$ and relations

$r^{2}=s^{2}=t^{2}=u^{2}=v^{2}=1=(rs)^{3}=(st)^{5}=(tu)^{6}=(tv)^{5}=(ru)^{4}=(rt)^{2}=(su)^{2}=(vr)^{2}=(vs)^{2}.$

Let $\Gamma_{W,S}$ be the associated graph. On the left below, we show the
edges in $Y_{W,S,+}$ as before on the right, we label each geometric edge
$\\{s_{i}s_{j}\\},s_{i},s_{j}\in S$, in the corresponding graph with $m_{ij}$,
if $(s_{i}s_{j})^{m_{ij}}=1$ is among the above relations and $m_{ij}\not=2$.
The remaining edges are labelled with infinity.

|
---|---
$\textstyle{s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\epsilon}}$$\textstyle{t\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\zeta}}$$\textstyle{v}$$\textstyle{{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\alpha}}$$\scriptstyle{y_{\gamma}}$$\textstyle{{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\beta}}$$\scriptstyle{y_{\delta}}$ |
---|---
$\textstyle{s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\textstyle{t\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\textstyle{v}$$\textstyle{{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$$\scriptstyle{4}$$\textstyle{{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\scriptstyle{\infty}$

We have $\hat{P}_{W,S}=\mathbb{Q}\Gamma_{W,S}$ and $\hat{I}_{W,S}$ is the
ideal of $\hat{P}_{W,S}$ with the following generators:

$\begin{array}[]{lll}C_{y_{\alpha}}=[rsr]-[r],&C_{{y}_{\gamma}}=[rur]-2[r],&C_{{y}_{\beta}}=[utu]-3[u],\\\\[10.00002pt]
C_{\bar{y}_{\alpha}}=[srs]-[s],&C_{\bar{y}_{\gamma}}=[uru]-2[u],&C_{\bar{y}_{\beta}}=[tut]-3[t]\\\\[10.00002pt]
C_{{y}_{\epsilon}}=[ststs]-3[sts]+[s],&C_{{y}_{\zeta}}=[tvtvt]-3[tvt]+[t],&\\\\[10.00002pt]
C_{\bar{y}_{\epsilon}}=[tstst]-3[tst]+[t],&C_{\bar{y}_{\zeta}}=[vtvtv]-3[vtv]+[v].&\\\\[10.00002pt]
\end{array}$

###  4.16. The Ring $\widetilde{R}$

In this section, we give the definition of the ring
$\widetilde{R}=\widetilde{R}_{W,S}$ for a Coxeter System. The ring has
relations given by polynomials satisfying a recurrence relation. To facilitate
a discussion of the representation theory motivating the theory in the next
section, we give the definition of the polynomials in the more general setting
of a ring with idempotents.

### 4.17.

Let $B$ be a (possibly non-unital) ring. For any idempotents $e,f\in B$ and
elements $a\in fBe$, $b\in eBf$, define elements $c_{n}(a,b,e,f)\in B$ for
$n\in\mathbb{Z}$ by the recurrence formulae

$c_{2n+2}(a,b,e,f)=ac_{2n+1}(a,b,e,f)-c_{2n}(a,b,e,f),\qquad
c_{2n+1}(a,b,e,f)=bc_{2n}(a,b,e,f)-c_{2n-1}(a,b,e,f)$

for $n\in\mathbb{Z}$ and initial conditions

$c_{0}(a,b,e,f)=0,\qquad c_{1}(a,b,e,f)=e.$

We have

$c_{2}(a,b,e,f)=a,\ c_{3}(a,b,e,f)=ba-e,\ c_{4}(a,b,e,f)=aba-2a,\
c_{5}(a,b,e,f)=baba-3ba+e,\dots$

Define also elements $C_{n}(a,b,e,f)\in B$ for $n\in\mathbb{N}_{\geq 2}$ as
follows. Let $C_{2}(a,b,e,f)=a$. For $n\geq 3$, let
$C_{n}(a,b,e,f):=\rho_{B,e,a,b}(C_{n}(t))$ where $C_{n}(t)\in\mathbb{Z}[t]$ is
the minimal polynomial over $\mathbb{Q}$ of (the algebraic integer)
$\displaystyle 4\cos^{2}\frac{\pi}{n}$, and $\rho_{B,e,a,b}$ denotes the the
ring homomorphism $\rho_{B,e,a,b}\colon\mathbb{Z}[t]\to eBe$ which maps
$1\mapsto e$ and $t\mapsto ba$ (note that the ring $eBe\ni e,ba$ has identity
element $e$). We see that

$C_{3}(a,b,e,f)=ba-e,\ C_{4}(a,b,e,f)=ba-2e,\ C_{5}(a,b,e,f)=baba-3ba+e,\
C_{6}(a,b,e,f)=ba-3e,\ \dots$

We write $c_{n}=c_{n,B}$ and $C_{n}=C_{n,B}$ if it is necessary to indicate
dependence on the ring $B$. Then for a ring homomorpism $\theta\colon B\to
B^{\prime}$, one has
$\theta(c_{n,B}(a,b,e,f))=c_{n,B^{\prime}}(\theta(a),\theta(b),\theta(e),\theta(f))$
and similarly with $c_{n}$ replaced by $C_{n}$ if $n\geq 2$.

### 4.18.

By [13], $c_{n}(a,b,e,f)=\prod_{N\in\mathbb{N}_{\geq 2},N\mid
n}C_{N}(a,b,e,f)$ for all $n\in\mathbb{N}_{\geq 2}$, where $N\mid n$ means $N$
is a divisor of $n$ in $\mathbb{Z}$ and the order of factors in the product on
the right is immaterial except that $C_{2}(a,b,e,f)$ must be the leftmost
factor if $n$ is even. In fact, if the pairwise distinct positive, non-unit
integer divisors, of $N$ are $N_{1},\ldots,N_{k}$, then
$c_{n}(a,b,e,f)=C^{\prime}_{N_{1}}\ldots C^{\prime}_{N_{k}}$ where
$C^{\prime}_{N}:=C_{N}(a,b,e,f)$ unless $n$ is even and $N=N_{i}$ where $1\leq
i<j\leq k$ and $N_{j}=2$, in which case $C^{\prime}_{N}:=C_{N}(b,a,f,e)$. The
equivalence of these various factorizations is trivial on noting that all
$C_{n}(a,b,e,f)$ with $n\geq 3$ lie in the commutative unital subring of $B$
generated by $e$ and $ba$, with $aC_{n}(a,b,e,f)=C_{n}(b,a,f,e)a$.

### 4.19.

We now take notation as in 4.1. Thus, $P_{W,S}$ is the path algebra over
$\mathbb{Z}$ of $\Gamma_{W,S}$, $I_{W,S}$ is an ideal of $P_{W,S}$ and
$R_{W,S}$ is the quotient ring $R_{W,S}:=P_{W,S}/I_{W,S}$.

By definition, $I_{W,S}$ is the two-sided ideal of $P_{W,S}$ generated by
elements $C_{[sr]}=C_{m_{r,s}}([srs])=C_{m_{r,s}}([rs],[sr],[s],[r])$ of 4.17,
for $(r,s)\in S\times S$ such that $2<m_{r,s}<\infty$. Let
$\widetilde{I}_{W,S}$ denote the two-sided ideal of $P_{W,S}$ generated by the
elements $c_{m_{r,s}}([rs],[sr],[s],[r])$ for $(r,s)\in S\times S$ such that
$2<m_{r,s}<\infty$. From 4.17, we have $\widetilde{I}_{W,S}\subseteq I_{W,S}$.
Define the quotient ring $\widetilde{R}_{W,S}:=P_{W,S}/\widetilde{I}_{W,S}$ of
$P_{W,S}$. The above gives a canonical surjective ring homomorphism
$\widetilde{R}_{W,S}\to R_{W,S}$.

###  4.20. Parameterizing Bases of $R_{W,S}$ and $\hat{R}_{W,S}$.

Recall from Tits solution to the word problem for $(W,S)$ that a sequence
$(s_{1},\ldots,s_{n})$ in $S$ is the unique reduced expression of the
corresponding product $s_{1}\cdots s_{n}\in W$ if and only if it contains no
consecutive subsequence $(s_{i},\ldots,s_{j})$, where $1\leq i<j\leq n$, of
length $j-i+1$ which either is of the form $(s,s)$ for some $s\in S$ or is an
alternating sequence $(s,r,s,\ldots)$ where $r,s\in S$ are distinct with
$m_{r,s}=j-i+1$. Let $W_{1}$ be the set of all such sequences with $n\geq 1$.

Let $\varphi\colon\mathbb{N}_{\geq 1}\to\mathbb{N}$ denote the Euler totient
function (that is, $\varphi(n)=|(\mathbb{Z}/n\mathbb{Z})^{*}|$ where
$(\mathbb{Z}/n\mathbb{Z})^{*}$ indicates the unit group. Note that if $n\geq
3$, then $\frac{\varphi(n)}{2}$ is the degree of $C_{n}(t)\in\mathbb{Z}[t]$.

Let $W_{1}^{\prime}$ be the set of all sequences $(s_{1},\ldots,s_{n})$ in
$S$, with $n\geq 1$, which contain no consecutive subsequence
$(s_{i},\ldots,s_{j})$ which is of the form $(s,r,s,\ldots)$ where $r,s\in S$,
$m_{r,s}<\infty$ and $j-i+1=1+\varphi(m_{r,s})$. Note that
$W^{\prime}_{1}\subseteq W_{1}$.

It is known from [13] that the elements $[s_{1}\ldots s_{n}]$ of
$\widetilde{R}_{W,S}$ for $(s_{1},\ldots,s_{n})\in W_{1}$ are pairwise
distinct and form a $\mathbb{Z}$-module basis of $\widetilde{R}_{W,S}$222A
different basis for $\widetilde{R}$, with some very favorable properties, is
also considered in [13].

######  Theorem 4.21.

  1. (1)

The elements $[s_{1}\ldots s_{n}]$ of $\hat{R}_{W,S}$ for
$(s_{1},\ldots,s_{n})\in W^{\prime}_{1}$ are pairwise distinct and form a
basis of $\hat{R}_{W,S}$ as vector space over $\mathbb{Q}$.

  2. (2)

The elements $[s_{1}\ldots s_{n}]$ of $R_{W,S}$ for $(s_{1},\ldots,s_{n})\in
W^{\prime}_{1}$ are pairwise distinct and form a $\mathbb{Z}$-module basis of
$R_{W,S}$.

###### Proof.

Part (1) follows from Lemmas 4.12 and Part (2) follows from Corollary 4.14. ∎

######  Example 4.22.

(1) Suppose that $m_{r,s}\in\\{2,3,4,6\\}$ for all $r\neq s$ in $S$. Since
$\varphi(m)=2$ for $m=3,4,6$, the sequences $(s_{1},\ldots,s_{n})\in
W_{1}^{\prime}$ are precisely the sequences of successive vertices of non-
backtracking paths in $\Gamma_{W,S}$. If, further, the Coxeter graph of
$(W,S)$ is a tree, then for any $r,s\in S$, there is a unique non-backtracking
path in $\Gamma_{W,S}$ from $r$ to $s$ and it follows that

$\dim_{\mathbb{Q}}([r]\hat{R}_{W,S}[s])=\mathrm{rank}_{\mathbb{Z}}([r]R_{W,S}[s])=1,$

and so

$\dim_{\mathbb{Q}}(\hat{R}_{W,S})=\mathrm{rank}_{\mathbb{Z}}(R_{W,S})=|S|^{2}.$

In particular, these facts all hold if $(W,S)$ is an irreducible Weyl group or
an irreducible affine Weyl group which is not of type $\widetilde{A}_{n}$ for
any $n\in\mathbb{N}_{\geq 1}$.

(2) Suppose that for each $r\neq s$ in $S$, $m_{r,s}$ is either $\infty$ or a
prime integer. Then $I_{W,S}=\widetilde{I}_{W,S}$ and hence
$R_{W,S}=\widetilde{R}_{W,S}$.

##  5\. Reflection Representations

In this section, we describe results of Dyer on the relation between standard
reflection representations of Coxeter systems in real vector spaces and lax
and strict reflection representations over non-commutative rings.

### 5.1.

Let $A$ be a commutative ring. Let $B$ be an $A$-algebra, and $M$ a left
$A$-algebra module for $B$, as defined in Section 1. For a group $G$, a
representation of $G$ on the $B$-module $M$ is by definition a group
homomorphism $\theta\colon G\to\mathrm{Aut}_{B}(M)$. We usually write
$gm:=(\theta(g))(m)$ and say simply that $G$ acts $B$-linearly on $M$ (leaving
tacit the condition that the action is also $A$-linear). Similar conventions
apply to right $G$ actions and right modules.

For the $A$-algebra $B$, the left (resp., right) regular $B$-module is $B$ as
$A$-module, with $B$-action by left (resp., right) multiplication. If $B$ is
unital and $M$ is its left regular module, then $\mathrm{End}_{B}(M)\cong
B^{\mathrm{op}}$, where $B^{\mathrm{op}}$ is the opposite algebra, and
$\mathrm{Aut}_{B}(M)$ identifies with the group of units of $B^{\mathrm{op}}$.
If $B$ is commutative unital, we often write
$\mathrm{GL}_{B}(M):=\mathrm{Aut}_{B}(M)$.

### 5.2.

Let $M$ be a left $B$-module, ${M}^{\bigvee}$ be a right $B$-module and
$\langle\,-,-\,\rangle\colon M\times{M}^{\bigvee}\to B$ be a $B$-bilinear map.
That is, for all $r,r^{\prime}\in B$, we require
$\langle\,rm+r^{\prime}m^{\prime},m^{\prime\prime}\,\rangle=r\langle\,m,m^{\prime\prime}\,\rangle+r^{\prime}\langle\,m^{\prime},m^{\prime\prime}\,\rangle$
for $m,m^{\prime}\in M$ and $m^{\prime\prime}\in{M}^{\bigvee}$, and
$\langle\,m^{\prime\prime},mr+m^{\prime}r^{\prime}\,\rangle=\langle\,m^{\prime\prime},m\,\rangle
r+\langle\,m^{\prime\prime},m^{\prime}\,\rangle r^{\prime}$ for
$m^{\prime\prime}\in M$ and $m,m^{\prime}\in{M}^{\bigvee}$. We also require
these conditions to hold with $r$ and $r^{\prime}$ in the coefficient ring,
$A$, of $B$. Suppose given families $(\alpha_{s})_{s\in S}$ in $M$,
$({\alpha}^{\bigvee}_{s})_{s\in S}$ in ${M}^{\bigvee}$ and $(e_{s})_{s\in S}$
in $B$ such that for all $s\in S$ $e_{s}^{2}=e_{s}$,
$e_{s}\alpha_{s}=\alpha_{s}$,
${\alpha}^{\bigvee}_{s}e_{s}={\alpha}^{\bigvee}_{s}$ and
$\langle\,\alpha_{s},{\alpha}^{\bigvee}_{s}\,\rangle=2e_{s}$. Define the
$S\times S$ matrix $A=(a_{r,s})_{r,s\in S}$ with entries in $B$ by
$a_{r,s}:=\langle\,\alpha_{r},{\alpha}^{\bigvee}_{s}\,\rangle\in e_{r}Be_{s}$.

Define $B$-module endomorphisms $\phi_{s}\in\mathrm{End}_{B}(M)$ of $M$ for
$s\in S$ by
$\phi_{s}(m)=m-\langle\,m,{\alpha}^{\bigvee}_{s}\,\rangle\alpha_{s}$ for all
$m\in M$. It is easy to check that $\phi_{s}^{2}=\mathrm{Id}_{M}$ for all
$s\in S$, so $\phi_{s}\in\mathrm{Aut}_{B}(M)$.

The following is a main result in Dyer [13]:

######  Proposition 5.3.

(Dyer [13]) Suppose notation is as above and that for all $(r,s)\in S\times S$
with $r\neq s$ and $m_{r,s}\neq\infty$, one has
$c_{m_{r,s}}(a_{s,r},a_{r,s},e_{r},e_{s})=0$. Then there is a left $B$-linear
left $W$-action on $M$ determined by
$sm=m-\langle\,m,{\alpha}^{\bigvee}_{s}\,\rangle\alpha_{s}$ for all $m\in M$
and $s\in S$ and a right $B$-linear left $W$-action on ${M}^{\bigvee}$
determined by
$sm^{\prime}=m^{\prime}-{\alpha}^{\bigvee}_{s}\langle\,\alpha_{s},m^{\prime}\,\rangle$
for all $m^{\prime}\in{M}^{\bigvee}$. These satisfy
$\langle\,wm,wm^{\prime}\,\rangle=\langle\,m,m^{\prime}\,\rangle$ for all
$m\in M$, $m^{\prime}\in{M}^{\bigvee}$ and $w\in W$.

### 5.4.

We call the matrix $(a_{r,s})_{r,s\in S}$ a non-commutative generalized Cartan
matrix (NCM) when the assumptions in 5.3 hold. In that case, the left
$W$-action on $M$ gives a representation $\phi\colon W\to\mathrm{Aut}_{B}(M)$,
and similarly for ${M}^{\bigvee}$. We call these lax reflection
representations of $(W,S)$ on $M$ and ${M}^{\bigvee}$. If, more strongly, one
has $C_{m_{r,s}}(a_{s,r},a_{r,s},e_{r},e_{s})=0$ for all $(r,s)\in S\times S$
with $r\neq s$ and $m_{r,s}\neq\infty$, we say these representations are
strict reflection representations and say that $A$ is a strict NCM.

In either case, the subset $\Phi:=\\{w\alpha_{s}\mid w\in W,s\in S\\}$ of $M$
is called the (corresponding) root system of $(W,S)$ in $M$, and
${\Phi}^{\bigvee}:=\\{w{\alpha}^{\bigvee}_{s}\mid w\in W,s\in S\\}$ is called
the coroot system of $(W,S)$ in ${M}^{\bigvee}$. For some basic properties of
lax reflection representations and their root systems, see [13].

######  Example 5.5.

Suppose above that $B=\mathbb{R}$, $M=V$ and ${M}^{\bigvee}={V}^{\bigvee}$ are
$\mathbb{R}$-vector spaces (unital $B$-modules) and $e_{r}=1_{B}$ for all
$r\in S$. Let $a_{r,s}:=\langle\,\alpha_{r},{\alpha}^{\bigvee}_{s}\,\rangle$.
The proposition produces lax reflection representations
$W\to\mathrm{GL}_{\mathbb{R}}(V)$ and
$W\to\mathrm{GL}_{\mathbb{R}}({V}^{\bigvee})$ provided that $a_{s,s}=2$ for
all $s\in S$, and, for all $r\neq s\in S$, $a_{r,s}=0$ if $m_{r,s}=2$ and
$a_{r,s}a_{s,r}=4\cos^{2}\frac{k_{r,s}\pi}{m_{r,s}}$ for some $1\leq
k_{r,s}\leq m_{r,s}/2$ if $3\leq m_{r,s}<\infty$. These reflection
representations are strict if and only if $k_{r,s}$ is relatively prime to
$m_{r,s}$ whenever $3\leq m_{r,s}<\infty$. We call the matrix
$A:=(a_{r,s})_{r,s}$ a strict (resp., lax) real reflection matrix, abbreviated
SRRM (resp., LRRM) for $(W,S)$ if it arises in ths way from a strict (resp.,
lax) reflection representation.

######  Remark.

(1) Representations as in the preceding example are mostly studied under
conditions which ensure they are strict and that the representation and its
root system have standard (convexity and positivity) properties. Essentially
the most general such conditions are the following:

  1. (1)

$a_{r,r}=2$ for all $r\in S$.

  2. (2)

$a_{r,s}\leq 0$ for all $r\neq s$ in $S$.

  3. (3)

$a_{r,s}=0$ if $r,s\in S$ and $m_{r,s}=2$.

  4. (4)

$a_{r,s}a_{s,r}=4\cos^{2}\frac{\pi}{m_{r,s}}$ if $r,s\in S$ and $3\leq
m_{r,s}<\infty$.

  5. (5)

$a_{r,s}a_{s,r}\geq 4$ if $r,s\in S$ and $m_{r,s}=\infty$.

  6. (6)

$\Pi$ and ${\Pi}^{\bigvee}$ are positively independent.

Here, a family $(\beta_{i})_{i\in I}$ of elements in a real vector space is
said to be positively independent if $\sum_{i\in I}a_{i}\beta_{i}=0$ for non-
negative real scalars $a_{i}$, almost all zero, implies $a_{i}=0$ for all
$i\in I$. In particular, linearly independent vectors are positively
independent. When (1)–(5) hold, we say that $A:=(a_{r,s})_{r,s\in S}$ is a
(possibly) non-crystallographic generalized Cartan matrix (NGCM) for $(W,S)$.
The finite NGCMs with integral entries are essentially the Generalized Cartan
Matrices (GCMs) appearing in the study of Kac-Moody Lie algebras [20] and
elsewhere. Note that NGCMs are SRRMs. See [15], [14], [10] and [13] for
further details on reflection representations attached to NGCMs.

(2) In some very special cases including finite irreducible Coxeter systems,
real reflection representations attached to SRRMs may be viewed as “Galois
twists” of those attached to NGCMs, but this is not the case in general (a
precise discussion of this would involve reflection representations defined
over subfields of $\mathbb{R}$).

(3) For any Coxeter system $(W^{\prime},S)$ for which there is a surjective
group homomorphism $W\to W^{\prime}$ which is the identity on $S$, a SRRM for
$(W^{\prime},S^{\prime})$ is a LRRM for $(W,S)$.

The following proposition summarizes part of the above discussion which is
used subsequently.

######  Proposition 5.6.

Let $A=(a_{r,s})_{r,s\in S}$ be a LRRM for $(W,S)$ and $V=V_{A}$ ,
${V}^{\bigvee}={V}^{\bigvee}_{A}$ be real vector spaces with bases
$\\{\alpha_{s}\mid s\in S\\}$ and $\\{{\alpha}^{\bigvee}_{s}\mid s\in S\\}$
respectively. Then there are $\mathbb{R}$-linear lax reflection
representations of $W$ on $V$ and ${V}^{\bigvee}$ such that for $s,r\in S$,
one has $s\alpha_{r}=\alpha_{r}-a_{r,s}\alpha_{s}$ and
$s{\alpha}^{\bigvee}_{r}={\alpha}^{\bigvee}_{r}-a_{s,r}{\alpha}^{\bigvee}_{s}$.
We call $(V_{A},{V}^{\bigvee}_{A})$ the standard paired (lax) real reflection
representations of $(W,S)$ with LRRM $A$. If $A$ is a SRRM, these
representations are strict.

### 5.7.

Let $A$ be a LRRM for $(W,S)$ and let
$(V,{V}^{\bigvee})=(V_{A},{V}^{\bigvee}_{A})$ be the standard paired real
reflection representations of $(W,S)$ with LRRM $A$. As constructed above, $V$
and ${V}^{\bigvee}$ have left $W$-actions, but we shall find it convenient to
regard $W$ as instead acting on the right of $V$, by defining $vw:=w^{-1}v$
for $v\in V$ and $w\in W$. We still regard the $W$-action on ${V}^{\bigvee}$
as a left action. Then ${V}^{\bigvee}\otimes_{\mathbb{R}}V$ has natural
commuting left and right $W$-actions, defined by

$w(v^{\prime}\otimes v)=(wv^{\prime})\otimes v,\qquad(v^{\prime}\otimes
v)w=v^{\prime}\otimes(vw),\qquad\text{\rm if $v^{\prime}\in{V}^{\bigvee}$,
$v\in V$, $w\in W$.}$

One has

$s({\alpha}^{\bigvee}_{r}\otimes\alpha_{t})={\alpha}^{\bigvee}_{r}\otimes\alpha_{t}-a_{s,r}{\alpha}^{\bigvee}_{s}\otimes\alpha_{t},\qquad({\alpha}^{\bigvee}_{r}\otimes\alpha_{t})s={\alpha}^{\bigvee}_{r}\otimes\alpha_{t}-a_{t,s}{\alpha}^{\bigvee}_{r}\otimes\alpha_{s},\qquad\text{\rm
for $r,s,t\in S$.}$

### 5.8.

Let $B:=\mathrm{Mat}^{\prime}_{S\times S}(\mathbb{R})$ denote the
$\mathbb{R}$-algebra of real $S\times S$-matrices with only finitely many non-
zero entries. Note that $B$ has a $\mathbb{R}$-basis $(e_{r,s})_{r,s\in S}$
where $e_{r,s}$ is the matrix unit (the matrix with
$(r^{\prime},s^{\prime})$-entry.$\delta_{r,r^{\prime}}\delta_{s,s^{\prime}}$
where $\delta_{j,k}$ denotes the Kronecker delta in $\mathbb{R}$).

We shall also regard $B$ as a $(B,B)$-bimodule in the natural way. Identify
${V}^{\bigvee}\otimes_{\mathbb{R}}V=B$ as $\mathbb{R}$-vector space so that
${\alpha}^{\bigvee}_{r}\otimes\alpha_{s}=e_{r,s}$. This endows $B$ with
structure of $\mathbb{R}$-algebra and with left and right $W$-actions which
commute, such that the left (resp., right) $W$-action is right (resp., left)
$B$-linear.

### 5.9.

Consider any ring ${R^{\prime}}:=P_{W,S}/I$ where $I$ is a two-sided ideal of
$P_{W,S}$ such that $\widetilde{I}_{W,S}\subseteq I$. The two cases of
greatest interest are (1) $I=I_{W,S}$ and ${R^{\prime}}=R_{W,S}$, and (2)
$I=\widetilde{I}_{W,S}$ and ${R^{\prime}}=\widetilde{R}_{W,S}$. For any
element $x$ of $P_{W,S}$, we denote its image in ${R^{\prime}}$ under the
canonical surjection $P_{W,S}\to{R^{\prime}}$ still by $x$. In particular,
this defines $[x_{1}\ldots x_{n}]\in{R^{\prime}}$ if $n>0$ and
$x_{1},\ldots,x_{n}\in S$ are the successive vertices of a path in
$\Gamma_{W,S}$. We also set $[x_{1}\ldots x_{n}]:=0_{{R^{\prime}}}$ if $n>0$
and $x_{1},\ldots,x_{n}\in S$ are not the successive vertices of a path in
$\Gamma_{W,S}$ (that is, if for some $1\leq i\leq n-1$, $m_{x_{i},x_{i+1}}\leq
2$). Note that the elements $[s]$ for $s\in S$ are pairwise orthogonal
idempotents in ${R^{\prime}}$ such that ${R^{\prime}}=\bigoplus_{r,s\in
S}[r]{R^{\prime}}[s]$ as abelian group.

######  Proposition 5.10.

Define the quotient ring ${R^{\prime}}=P_{W,S}/I$ where
$I\supseteq\widetilde{I}_{W,S}$ is a two-sided ideal of $P_{W,S}$. Let $M$
(resp., ${M}^{\bigvee}$) denote the left regular ${R^{\prime}}$-module
${R^{\prime}}$.

  1. (1)

There is a (unique) left $R^{\prime}$-linear lax refelction representation of
$W$ on $M$ such that

$s[r]=\begin{cases}-[s],&\text{\rm if $r=s$}\\\ [r]+[rs],&\text{\rm if $r\neq
s$.}\end{cases}\qquad\text{\rm for $r,s\in S$.}$

and a (unique) right $R^{\prime}$-linear lax reflection representation of $W$
on ${M}^{\bigvee}$ such that

$s[r]=\begin{cases}-[s],&\text{\rm if $r=s$}\\\ [r]+[sr],&\text{\rm if $r\neq
s$.}\end{cases}\qquad\text{\rm for $r,s\in S$.}$

  2. (2)

Base change $-\otimes_{\mathbb{Z}}\mathbb{R}$ gives a (possibly non-unital)
$\mathbb{R}$-algebra $R^{\prime}\otimes_{\mathbb{Z}}\mathbb{R}$ and left
(resp., right) $R^{\prime}\otimes_{\mathbb{Z}}\mathbb{R}$-linear lax left
reflection actions of $W$ on the left and right regular
$R^{\prime}\otimes_{\mathbb{Z}}\mathbb{R}$-modules
$M\otimes_{\mathbb{Z}}\mathbb{R}$ and
${M}^{\bigvee}\otimes_{\mathbb{Z}}\mathbb{R}$ respectively.

  3. (3)

If $I\supseteq\widetilde{I}_{W,S}$, then the lax reflection representations in
(1)–(2) are strict.

###### Proof.

We prove (1). Define families $(e_{s})_{s\in S}$ in ${R^{\prime}}$,
$(\alpha_{s})_{s\in S}$ in $M$ and $({\alpha}^{\bigvee}_{s})_{s\in S}$ in
${M}^{\bigvee}$ by setting $\alpha_{s}={\alpha}^{\bigvee}_{s}:=[s]$ for all
$s\in S$. A ${R^{\prime}}$-bilinear map $\langle\,-,-\,\rangle\colon
M\times{M}^{\bigvee}\to{R^{\prime}}$ is uniquely determined by the values
$\langle\,[r],[s]\,\rangle$ for $r,s\in S$, which may be arbitrarily assigned
subject only to the conditions that $\langle\,[r],[s]\,\rangle\in
e_{r}{R^{\prime}}e_{s}$ for all $r,s$. Hence there is a unique
${R^{\prime}}$-bilinear map $\langle\,-,-\,\rangle\colon
M\times{M}^{\bigvee}\to{R^{\prime}}$ such that

$\langle\,\alpha_{r},{\alpha}^{\bigvee}_{s}\,\rangle=\begin{cases}2[r],&\text{\rm
if $r=s$}\\\ -[rs],&\text{\rm if $r\neq s$.}\end{cases}$

Note that $\langle\,\alpha_{r},{\alpha}^{\bigvee}_{s}\,\rangle=0$ if
$m_{rs}=2$ by our conventions. The conditions of Proposition 5.3 are
satisfied, by definition of $\widetilde{I}_{W,S}$ and the assumption that
$\widetilde{I}_{W,S}\subseteq I$. Hence the proposition gives lax
representations as required in (1). Part (2) follows easily by base change and
(3) is immediate from the definitions. ∎

### 5.11.

Regard the left $R^{\prime}$-linear $W$-action just defined on $M$ as a right
action by setting $mw:=w^{-1}m$ for $w\in W$ and $m\in M$, and henceforward
write $M$ and ${M}^{\bigvee}$ just as $R^{\prime}$. Then $R^{\prime}$ admits a
right $R^{\prime}$-linear left reflection action of $W$, and a left
$R^{\prime}$-linear right reflection action of $W$. Moreover, these actions
commute, as one can see by direct computation or from the following lemma.

######  Lemma 5.12.

Let $C$ be any idempotent ring (i.e. $C^{2}=C$). Let $\theta\colon C\to C$
(resp., $\tau\colon C\to C$) be an endomorphism of the left (right) regular
$C$ module. Then $\tau$ and $\theta$ commute in
$\mathrm{End}_{\mathbb{Z}}(C)$.

###### Proof.

Let $c\in C$. Since $c\in C=C^{2}$, we may write $c=\sum_{i=1}^{n}c_{i}d_{i}$
for some $n\in\mathbb{N}$ and $c_{i},d_{i}\in C$. Then

$\tau(\theta(c))=\tau(\theta(\sum_{i=1}^{n}c_{i}d_{i}))=\tau(\sum_{i=1}^{n}c_{i}\theta(d_{i}))=\sum_{i=1}^{n}\tau(c_{i})\theta(d_{i})=\theta(\sum_{i=1}^{n}\tau(c_{i})d_{i})=\theta(\tau(\sum_{i=1}^{n}c_{i}d_{i})=\theta(\tau(c)).\qed$

### 5.13.

Let $A:=(a_{r,s})_{r,s\in S}$ be an $S\times S$-indexed family in
$\mathbb{R}$. Then there is a unique ring homomorphism
$\theta^{\prime}_{A}\colon P_{W,S}\to B$ such that for any path in
$\Gamma_{W,S}$ with successive vertices $s_{1},\ldots,s_{n}$, one has
$\theta^{\prime}_{A}([s_{1}\cdots s_{n}])=(-1)^{n-1}a_{s_{1},s_{2}}\cdots
a_{s_{n-1},s_{n}}e_{s_{1},s_{n}}$ (where here and below,
$a_{s_{1},s_{2}}\cdots a_{s_{n-1},s_{n}}=1_{\mathbb{R}}$ by convention if
$n=1$). Note that $\theta^{\prime}_{A}$ doesn’t depend on the values $a_{r,s}$
with $m_{r,s}\leq 2$.

######  Proposition 5.14.

Let $A$ and $\theta^{\prime}_{A}$ be as above.

  1. (1)

If $A$ is a LRRM for $(W,S)$, then $\theta^{\prime}_{A}$ factors as
${P}_{W,S}\twoheadrightarrow\widetilde{R}_{W,S}\xrightarrow{\theta_{A,\widetilde{R}}}B$
for a unique ring homomoprhism $\theta_{A,\widetilde{R}}$.

  2. (2)

If $A$ is a SRRM for $(W,S)$, then $\theta_{A,\widetilde{R}}$ factors as
$\widetilde{R}_{W,S}\twoheadrightarrow R_{W,S}\xrightarrow{\theta_{A,R}}B$ for
a unique ring homomoprhism $\theta_{A,R}$.

  3. (3)

Suppose $A$ is a LRRM. Let $(S_{i})_{i\in I}$ be the equivalence classes for
the finest equivalence relation $\sim$ on $S$ such that $r\sim s$ if
$a_{r,s}\neq 0$. Then
$\mbox{Span}(\mathrm{Img}(\theta_{A,\widetilde{R}}))=\bigoplus_{i\in
I}\mathrm{Mat}^{\prime}_{S_{i}\times S_{i}}(\mathbb{R})\subseteq B$.

  4. (4)

If $A$ is a SRRM, then
$\mbox{Span}(\mathrm{Img}(\theta_{A,{R^{\prime}}}))=\bigoplus_{i\in
I}\mathrm{Mat}^{\prime}_{S_{i}\times S_{i}}(\mathbb{R})$ where
$(W_{i},S_{i})_{i\in I}$ are the irreducible compoenents of the Coxeter system
$(W,S)$.

###### Proof.

We prove (1). Let $A$ be a LRRM. Since
$\widetilde{R}_{W,S}=P_{W,S}/\widetilde{I}_{W,S}$, it will suffice to show
that $\ker\theta^{\prime}_{A}\supseteq\widetilde{I}_{W,S}$. Recall that
$\widetilde{I}_{W,S}$ is generated by elements $c_{m}([rs],[sr],[s],[r])\in
P_{W,S}$ where $r,s\in S$ with $3\leq m:=m_{r,s}<\infty$. Let $x:=e_{r,r}\in
B$ if $m$ is odd and $x:=e_{s,r}\in B$ if $m$ is even. It is easily seen that

$\begin{split}\theta^{\prime}_{A,}(c_{m,P}([rs],[sr],[s],[r])&=c_{m,B}(-a_{r,s}e_{r,s},-a_{sr}e_{s,r},e_{s,s},e_{r,r})=c_{m,\mathbb{R}}(-a_{r,s},-a_{sr},1_{\mathbb{R}},{1}_{\mathbb{R}})x\end{split}.$

Since $A$ is a LRRM, we may write $a_{r,s}a_{s,r}=4\cos^{2}\frac{k\pi}{n}$
where $k\in\mathbb{N}$ and $n\in\mathbb{N}_{\geq 2}$ are relatively prime,
$n|m$ and $1\leq k<m$. We have
$C_{n,\mathbb{R}}(-a_{r,s},-a_{sr},1_{\mathbb{R}},{1}_{\mathbb{R}})=C_{n}\left(4\cos^{2}\frac{k\pi}{n}\right)=0$.
So $c_{m,-\mathbb{R}}(-a_{r,s},-a_{sr},1_{\mathbb{R}},{1}_{\mathbb{R}})=0$ by
4.18 and hence $\theta^{\prime}_{A,}(c_{m,P}([rs],[sr],[s],[r])=0$ as
required.

The proof of (2) is similar but simpler. Let $A$ be a SRRM. Since
$R_{W,S}=P_{W,S}/I_{W,S}$, it will suffice to show that
$\ker\theta^{\prime}_{A}\supseteq I_{W,S}$. Recall that $I_{W,S}$ is generated
by elements $C_{m}([rs],[sr],[s],[r])\in P_{W,S}$ where $r,s\in S$ with $3\leq
m:=m_{r,s}<\infty$. Since $\varphi(m)$ is even, we have

$\begin{split}\theta^{\prime}_{A,}(C_{m,P}([rs],[sr],[s],[r])&=C_{m,B}(-a_{r,s}e_{r,s},-a_{sr}e_{s,r},e_{s,s},e_{r,r})=c_{m,\mathbb{R}}(-a_{r,s},-a_{sr},1_{\mathbb{R}},{1}_{\mathbb{R}})e_{r,r}\end{split}.$

Since $A$ is a SRRM, we may write $a_{r,s}a_{s,r}=4\cos^{2}\frac{k\pi}{m}$
where the integer $k$ is relatively prime to $m$ and $1\leq k<m$. We then have
$C_{m,-\mathbb{R}}(-a_{r,s},-a_{sr},1_{\mathbb{R}},{1}_{\mathbb{R}})=h_{m}\left(4\cos^{2}\frac{k\pi}{m}\right)=0$
and hence $\theta^{\prime}_{A,}(c_{m,P}([rs],[sr],[s],[r])=0$ as required.

In (3)–(4), we regard $\mathrm{Mat}^{\prime}_{J\times J}(\mathbb{R})$, for
$J\subseteq S$, as a $\mathbb{R}$-subalgebra of $B$ in the natural way
(identifying its matrix units $e_{j,k}$, for $j,k\in J$, with $e_{j,k}\in B$).
To prove (3), observe that by (1),
$\mathrm{Img}(\theta_{A,\widetilde{R}})=\mathrm{Img}(\theta^{\prime}_{A})$.
This image has the same $\mathbb{R}$-linear span as the set of matrix units
$e_{r,s}$ such that there is some path in $\Gamma_{W,S}$ with successive
vertices $r=s_{1},\ldots,s_{n}=s$ such that $a_{s_{i},s_{i+1}}\neq 0$ for
$i=1,\ldots,n-1$. Since $A$ is a LRRM, for any $t\neq u\in S$, we have
$a_{t,u}\neq 0\iff a_{u,t}\neq 0\implies m_{t,u}\neq 2$, and (3) follows.

Part (4) follows from (3) since if $A$ is a SRRM, then
$\mathrm{Img}(\theta_{A,R})=\mathrm{Img}(\theta_{A,\widetilde{R}})$ and for
distinct $t,u\in S$, we have $a_{t,u}\neq 0\iff a_{u,t}\neq 0\iff m_{t,u}\neq
2$. ∎

### 5.15.

By base change $-\otimes_{\mathbb{Z}}\mathbb{R}$, we obtain from the ring
$\widetilde{R}_{W,S}$ and ring homomorphism
$\theta_{A,\widetilde{R}}\colon\widetilde{R}_{W,S}\to B$ into the
$\mathbb{R}$-algebra $B$, a $\mathbb{R}$-algebra
$\widetilde{{R}}^{\prime}:=\widetilde{R}_{W,S}\otimes_{\mathbb{Z}}\mathbb{R}$
and $\mathbb{R}$-algebra homomorphism
$\theta_{A,\widetilde{R}^{\prime}}\colon\widetilde{R}^{\prime}\to B$.
Similarly, define the $\mathbb{R}$-algebra
${{R}}^{\prime}:={R_{W,S}}\otimes_{\mathbb{Z}}\mathbb{R}$ and
$\mathbb{R}$-algebra homomorphism $\theta_{A,R^{\prime}}\colon R^{\prime}\to
B$. We have natural inclusion homomorphisms
$\widetilde{R}_{W,S}\to\widetilde{R}^{\prime}$ and $R_{W,S}\to R^{\prime}$ of
rings, given in each case by $r\to r\otimes_{\mathbb{Z}}1$, since
$\widetilde{R}_{W,S}$ and $R_{W,S}$ are free as $\mathbb{Z}$-modules.

Note that if $C$ denotes one of the rings (or $\mathbb{R}$-algebras)
$\widetilde{R}_{W,S}$, $\widetilde{R}^{\prime}$, $R_{W,S}$, $R^{\prime}$ or
$B$, we have defined on $C$ commuting left and right $W$ actions, such that
the left (resp., right) $W$-action is right (resp., left) $C$-linear. We
regard each such $C$ as a $W\times W^{\mathrm{op}}$-set in the natural way.
The two inclusion homomorphisms
$\widetilde{R}_{W,S}\hookrightarrow\widetilde{R}^{\prime}$ and
$R_{W,S}\hookrightarrow R^{\prime}$, and the canonical surjective homorphisms
$\widetilde{R}_{W,S}\twoheadrightarrow R$ and
$\widetilde{R}^{\prime}_{W,S}\twoheadrightarrow R^{\prime}$ are obviously
$W\times W^{\mathrm{op}}$-equivariant.

######  Theorem 5.16.

Let notations and assumptions be as above.

  1. (1)

If $A$ is a LRRM for $(W,S)$, the ring homomorphism
$\theta_{A,{\widetilde{R}}}\colon\widetilde{R}_{W,S}\to B$ and
$\mathbb{R}$-algebra homomorphism
$\theta_{A,{\widetilde{R}}^{\prime}}\colon\widetilde{R}^{\prime}\to B$ are
$W\times W^{\mathrm{op}}$-equivariant.

  2. (2)

If $A$ is a SRRM for $(W,S)$, the ring homomorphism $\theta_{A,{R}}\colon
R_{W,S}\to B$ and $\mathbb{R}$-algebra homomorphism
$\theta_{A,{R}^{\prime}}\colon R^{\prime}\to B$ are $W\times
W^{\mathrm{op}}$-equivariant.

  3. (3)

Suppose that $(W,S)$ is irreducible and $A$ is a SRRM. Then
$\theta_{A,R^{\prime}}$ is surjective.

  4. (4)

The $W\times W^{\mathrm{op}}$ actions on $R_{W,S}$, $R^{\prime}$
$\widetilde{R}_{W,S}$, and $\widetilde{R}^{\prime}$ are all faithful.

###### Proof.

For all the various left and right $W$-sets $X$ involved in this proof, we
write the various action maps $W\times X\to X$ as $(w,x)\mapsto wx$ and
$X\times W\to X$ as $(x,w)\mapsto xw$ for notational simplicity. To prove (1),
it will suffice to prove the assertion for $\theta_{A,\widetilde{R}}$, as that
for $\theta_{A,\widetilde{R}^{\prime}}$ then follows by base change. We show
first that for $r,s\in S$, we have
$\theta_{A,{\widetilde{R}}}([r]s)=\left(\theta_{A,{\widetilde{R}}}([r])\right)s$.
If $s=r$, then

$\theta_{A,{\widetilde{R}}}([r]s)=\theta_{A,{\widetilde{R}}}(-[r])=-e_{r,r}=e_{r,r}r=\theta_{A,{\widetilde{R}}}([r])s$

while if $s\neq r$, then

$\theta_{A,{\widetilde{R}}}([r]s)=\theta_{A,{\widetilde{R}}}([r]+[rs])=e_{r,r}-a_{r,s}e_{r,s}=e_{r,r}s=\theta_{A,{\widetilde{R}}}([r])s,$

as required. Now for any $x\in R$ and $r,s\in S$ we have

$\begin{split}\theta_{A,{\widetilde{R}}}((x\cdot[r])s)&=\theta_{A,{\widetilde{R}}}(x\cdot([r]s))=\theta_{A,{\widetilde{R}}}(x)\cdot\theta_{A,{\widetilde{R}}}([r]s)=\theta_{A,{\widetilde{R}}}(x)\cdot(\theta_{A,{\widetilde{R}}}([r])s)\\\
&=\left(\theta_{A,{\widetilde{R}}}(x)\cdot\theta_{A,{\widetilde{R}}}([r])\right)s=\left(\theta_{A,{\widetilde{R}}}(x\cdot[r])\right)s\end{split}$

Since $x=\sum_{r\in R}x\cdot[r]$ where $x\cdot[r]=0$ for almost all $r\in S$,
this impies

$\begin{split}\theta_{A,{\widetilde{R}}}(xs)&=\theta_{A,{\widetilde{R}}}\left(\sum_{r\in
S}\left(x\cdot[r]\right)s\right)=\sum_{r\in
S}\theta_{A,{\widetilde{R}}}\left((x\cdot[r])s\right)\\\ &=\sum_{r\in
S}\theta_{A,{\widetilde{R}}}(x\cdot[r])s=\theta_{A,{\widetilde{R}}}\left(\sum_{r\in
S}(x\cdot[r])\right)s=\theta_{A,{\widetilde{R}}}(x)s\end{split}$

Finally, by induction on the length of $w\in W$, it follows
$\theta_{A,{\widetilde{R}}}(xw)=\theta_{A,{\widetilde{R}}}(x)w$ for all
$x\in\widetilde{R}_{W,S}$ and $w\in W$. Similarly or by symmetry,
$\theta_{A,{\widetilde{R}}}(wx)=w\theta_{A,{\widetilde{R}}}(x)$ for all
$x\in\widetilde{R}_{W,S}$ and $w\in W$completing the proof of (1).

Part (2) follows from (1) since if $A$ is a SRRM, we have the factorization of
$\theta_{A,\widetilde{R}}$ in Proposition 5.14 where the canonical surjection
$\widetilde{R}_{W,S}\twoheadrightarrow R_{W,S}$ is $W\times
W^{\mathrm{op}}$-equivariant. Part (3) follows from Proposition 5.14 (4).

We prove (4). There is no loss of generality in taking $(W,S)$ to be
irreducible. Let $A$ be a NGCM for $(W,S)$. It is known that the $W$ actions
on $V_{A}$ and ${V}^{\bigvee}_{A}$ are faithful, so the induced $W\times
W^{\mathrm{op}}$-action on ${V}^{\bigvee}_{A}\otimes_{\mathbb{R}}V_{A}\cong B$
is faithful. But $B$ is a quotient of $R^{\prime}$, so the $W\times
W^{\mathrm{op}}$-action on $R^{\prime}$ is faithful. Since that action arises
by base change from that on $R_{W,S}$, the action on $R_{W,S}$ is faithful.
Since $R_{W,S}$ (resp., $R^{\prime}$) is a quotient of $\widetilde{R}_{W,S}$
(resp., $\widetilde{R}^{\prime}$), all the required $W\times
W^{\mathrm{op}}$-actions are faithful. ∎

The above theorem has many consequences for special classes of Coxeter groups,
such as finite Coxeter groups groups and affine Weyl groups, amongst which we
observe here only the following.

######  Corollary 5.17.

Suppose that $(W,S)$ is an irreducible finite Weyl group with Cartan matrix
$A$. Then $R^{\prime}$ is $W\times W^{\mathrm{op}}$-equivariantly isomorphic
to the matrix ring $B=\mathrm{Mat}_{S\times S}(\mathbb{R})$ as
$\mathbb{R}$-algebra, where $B$ may be $W\times W^{\mathrm{op}}$-equivariantly
identified with ${V}^{\bigvee}_{A}\otimes_{\mathbb{R}}V_{A}$ so that
${\alpha}^{\bigvee}_{r}\otimes\alpha_{s}=e_{r,s}$.

###### Proof.

We prove this under the more general hypotheses that $(W,S)$ is of finite
rank, that the Coxeter graph of $(W,S)$ is a tree and that
$m_{r,s}\in\\{2,3,4,6\\}$ for all $r\neq s$ in $S$.333The finite rank
assumption can also be omitted but we leave the additional arguments to show
this to the reader. By (2) and Example 4.22(1), $\theta_{A,R^{\prime}}\colon
R^{\prime}\to B=\mathrm{Mat}_{S\times S}(\mathbb{R})$ is a surjective,
$W\times W^{\mathrm{op}}$-equivariant homomorphism of $\mathbb{R}$-algebras of
the same finite dimension $|S|^{2}$. Hence it is an isomorphism as asserted. ∎

##  6\. Free Products of Rings

In this section, we give a summary of definitions and results on the free
product of rings from Cohn [5], [6], and [7]. We apply the results to show
that a ring associated to an edge of the graph of a Coxeter system is a free
product of rings.

### 6.1.

Let $B$ be an associative ring with a unit element and let $A$ be a (not
necessarily commutative) subring of $B$ containing the unit element, then we
say $B$ is an $A$ ring. (Note that if $A$ is commutative and $B$ is a unital
$A$-algebra in which $A$ embeds by $a\mapsto a1_{B}$, then $B$ is an $A$
ring). A homomorphism of $A$ rings $f:B\to C$ is an $A$-bimodule homomorphism
that sends the unit element of $B$ to the unit element of $C$.

######  Definition 6.2.

(Cohn [5], Section 3) The $A$ ring $R$ is said to be the free product of the
$A$ rings $\\{R_{i}|i\in I\\}$ ( over $A$) if $\\{R_{i}|i\in I\\}$ forms a
family of subrings of $R$ such that

  1. (i)

$R_{i}\cap R_{j}=A$ for $i\not=j$,

  2. (ii)

if $X_{i}$ is a set of generators of $R_{i}$, $i\in I$, then
$\bigcup_{I}X_{i}$ is a set of generators of $R$,

  3. (iii)

If $C_{i},i\in I$ is a set of defining relations of $R_{i},i\in I$ (in terms
of the set of generators $X_{i}$) then $\bigcup_{I}C_{i}$ is a set of defining
relations of $R$ (in terms of the set of generators $\bigcup_{I}X_{i}$).

The free product of $A$ rings does not always exist, but when it does it
coincides with the universal product which is always guaranteed to exist, see
Cohn [5], Theorem 3.1. When the free product of the $A$-rings $\\{R_{i}|i\in
I\\}$ exists, we will denote it by $\displaystyle\\{\ast_{A}R_{i}\\}_{i\in I}$
or $R_{1}\ast_{A}R_{2}$ if $I=\\{1,2\\}$.

######  Definition 6.3.

An $A$ ring $U$ is called the universal product of the $A$ rings $\\{R_{i}|\
i\in I\\}$ if there exist homomorphisms $\phi_{i}:R_{i}\to U$ such that

  1. (i)

$U$ is generated by the subrings $\phi_{i}(R_{i})$,

  2. (ii)

given any $A$ ring, $C$, and any family of homomorphisms $\psi_{i}:R_{i}\to
C$, there exists a homomorphism $\psi:U\to C$ satisfying
$\psi\circ\phi_{i}=\psi_{i}$ for each $i\in I$.

One can see that the universal product of $A$ rings is determined up to
isomorphism by the universal mapping property. It is not difficult to see that
the free product, when it exists, has the same universal mapping property and
thus must be isomorphic to the universal product. The following theorem gives
the precise relationship between the two:

######  Theorem 6.4.

(Cohn [5]) Let $R_{i},i\in I$ be a family of $A$ rings and
$\\{U,\phi_{i}:R_{i}\to U|\ i\in I\\}$ their universal product. Then the free
product of the $R_{i},i\in I$ exists if and only if

  1. (i)

the canonical homomorphism $\phi_{i}:R_{i}\to U$ is a monomorphism for each
$i\in I$,

  2. (ii)

$\phi_{i}(R_{i})\cap\phi_{j}(R_{j})=A$ for $i\not=j$.

When these two conditions hold, $U$ is in fact the free product of the rings
$R_{i}$.

In Cohn [7] (also see Cohn [6] and Serre [23], Exercise 1.2. for special
cases), the construction of the free product of a family of $A$ rings, using
tensor products of bimodules, is described. The free product exists when some
conditions of flatness are satisfied, see Cohn [5], Theorem 4.5. In
particular, when the base ring, $A$, is a field the free product of any family
of $A$ rings exists, Cohn [5], Corollary to Theorem 4.7.

Using the construction of the free product via tensor products of bimodules,
Cohn proves the following:

######  Theorem 6.5.

(Cohn [7], Theorem 2.5) If $\\{R_{i}|i\in I\\}$ is any family of $K$-rings
without zero divisors, where $K$ is a field, then their free product has no
zero divisors.

We use Cohn’s results to gain insight into the structure of the following
rings, which will in turn help us to understand the structure of the path
algebras we have associated to Coxeter systems.

######  Theorem 6.6.

Let $f(t)$ be a non-constant polynomial in $\mathbb{Q}[t]$ with non-zero
constant term, and let $\mathbb{Q}[y,\bar{y}]$ be the ring of polynomials over
$\mathbb{Q}$ in the non-commuting variables $y$ and $\bar{y}$. Let $R_{f}$
denote the quotient ring;

$R_{f}=\mathbb{Q}[y,\bar{y}]/\langle f(y\bar{y}),f(\bar{y}y)\rangle.$

Then $R_{f}$ is isomorphic to the ring

$\mathbb{Q}[x_{y},x_{y}^{-1}]\ast_{\mathbb{Q}}K,$

where $\mathbb{Q}[x_{y},x_{y}^{-1}]$ is the ring of Laurent polynomials in the
variable $x_{y}$ over $\mathbb{Q}$ and $K$ is the ring extension
$\mathbb{Q}[t]/\langle f(t)\rangle$ of $\mathbb{Q}$. Furthermore, if $f(t)$ is
irreducible, then $R_{f}$ is a domain.

###### Proof.

If $f(t)$ is irreducible, then $K$ is a field extension of $\mathbb{Q}$ and
hence, is a domain. That the ring of Laurent polynomials,
$\mathbb{Q}[x_{y},x_{y}^{-1}]$, is a domain is well known. By Theorem 6.5, the
ring $\mathbb{Q}[x_{y},x_{y}^{-1}]\ast_{\mathbb{Q}}K$ does not have zero
divisors, since both $\mathbb{Q}[x_{y},x_{y}^{-1}]$ and
$K=\mathbb{Q}[t]/\langle f(t)\rangle$ are domains. Hence it is enough to prove
that $R_{f}$ is isomorphic to the free product

$\mathbb{Q}[x_{y},x_{y}^{-1}]\ast_{\mathbb{Q}}K.$

First we claim that $y$ has an inverse in the ring
$R_{f}=\mathbb{Q}[y,\bar{y}]/\langle f(y\bar{y}),f(\bar{y}y)\rangle$. Since
$f$ is non-constant with a non-zero constant term,
$f(t)=th(t)+\kappa=h(t)t+\kappa$ for some $\kappa\in\mathbb{Q},\kappa\not=0$
and some $h(t)\in\mathbb{Q}[t]$. Without loss of generality, we can assume
that $\kappa=-1$. Thus,

$y\bar{y}h(y\bar{y})=1_{R_{f}}=h(y\bar{y})y\bar{y},$

and $\displaystyle(y\bar{y})^{-1}=h(y\bar{y})$ in the ring $R_{f}$. By
symmetry, $\bar{y}y$ has an inverse in $R_{f}$ and
$\displaystyle(\bar{y}y)^{-1}=h(\bar{y}y)$. Using our expression for
$(y\bar{y})^{-1}$, we see that $\displaystyle\bar{y}h(y\bar{y})$ is a right
inverse for $y$ and using our expression for $(\bar{y}y)^{-1}$, we see that

$1_{R_{f}}=h(\bar{y}y)\bar{y}y=\bar{y}h(y\bar{y})y,$

which shows that $\displaystyle\bar{y}h(y\bar{y})$ is also a left inverse for
$y$ in $R_{f}$. This proves our claim.

Now let
$\hat{\phi}:\mathbb{Q}[y,\bar{y}]\to\mathbb{Q}[x_{y},x_{y}^{-1}]\ast_{\mathbb{Q}}K$
be the unique $\mathbb{Q}$ algebra homomorphism with $\hat{\phi}(y)=x_{y}$ and
$\hat{\phi}(\bar{y})=x_{y}^{-1}t$. This induces a well defined quotient
homomorphism, $\phi:R_{f}\to\mathbb{Q}[x_{y},x_{y}^{-1}]\ast_{\mathbb{Q}}K$
since
$\hat{\phi}(f(y\bar{y}))=f(\hat{\phi}(y\bar{y}))=f(x_{y}x_{y}^{-1}t)=f(t)=0$
and
$\hat{\phi}(f(\bar{y}y))=f(\hat{\phi}(\bar{y}y))=f(x_{y}^{-1}tx_{y})=x_{y}^{-1}f(t)x_{y}=0$.

To show that the map $\phi$ defined above is an isomorphism, we construct its
inverse. We let $\hat{\psi}_{1}:\mathbb{Q}[t]\to R_{f}$ be the $\mathbb{Q}$
algebra homomorphism such that $\hat{\psi}_{1}(t)=y\bar{y}$. Since
$\hat{\psi}_{1}(f(t))=f(\hat{\psi}_{1}(t))=f(y\bar{y})=0$, we get a well
defined quotient homomorphism $\psi_{1}:K\to R_{f}$. Let
$\psi_{2}:\mathbb{Q}[x_{y},x_{y}^{-1}]\to R_{f}$ be the $\mathbb{Q}$ algebra
homomorphism with $\psi_{2}(x_{y})=y$ and
$\displaystyle\psi_{2}(x_{y}^{-1})=\bar{y}h(y\bar{y}).$ This is a well defined
homomorphism since $\psi_{2}(x_{y}^{-1})=(\psi_{2}(x_{y}))^{-1}$. The
universal mapping property of the free product gives a ring homomorphism
$\psi:\mathbb{Q}[x_{y},x_{y}^{-1}]\ast_{\mathbb{Q}}K\to R_{f}$ which restricts
to $\psi_{1}$ and $\psi_{2}$ on $K$ and $\mathbb{Q}[x_{y},x_{y}^{-1}]$
respectively.

We finish the proof by showing that the homomorphism $\psi$ is the inverse of
the homomorphism $\phi$. We need only verify that $\psi\circ\phi$ is the
identity on the generators of $R_{f}$ and that $\phi\circ\psi$ is the identity
on the generators of $\mathbb{Q}[x_{y},x_{y}^{-1}]\ast_{\mathbb{Q}}K$. We have
$\psi(\phi(y))=\psi(x_{y})=y$ and

$\psi(\phi(\bar{y}))=\psi(x_{y}^{-1}t)=\bar{y}h(y\bar{y})y\bar{y}=y^{-1}y\bar{y}=\bar{y}.$

Thus $\psi\circ\phi$ is the identity map on $R_{f}$. On the other hand,
$\phi(\psi(t))=\phi(y\bar{y})=x_{y}x_{y}^{-1}t=t$ and
$\phi(\psi(x_{y}))=\phi(y)=x_{y}$. Finally we have

$\phi(\psi(x_{y}^{-1}))=\phi\left(\bar{y}h(y\bar{y})\right)=x_{y}^{-1}th(x_{y}x_{y}^{-1}t)=x_{y}^{-1}th(t)=x_{y}^{-1},$

since $f(t)=th(t)-1$ and hence $\displaystyle
th(t)=1_{\mathbb{Q}[x_{y},x_{y}^{-1}]\ast_{\mathbb{Q}}K}$. This shows that the
rings are isomorphic and completes our proof. ∎

### 6.7.

Consider now a graph $\Gamma_{W,S}$ associated to a Coxeter system $(W,S)$
with generators $S=\\{s_{1},s_{2},\dots,s_{N}\\}$ and Coxeter matrix
$(m_{ij})_{\\{1\leq i,j\leq N\\}}$. Let
$Y_{W,S}=Y_{W,S,+}\bigcupdot\overline{Y_{W,S,+}}$ be an orientation of
$\Gamma_{W,S}$. To each $y_{ij}=[s_{i}s_{j}]\in Y_{W,S,+}$, we associate a
quotient ring, $R_{y_{ij}}$, of the polynomial ring
$\mathbb{Q}[x_{y_{ij}},\bar{x}_{{y}_{ij}}]$ in the non-commuting variables
$x_{y_{ij}}$ and $\bar{x}_{{y}_{ij}}$ , as follows:

$R_{y_{ij}}=\begin{cases}\mathbb{Q}[x_{y_{ij}},\bar{x}_{{y}_{ij}}]&\mbox{if}\
m_{ij}=\infty\\\ \mathbb{Q}[x_{y_{ij}},\bar{x}_{{y}_{ij}}]/\langle
C_{m_{ij}}(x_{y_{ij}}\bar{x}_{y_{ij}}),C_{m_{ij}}(\bar{x}_{y_{ij}}x_{y_{ij}})\rangle&\mbox{if}\
m_{ij}<\infty\end{cases}$

######  Corollary 6.8.

Let $\Gamma_{W,S}$ be a graph associated to a Coxeter system $(W,S)$ with
generators $S=\\{s_{1},s_{2},\dots,s_{N}\\}$ and Coxeter matrix
$(m_{ij})_{\\{1\leq i,j\leq N\\}}$. Let $R_{y_{ij}}$ be the ring associated to
the edge $y_{ij}=[s_{i}s_{j}]\in Y_{W,S,+}$. Then we have the following:

$R_{y_{ij}}\cong\begin{cases}\mathbb{Q}[x_{y_{ij}},\bar{x}_{y_{ij}}]&\mbox{if}\
m_{ij}=\infty\\\
\mathbb{Q}[x_{y_{ij}},x_{y_{ij}}^{-1}]\ast_{\mathbb{Q}}K&\mbox{if}\
m_{ij}<\infty\end{cases},$

where $\mathbb{Q}[x_{y_{ij}},x_{y_{ij}}^{-1}]$ is the ring of Laurent
polynomials in $x_{y_{ij}}$ over $\mathbb{Q}$ and $K\cong\mathbb{Q}[t]/\langle
C_{m_{ij}}(t)\rangle$ is a field extension of $\mathbb{Q}$. Furthermore,
$R_{y_{ij}}$ is a domain for all $y_{ij}\in Y_{W,S,+}$.

###### Proof.

This follows directly from Theorem 6.6 and the fact that the ring
$\mathbb{Q}[x_{y_{ij}},\bar{x}_{y_{ij}}]$ in the non-commuting variables
$x_{y_{ij}}$ and $\bar{x}_{y_{ij}}$ is a domain (see, for example, Cohn [8],
Section 5.3). ∎

##  7\. A faithful Representation of $\hat{R}_{W,S}$.

In this section, we show that the algebra $\hat{R}_{W,S}$(defined in Section
4) associated to a finite rank Coxeter system is isomorphic to a subring of a
matrix ring over a non-commutative ring.

### 7.1.

Throughout the section $\Gamma_{W,S}$ denotes a graph associated to a Coxeter
system $(W,S)$ with generators $S=\\{s_{1},\dots,s_{N}\\}$ and Coxeter matrix
$(m_{ij})_{\\{1\leq i,j\leq N\\}}$. We fix an orientation,
$Y_{W,S}=Y_{W,S,+}\bigcupdot\overline{Y_{W,S,+}}$, for our graph
$\Gamma_{W,S}$. We will show that the associated algebra $\hat{R}_{W,S}$ can
be imbedded as a $\mathbb{Q}$ algebra into a matrix ring $M_{N+1}(Q)$, where
$Q$ is a quotient of a non-commutative polynomial ring over $\mathbb{Q}$ in
several variables. This is turn will give us some insight into the zero
divisors in $\hat{R}_{W,S}$ and its subring $R_{W,S}$.

###  7.2. A Graph Extension

Let $(W^{\bullet},S^{\bullet})$ denote the Coxeter system with Coxeter
generators $S^{\bullet}$ $=\\{s_{1},\dots,s_{N},s_{N+1}\\}$ and Coxeter matrix
$(m^{\bullet}_{ij})_{1\leq i,j\leq N+1}$, such that
$m_{ij}^{\bullet}=m_{ij},1\leq i,j\leq N$,
$m^{\bullet}_{\underline{N+1}i}=m^{\bullet}_{i\underline{N+1}}=3,1\leq i\leq
N$ and $m^{\bullet}_{\underline{N+1}\ \underline{N+1}}=1$. (We denote $N+1$ by
$\underline{N+1}$ in subscripts where confusion might be caused by ambiguity.)
Since $W$ can be identified with the parabolic subgroup $W^{\bullet}_{[N]}$ of
$W^{\bullet}$ corresponding to the Coxeter system
$(W^{\bullet}_{[N]},S^{\bullet}_{[N]})$, by Lemma 4.4, we have an inclusion of
the graph $\Gamma_{W,S}$ in the graph $\Gamma_{W^{\bullet},S^{\bullet}}$ and
imbeddings of the associated non-unital $\mathbb{Q}$ algebras
${i}_{[N],[N+1]}:\hat{P}_{W,S}\to\hat{P}_{W^{\bullet},S^{\bullet}}$ and
$\tilde{i}_{[N],[N+1]}:\hat{R}_{W,S}\to\hat{R}_{W^{\bullet},S^{\bullet}}$. For
simplicity of notation, we will identify the vertices and edges of the graph
$\Gamma_{W,S}$ with their images in the graph of
$\Gamma_{W^{\bullet},S^{\bullet}}$. We can extend the orientation of
$\Gamma_{W,S}$ to an orientation for $\Gamma_{W^{\bullet},S^{\bullet}}$ by
letting
$Y_{W^{\bullet},S^{\bullet},+}=Y_{W,S,+}\bigcupdot\\{y_{\underline{N+1}i}|1\leq
i\leq N\\}$.

######  Example 7.3.

For our running example, we let $S^{\bullet}=\\{r,s,t,u,v,w\\}$ with new
vertex $w$. We show the positively oriented edges,
$Y_{W^{\bullet},S^{\bullet},+}$, of the graph
$\Gamma_{W^{\bullet},S^{\bullet},+}$ below. The vertices and edges in
$\Gamma_{W^{\bullet},S^{\bullet}}\backslash\Gamma_{W,S}$ are shown in blue.
Alongside, we show the new vertex and new geometric edges of
$\Gamma_{W^{\bullet},S^{\bullet}}$ with their labels in blue.

|
---|---
$\textstyle{{\color[rgb]{0,0,1}w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\epsilon}}$$\textstyle{t\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\zeta}}$$\textstyle{v}$$\textstyle{{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\alpha}}$$\scriptstyle{y_{\gamma}}$$\textstyle{{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{\beta}}$$\scriptstyle{y_{\delta}}$ |
---|---
$\textstyle{{\color[rgb]{0,0,1}w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\color[rgb]{0,0,1}3}$$\scriptstyle{\color[rgb]{0,0,1}3}$$\scriptstyle{\color[rgb]{0,0,1}3}$$\scriptstyle{\color[rgb]{0,0,1}3}$$\scriptstyle{\color[rgb]{0,0,1}3}$$\textstyle{s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\textstyle{t\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\textstyle{v}$$\textstyle{{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$$\scriptstyle{4}$$\textstyle{{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\scriptstyle{\infty}$

###  7.4. Isomorphism of $\hat{R}_{W^{\bullet},S^{\bullet}}$ and a Matrix Ring

For $y\in Y_{W,S,+}$, let $R_{y}$ be the associated ring described at the end
of Section 6. By Definition 6.2, the free product
$\left\\{\ast_{\mathbb{Q}}R_{y}\right\\}_{y\in Y_{W,S,+}}$ is isomorphic to
the quotient ring, $Q$, of the polynomial ring in the non-commuting variables
$\left\\{x_{y},\bar{x}_{y}|\ y\in Y_{W,S,+}\right\\}$;

$Q=\mathbb{Q}[\left\\{x_{y_{ij}},\bar{x}_{y_{ij}}\right\\}_{y_{ij}\in
Y_{W,S,+}}]/\left\langle\left\\{C_{m_{ij}}(x_{y_{ij}}\bar{x}_{y_{ij}}),C_{m_{ij}}(\bar{x}_{y_{ij}}x_{y_{ij}})\right\\}_{\left\\{y_{ij}\in
Y_{W,S,+}|\ m_{ij}<\infty\right\\}}\right\rangle.$

By Theorem 6.5 and Corollary 6.8, $Q$ is a domain.

We will denote the coset, $p+\hat{I}_{W^{\bullet},S^{\bullet}}$, of
$p\in\hat{P}_{W^{\bullet},S^{\bullet}}$ in $\hat{R}_{W^{\bullet},S^{\bullet}}$
by $p^{\sim}$ in what follows. We first single out some elements of
$\hat{R}_{W^{\bullet},S^{\bullet}}$ which will play a central role in defining
a homomorphism from $\hat{R}_{W^{\bullet},S^{\bullet}}$ to $M_{N+1}(Q)$. For
$1\leq i\leq N+1$, we let

(7.4.1)
$P_{i}^{j}=\begin{cases}[s_{i}s_{N+1}s_{j}]^{\sim}=(y_{i\underline{N+1}}y_{\underline{N+1}j})^{\sim}&\mbox{if}\
1\leq i,j\leq N\\\
[s_{i}s_{N+1}]^{\sim}=(y_{i\underline{N+1}})^{\sim}&\mbox{if}\ 1\leq i\leq
N,j=N+1\\\ [s_{N+1}s_{j}]^{\sim}=y_{\underline{N+1}j}^{\sim}&\mbox{if}\
i=N+1,1\leq j\leq N\\\ [s_{N+1}]^{\sim}&\mbox{if}\ i=j=N+1\\\ \end{cases}$

######  Note.

In what follows we use the notation $\dot{o}(y)$ and $\dot{t}(y)$ to denote
the index of $o(y)$ and $t(y)$ respectively in the set $[N+1]$, for $y\in
Y_{W^{\bullet},S^{\bullet}}$, i.e. if $y=[s_{i},s_{j}]$, then $\dot{o}(y)=i$
and $\dot{t}(y)=j$.

######  Lemma 7.5.

We have the following identities in $\hat{R}_{W^{\bullet},S^{\bullet}}$, where
$s_{i},s_{j},s_{k},s_{l}\in S^{\bullet}$:

$\displaystyle P_{i}^{i}$ $\displaystyle=[s_{i}]^{\sim}$ $\displaystyle
P_{i}^{j}P_{k}^{l}$ $\displaystyle=0^{\sim}\ \mbox{if}\ j\not=k$
$\displaystyle P_{i}^{j}P_{k}^{l}$ $\displaystyle=P_{i}^{l}\ \mbox{if}\ j=k$

###### Proof.

Note that $C_{y_{ij}}^{\sim}=[s_{i}s_{j}s_{i}]^{\sim}-[s_{i}]^{\sim}=0^{\sim}$
if either $i$ or $j$ are equal to $N+1$. Hence, if $i\not=N+1$, then
$P_{i}^{i}=[s_{i}s_{N+1}s_{i}]^{\sim}=[s_{i}]^{\sim}$. If $i=N+1$, then
$P_{i}^{i}=[s_{N+1}]^{\sim}$. This proves the first identity.

Consider now $P_{i}^{j}P_{k}^{l}$. It is clear that the product is $0^{\sim}$
if $j\not=k$. Suppose that $1\leq i,j,k\leq N$, then

$P_{i}^{j}P_{j}^{k}=([s_{i}s_{N+1}s_{j}][s_{j}s_{N+1}s_{k}])^{\sim}=([s_{i}s_{N+1}][s_{N+1}s_{j}s_{N+1}][s_{N+1}s_{k}])^{\sim}=[s_{i}s_{N+1}s_{k}]^{\sim}=P_{i}^{k}$

by our opening remark. The other cases follow similarly and are left to the
reader. ∎

### 7.6.

For each $i\in\\{1,2,\dots,N+1\\}$, we use the universal mapping property of
the polynomial ring to define a ring homomorphism
$\hat{\psi}_{i}:\mathbb{Q}[\\{x_{y},\bar{x}_{y}\\}_{y\in
Y_{W,S,+}}]\to[s_{i}]^{\sim}\hat{R}_{W^{\bullet},S^{\bullet}}[s_{i}]^{\sim}\subset\hat{R}_{W^{\bullet},S^{\bullet}}$,
such that $\hat{\psi}_{i}(1_{Q})=[s_{i}]^{\sim}$,
$\hat{\psi}_{i}(x_{y})=P_{i}^{\dot{o}(y)}y^{\sim}P_{\dot{t}(y)}^{i}$ and
$\hat{\psi}_{i}(\bar{x}_{y})=P_{i}^{\dot{o}(\bar{y})}\bar{y}^{\sim}P_{\dot{t}(\bar{y})}^{i}$
for $y\in Y_{W,S,+}$. Since $o(\bar{y})=t(y)$ and $t(\bar{y})=o(y)$, we have

$P_{i}^{\dot{o}(y)}y^{\sim}P_{\dot{t}(y)}^{i}P_{i}^{\dot{o}(\bar{y})}\bar{y}^{\sim}P_{\dot{t}(\bar{y})}^{i}=P_{i}^{\dot{o}(y)}(y\bar{y})^{\sim}P_{\dot{t}(\bar{y})}^{i}.$

Hence
$\hat{\psi}_{i}((x_{y}\bar{x}_{y})^{k})=P_{i}^{\dot{o}(y)}\left((y\bar{y})^{k}\right)^{\sim}P_{\dot{t}(\bar{y})}^{i}$,
by Lemma 7.5 . Thus for $y_{ij}\in Y_{W,S,+}$, $m_{ij}<\infty$, we have

$\hat{\psi}_{i}\left(C_{m_{ij}}(x_{y_{ij}}\bar{x}_{y_{ij}})\right)=C_{m_{ij}}\left(P_{i}^{\dot{o}(y)}(y\bar{y}\right)^{\sim}P_{\dot{t}(\bar{y})}^{i})=P_{i}^{\dot{o}(y)}C_{m_{ij}}\left((y\bar{y})^{\sim}\right)P_{\dot{t}(\bar{y})}^{i}=0^{\sim}$

Therefore the homomorphism $\hat{\psi}_{i}$ gives us a well defined quotient
homomorphism

$\psi_{i}:Q\to[s_{i}]^{\sim}\hat{R}_{W^{\bullet},S^{\bullet}}[s_{i}]^{\sim}\subset\hat{R}_{W^{\bullet},S^{\bullet}}$

acting as follows on the generators of $Q$;

$\displaystyle\psi_{i}(1_{Q})$ $\displaystyle=[s_{i}]^{\sim}$
$\displaystyle{\psi}_{i}(x_{y})$
$\displaystyle=P_{i}^{\dot{o}(y)}y^{\sim}P_{\dot{t}(y)}^{i},$ $\displaystyle
y\in Y_{W,S,+}.$ $\displaystyle{\psi}_{i}(\bar{x}_{y})$
$\displaystyle=P_{i}^{\dot{o}(\bar{y})}\bar{y}^{\sim}P_{\dot{t}(\bar{y})}^{i},$
$\displaystyle y\in Y_{W,S,+}.$

######  Lemma 7.7.

Let
$\psi_{i}:Q\to[s_{i}]^{\sim}\hat{R}_{W^{\bullet},S^{\bullet}}[s_{i}]^{\sim}\subset\hat{R}_{W^{\bullet},S^{\bullet}}$
and
$\psi_{j}:Q\to[s_{j}]^{\sim}\hat{R}_{W^{\bullet},S^{\bullet}}[s_{j}]^{\sim}\subset\hat{R}_{W^{\bullet},S^{\bullet}}$,
$1\leq i,j\leq N+1$ be as defined above. We have
$P_{j}^{i}\psi_{i}(q)P_{i}^{j}=\psi_{j}(q)$ for all $q\in Q,1\leq i,j\leq
N+1$.

###### Proof.

It suffices to show equality on the generators $1_{Q},x_{y},\bar{x}_{y},y\in
Y_{W,S,+}$ of $Q$. Using the identities in Lemma 7.5, we get:

$P_{j}^{i}\psi_{i}(1_{Q})P_{i}^{j}=P_{j}^{i}[s_{i}]^{\sim}P_{i}^{j}=P_{j}^{i}P_{i}^{j}=P_{j}^{j}=[s_{j}]^{\sim}=\psi_{j}(1_{Q}),$
$P_{j}^{i}\psi_{i}(x_{y})P_{i}^{j}=P_{j}^{i}P_{i}^{\dot{o}(y)}y^{\sim}P_{\dot{t}(y)}^{i}P_{i}^{j}=P_{j}^{\dot{o}(y)}y^{\sim}P_{\dot{t}(y)}^{j}=\psi_{j}(x_{y}),$

and

$P_{j}^{i}\psi_{i}(\bar{x}_{y})P_{i}^{j}=P_{j}^{i}P_{i}^{\dot{o}(\bar{y})}\bar{y}^{\sim}P_{\dot{t}(\bar{y})}^{i}P_{i}^{j}=P_{j}^{\dot{o}(\bar{y})}\bar{y}^{\sim}P_{\dot{t}(\bar{y})}^{j}=\psi_{j}(\bar{x}_{y}).$

This proves the lemma. ∎

### 7.8.

We let $\Psi:M_{N+1}(Q)\to\hat{R}_{W^{\bullet},S^{\bullet}}$ be the
homomorphism of $\mathbb{Q}$ vector spaces which acts on basis elements of
$M_{N+1}(Q)$ of the form $qe_{ij}$ as follows:

$\Psi(qe_{ij})=\psi_{i}(q)P_{i}^{j},$

where $q\in Q$ and $e_{ij}$ is the $(N+1)\times(N+1)$ matrix with $1$ in the
$(i,j)$ position and zeros elsewhere.

######  Lemma 7.9.

$\Psi$ is a ring homomorphism.

###### Proof.

To verify this, we need only show that
$\Psi((q_{1}e_{ij})(q_{2}e_{kl}))=\Psi(q_{1}e_{ij})\Psi(q_{2}e_{kl})$ for
$q_{1},q_{2}\in Q$ and $1\leq i,j,k,l\leq N+1$. If $j\not=k$, it is easy to
see that both sides are zero and are thus equal. If $j=k$, then, using Lemma
7.7, we get

$\displaystyle\Psi((q_{1}e_{ij})(q_{2}e_{jl}))$
$\displaystyle=\Psi(q_{1}q_{2}e_{il})=\psi_{i}(q_{1})\psi_{i}(q_{2})P_{i}^{l}$
$\displaystyle=\psi_{i}(q_{1})P_{i}^{j}P_{j}^{i}\psi_{i}(q_{2})P_{i}^{j}P_{j}^{i}P_{i}^{l}$
$\displaystyle=\psi_{i}(q_{1})P_{i}^{j}\psi_{j}(q_{2})P_{j}^{l}$
$\displaystyle=\Psi(q_{1}e_{ij})\Psi(q_{2}e_{jl}).$

∎

We show that both rings are isomorphic by showing that $\Psi$ has an inverse.
We define a homomorphism,

$\phi:\hat{P}_{W^{\bullet},S^{\bullet}}\to M_{N+1}(Q),$

which acts as follows on the vertices and edges of
$\Gamma_{W^{\bullet},S^{\bullet}}$ as follows:

$\displaystyle\phi([s_{i}])$ $\displaystyle=e_{ii},$ $\displaystyle 1\leq
i\leq N+1,$ $\displaystyle\phi(y_{ij})$ $\displaystyle=x_{y_{ij}}e_{ij},$
$\displaystyle 1\leq i\not=j\leq N,y_{ij}\in Y_{W^{\bullet},S^{\bullet},+}$
$\displaystyle\phi(y_{ij})$ $\displaystyle=\bar{x}_{\bar{y}_{ij}}e_{ij},$
$\displaystyle 1\leq i\not=j\leq
N,y_{ij}\in\overline{Y_{W^{\bullet},S^{\bullet},+}}$
$\displaystyle\phi(y_{ij})$ $\displaystyle=e_{ij},$ $\displaystyle
N+1\in\\{i,j\\},i\not=j.$

The homomorphism extends to a well defined homomorphism on
$\hat{P}_{W^{\bullet},S^{\bullet}}$ since $\phi$ respects the multiplicative
relations on the basis of paths $\mathfrak{P}_{W^{\bullet},S^{\bullet}}$ in
$\hat{P}_{W^{\bullet},S^{\bullet}}$ namely:

$\displaystyle\phi(y_{ij})\phi(y_{kl})$ $\displaystyle=\phi(0)=0$
$\displaystyle\mbox{if}\ j\not=k,y_{ij},y_{kl}\in
Y_{W^{\bullet},S^{\bullet}}.$
$\displaystyle\phi(y_{ij})\phi([s_{k}])=\phi([s_{k}])\phi(y_{ji})$
$\displaystyle=\phi(0)=0$ $\displaystyle\mbox{if}\ j\not=k,y_{ij}\in
Y_{W^{\bullet},S^{\bullet}}.$
$\displaystyle\phi(y_{ij})\phi([s_{j}])=\phi([s_{i}])\phi(y_{ij})$
$\displaystyle=\phi(y_{ij})$ $\displaystyle\mbox{if}\ y_{ij}\in
Y_{W^{\bullet},S^{\bullet}}.$

The following lemma shows that the quotient homomorphism
$\Phi:\hat{R}_{W^{\bullet},S^{\bullet}}\to M_{N+1}(Q)$ given by

$\Phi(p^{\sim})=\phi(p),\ p\in\hat{P}_{W^{\bullet},S^{\bullet}}$

is well defined:

######  Lemma 7.10.

Let $\phi:\hat{P}_{W^{\bullet},S^{\bullet}}\to M_{N+1}(Q)$ be as defined
above. Then, $\hat{I}_{W^{\bullet},S^{\bullet}}\subseteq\ker\phi$.

###### Proof.

It suffices to prove that the generators of
$\hat{I}_{W^{\bullet},S^{\bullet}}$ are in $\ker\phi$. For $1\leq i\not=j\leq
N$, $y_{ij}\in Y_{W^{\bullet},S^{\bullet},+}$ we have

$\phi(y_{ij}y_{ji})=\phi(y_{ij})\phi(y_{ji})=x_{y_{ij}}e_{ij}\bar{x}_{y_{ij}}e_{ji}=x_{y_{ij}}\bar{x}_{{y}_{ij}}e_{ii},$

and if $m_{ij}<\infty$,

$\phi(C_{y_{ij}})=C_{m_{ij}}(\phi(y_{ij})\phi(\bar{y}_{ij}))=C_{m_{ij}}(x_{y_{ij}}\bar{x}_{{y}_{ij}})e_{ii}=0.$

Similarly, if $1\leq i\not=j\leq N$,
$y_{ij}\in{Y_{W^{\bullet},S^{\bullet},+}}$, $m_{ij}<\infty$, we have

$\phi(C_{\bar{y}_{ij}})=C_{m_{ij}}(\phi(\bar{y}_{ij})\phi(y_{ij}))=C_{m_{ij}}(\bar{x}_{{y}_{ij}}x_{y_{ij}})e_{jj}=0.$

If $1\leq i\leq N$ and $j=N+1$, we have
$C_{y_{ij}}=y_{ij}\bar{y}_{ij}-[s_{i}]$ and

$\phi(C_{y_{ij}})=\phi(y_{ij})\phi(\bar{y}_{ij})-e_{ii}=e_{i\underline{N+1}}e_{\underline{N+1}i}-e_{ii}=0.$

Likewise, if $i=N+1$ and $1\leq j\leq N$, we have

$\phi(C_{y_{ij}})=e_{\underline{N+1}j}e_{j\underline{N+1}}-e_{\underline{N+1}\
\underline{N+1}}=0.$

Thus all generators of $\hat{I}_{W^{\bullet},S^{\bullet}}$ are in $\ker\phi$
and hence $\phi(\hat{I}_{W^{\bullet},S^{\bullet}})=0$. ∎

######  Lemma 7.11.

Let $\Phi:\hat{R}_{W^{\bullet},S^{\bullet}}\to M_{N+1}(Q)$ and
$\Psi:M_{N+1}(Q)\to\hat{R}_{W^{\bullet},S^{\bullet}}$ be defined as above.
Then $\Psi\circ\Phi$ is the identity map on
$\hat{R}_{W^{\bullet},S^{\bullet}}$.

###### Proof.

It is enough to show that this is true on the generators
$[s_{i}]^{\sim},s_{i}\in S$ and $y_{ij}^{\sim},y_{ij}\in
Y_{W^{\bullet},S^{\bullet}}$. Let $s_{i}\in S$, then

$\Psi\circ\Phi([s_{i}]^{\sim})=\Psi(\phi([s_{i}])=\Psi(e_{ii})=\psi_{i}(1_{Q})P_{i}^{i}=[s_{i}]^{\sim}.$

If $y_{ij}\in Y_{W^{\bullet},S^{\bullet},+},1\leq i\not=j\leq N$, we have

$\displaystyle\Psi\circ\Phi(y_{ij}^{\sim})$
$\displaystyle=\Psi(\phi(y_{ij}))=\Psi(x_{y_{ij}}e_{ij})$
$\displaystyle=\psi_{i}(x_{y_{ij}})P_{i}^{j}=P_{i}^{i}y_{ij}^{\sim}P_{j}^{i}P_{i}^{j}$
$\displaystyle=y_{ij}^{\sim}.$

If $y_{ij}\in Y_{W^{\bullet},S^{\bullet},+},1\leq i\not=j\leq N$, we have

$\displaystyle\Psi\circ\Phi(\bar{y}_{ij}^{\sim})$
$\displaystyle=\Psi(\phi(\bar{y}_{ij}))=\Psi(\bar{x}_{y_{ij}}e_{ji})$
$\displaystyle=\psi_{j}(\bar{x}_{y_{ij}})P_{j}^{i}=P_{j}^{j}\bar{y}_{ij}^{\sim}P_{i}^{j}P_{j}^{i}$
$\displaystyle=\bar{y}_{ij}^{\sim}.$

Finally, if $i=N+1$ or $j=N+1$, then

$\displaystyle\Psi\circ\Phi(y_{ij}^{\sim})$
$\displaystyle=\Psi(\phi(y_{ij}))=\Psi(e_{ij})$
$\displaystyle=\psi_{i}(1_{Q})P_{i}^{j}=[s_{i}s_{j}]^{\sim}=y_{ij}^{\sim}$

∎

######  Lemma 7.12.

Let $\Phi:\hat{R}_{W^{\bullet},S^{\bullet}}\to M_{N+1}(Q)$ and
$\Psi:M_{N+1}(Q)\to\hat{R}_{W^{\bullet},S^{\bullet}}$ be defined as above.
Then $\Phi\circ\Psi$ is the identity map on $M_{N+1}(Q)$.

###### Proof.

It is enough to show that $\Phi\circ\Psi$ acts as the identity on the
generators of $M_{N+1}(Q)$ as a ring;
$\\{x_{y}e_{ij},\bar{x}_{y}e_{ij},e_{ij}|\ 1\leq i,j\leq N+1,y\in
Y_{W,S,+}\\}$. For $1\leq i,j\leq N+1$, we have

$\displaystyle\Phi(\Psi(e_{ij}))=\Phi(P_{i}^{j})$
$\displaystyle=\begin{cases}\Phi([s_{i}]^{\sim})&\mbox{if}\ i=j\\\
\Phi((y_{i\underline{N+1}}y_{\underline{N+1}j})^{\sim})&\mbox{if}\ 1\leq
i\not=j\leq N\\\ \Phi(y_{\underline{N+1}j}^{\sim})&\mbox{if}\ i=N+1\
\mbox{and}\ 1\leq j\leq N\\\ \Phi(y_{i\underline{N+1}}^{\sim})&\mbox{if}\
j=N+1\ \mbox{and}\ 1\leq i\leq N\\\ \end{cases}$
$\displaystyle=\begin{cases}e_{ii}&\mbox{if}\ i=j\\\
e_{i\underline{N+1}}e_{\underline{N+1}j}=e_{ij}&\mbox{if}\ 1\leq i\not=j\leq
N\\\ e_{\underline{N+1}j}&\mbox{if}\ i=N+1\ \mbox{and}\ 1\leq j\leq N\\\
e_{i\underline{N+1}}&\mbox{if}\ j=N+1\ \mbox{and}\ 1\leq i\leq N\\\
\end{cases}$

Thus $\Phi(\Psi(e_{ij}))=\Phi(P_{i}^{j})=e_{ij},1\leq i,j\leq N+1$.

From the above calculation we have that $\Phi(P_{i}^{j})=e_{ij},1\leq i,j\leq
N+1$. Thus if $1\leq i,j\leq N+1,y\in Y_{W,S,+}$, we have

$\displaystyle\Phi\left(\Psi(x_{y}e_{ij})\right)$
$\displaystyle=\Phi\left(\psi_{i}(x_{y})P_{i}^{j}\right)$
$\displaystyle=\Phi\left(P_{i}^{\dot{o}(y)}y^{\sim}P_{\dot{t}(y)}^{i}P_{i}^{j}\right)$
$\displaystyle=e_{i\dot{o}(y)}\phi(y)e_{\dot{t}(y)i}e_{ij}$
$\displaystyle=e_{i\dot{o}(y)}x_{y}e_{\dot{o}(y)\dot{t}(y)}e_{\dot{t}(y)i}e_{ij}$
$\displaystyle=x_{y}e_{ij}.$

For $1\leq i,j\leq N+1,y\in Y_{W,S,+}$, we have

$\displaystyle\Phi\left(\Psi(\bar{x}_{y}e_{ij})\right)$
$\displaystyle=\Phi\left(\psi_{i}(\bar{x}_{y})P_{i}^{j}\right)$
$\displaystyle=\Phi\left(P_{i}^{\dot{o}(\bar{y})}\bar{y}^{\sim}P_{\dot{t}(\bar{y})}^{i}P_{i}^{j}\right)$
$\displaystyle=e_{i\dot{o}(\bar{y})}\phi(\bar{y})e_{\dot{t}(\bar{y})i}e_{ij}$
$\displaystyle=e_{i\dot{o}(\bar{y})}\bar{x}_{y}e_{\dot{o}(\bar{y})\dot{t}(\bar{y})}e_{\dot{t}(\bar{y})i}e_{ij}$
$\displaystyle=\bar{x}_{y}e_{ij}.$

This concludes our proof. ∎

##  8\. Zero Divisors in $R_{W,S}$

In this section, we put together the results from the previous sections, to
get an understanding of zero divisors in the ring $R_{W,S}$, which is the main
topic of interest of the paper.

### 8.1.

We let $(W,S)$ be a Coxeter system with $S=\\{s_{i}\\}_{i\in I}$, not
necessarily finite, and Coxeter matrix $(m_{ij})_{i,j\in I}$. We use the
notation established in Section 4.1 throughout. For $p\in\hat{P}_{W,S}$, we
let $x^{\sim}=p+\hat{I}_{W,S}$ denote the coset of $p$ in the quotient ring
$\hat{R}_{W,S}$.

Although $\hat{R}_{W,S}$ is clearly not a domain, we have the following
result:

######  Theorem 8.2.

Let $(W,S)$ be a Coxeter system with $S=\\{s_{i}\\}_{i\in I}$, and Coxeter
matrix $(m_{ij})_{i,j\in I}$. Let $x_{1}\in[s_{i}]{\hat{R}_{W,S}}[s_{j}]$ and
$x_{2}\in[s_{j}]{\hat{R}_{W,S}}[s_{k}]$, be elements of ${\hat{R}_{W,S}}$ such
that $x_{1}x_{2}=0$, then either $x_{1}=0$ or $x_{2}=0$.

###### Proof.

If $S$ is infinite, then $x_{1}=p_{1}^{\sim}$ and $x_{2}=p_{2}^{\sim}$, where
$p_{1},p_{2}\in\hat{P}_{W,S}$ and are linear combinations of paths involving
only a finite number of vertices and edges. Thus, using the direct limit
structure of $\hat{R}_{W,S}$ from Lemma 4.11, we have $x_{1}$ and $x_{2}$ are
both in $\tilde{i}_{J,I}(\hat{R}_{W_{J},S_{J}})$ for some parabolic subgroup
$(W_{J},S_{J})$ where $J\subset I$ is finite. Since $\tilde{i}_{J,I}$ is a
monomorphism, we can assume without loss of generality that $S$ is finite.

When $S$ is finite with cardinality $N$, by Lemmas 7.11 and 7.12, we have a
monomorphism $\Psi:{\hat{R}_{W,S}}\to M_{N+1}(Q)$ which maps
$[s_{i}]^{\sim}{\hat{R}_{W,S}}[s_{j}]^{\sim}$ into $Qe_{ij}$. Furthermore, $Q$
is a domain by Theorem 6.5. Thus it is enough to show that if
$(q_{1}e_{ij})(q_{2}e_{jk})=0$ where $q_{1},q_{2}\in Q$, then either $q_{1}=0$
or $q_{2}=0$. This is obvious since $Q$ is a domain. ∎

######  Corollary 8.3.

Let let $(W,S)$ be a Coxeter system with $S=\\{s_{i}\\}_{i\in I}$, and Coxeter
matrix $(m_{ij})_{i,j\in I}$. Let $x_{1}\in[s_{i}]R_{W,S}[s_{j}]$ and
$x_{2}\in[s_{j}]R_{W,S}[s_{k}]$ be elements of ${R_{W,S}}$ such that
$x_{1}x_{2}=0$, then either $x_{1}=0$ or $x_{2}=0$.

###### Proof.

By Lemma 4.13, we have a monomorphism $\tilde{i}:R_{W,S}\to\hat{R}_{W,S}$ on
the quotient algebras. Therefore the result follows from Theorem 8.2. ∎

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