# Semi-Invariant Rings: UFD and Codimension one Orbits

Charles Paquette 1,2, Deepanshu Prasad 1 AND David Wehlau 1,2 1Department of
Mathematics and Statistics, Queen’s University, Kingston ON, Canada
2Department of Mathematics and Computer Science, Royal Military College of
Canada, Kingston ON, Canada<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a finite dimensional associative $\mathbb{K}$-algebra over an
algebraically closed field $\mathbb{K}$ of characteristic zero. To $A$, we can
associate its basic form that is given by a quiver $Q=(Q_{0},Q_{1})$ with an
admissible ideal $R$. For a dimension vector $\beta$, we consider an
irreducible component $\mathcal{C}$ of the module variety of
$\beta$-dimensional representations of $A$. The reductive group ${\rm
GL}_{\beta}(\mathbb{K}):=\prod_{i\in Q_{0}}{\rm GL}_{\beta_{i}}(\mathbb{K})$
acts on $\mathcal{C}$ by change of basis, and has a unique closed orbit. We
consider the corresponding ring of semi-invariants
$\operatorname{SI}(Q,\mathcal{C})$. We prove that if $\mathcal{C}$ is
factorial and has maximal orbits of codimension one, then
$\operatorname{SI}(Q,\mathcal{C})$ is a complete intersection and is not
multiplicity free. If $\mathcal{C}$ is not factorial, then this conclusion
does not necessarily hold. We present examples showing that the codimension of
the complete intersection can be arbitrarily large. Finally, we interpret our
results in the case of hereditary algebras.

##  1\. Introduction

We let $A$ denote a finite dimensional (associative and unital) algebra over
an algebraically closed field $\mathbb{K}$ of characteristic zero. It is well
known that up to Morita equivalence, we may assume that $A$ is an admissible
quotient of a path algebra of a finite quiver. Thus, there exists a quiver
$Q=(Q_{0},Q_{1})$, which is uniquely determined by $A$, and an admissible
ideal $R$ of $\mathbb{K}Q$ such that $A\cong\mathbb{K}Q/R$. We will identify
left $A$-modules with representations of $Q$ satisfying the relations from
$R$.

Let $\beta$ be a dimension vector, that is, an element $(\beta_{i})_{i\in
Q_{0}}$ in the positive part $(\mathbb{Z}_{\geq 0})^{Q_{0}}$ of the
Grothendieck group $K_{0}(A)$ of $A$. We consider the representation space
$\operatorname{{\rm rep}}_{\beta}(Q,R)$, which parametrizes the
$\beta$-dimensional representations of $(Q,R)$. This is an affine variety,
which is not necessarily irreducible. The general linear group
$\operatorname{GL}_{\beta}(\operatorname{\mathbb{K}})\coloneqq\prod_{i\in
Q_{0}}\operatorname{GL}_{\beta_{i}}(\operatorname{\mathbb{K}})$ acts by change
of bases, so that the orbits corresponds to the isomorphism classes. This
provides a geometric approach to study the representations of $(Q,R)$.

In [Kin94], King introduced moduli spaces of representations of quivers (with
relations). In his work, King developed an algebraic notion of stability,
which allows one to interpret Mumford’s stability in the setting of quiver
representations. The multiplicative characters of
$\operatorname{GL}_{\beta}(\operatorname{\mathbb{K}})$ are in bijection with
$\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^{Q_{0}},\mathbb{Z})$, the space of
weights. For a stability parameter (or weight)
$\theta\in\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^{Q_{0}},\mathbb{Z})$, one
considers the $\theta$-semistable representations in some irreducible
component $\mathcal{C}$ of $\operatorname{{\rm rep}}_{\beta}(Q,R)$. Up to an
equivalence relation (called $S$-equivalence), these representations form a
projective variety $\mathcal{M}(\mathcal{C})^{\theta-ss}$, the moduli space of
$\theta$-semistable representations in $\mathcal{C}$. These moduli spaces are
projective varieties corresponding to certain subrings of the so-called rings
of semi-invariants of $\mathcal{C}$. The study of the semi-invariant ring
$\operatorname{SI}(Q,\mathcal{C})$ provides a way to understand these moduli
spaces. Specifically King showed that

(1) $\mathcal{M}(\mathcal{C})^{\theta-
ss}\coloneqq\mathrm{Proj}\left(\bigoplus_{n\geq
0}\operatorname{SI}(Q,\mathcal{C})_{n\theta}\right)$

where $\bigoplus_{n\geq 0}\operatorname{SI}(Q,\mathcal{C})_{n\theta}$ is a
direct sum of weight spaces forming a graded subring of
$\operatorname{SI}(Q,\mathcal{C})$, is a certain GIT-quotient.

A famous result of Sato-Kimura states that:

######  Theorem (Sato-Kimura).

Let $G$ be a connected linear algebraic group and $V$ be a rational
representation of $G$ such that the action has a dense open orbit. Then, the
semi-invariant ring is a polynomial ring. (see [DW17, Lemma 9.4.2] or [KS77,
Proposition 5, page-60]))

This theorem applies when $R=0$, since $\operatorname{{\rm rep}}_{\beta}(Q)$
is an affine space. Therefore, in this case when $\operatorname{{\rm
rep}}_{\beta}(Q)$ contains an open $\operatorname{GL}_{\beta}$-orbit, the ring
of semi-invariants of the $\beta$-dimensional representations is a polynomial
ring. In general, if $\mathcal{C}$ contains an open
$\operatorname{GL}_{\beta}$-orbit, then $\operatorname{SI}(Q,\mathcal{C})$ is
not necessarily a polynomial ring. However, one can adapt the proof of Sato
and Kimura (see also [CKW13]) to prove that if ${\mathcal{C}}$ is an orbit
closure, then the ring of semi-invariants is multiplicity-free, that is, each
weight space of $\operatorname{SI}(Q,\mathcal{C})$ is at most one dimensional.

A next natural step is to consider the case where $\mathcal{C}$ contains a
codimension one orbit. In this case, if $\mathcal{C}$ is not an orbit closure,
then $\mathcal{C}$ contains an open (dense) set of codimension one orbits.
This leads us to study the semi-invariant ring
$\operatorname{SI}(Q,\mathcal{C})$ when $\mathcal{C}$ contains an open dense
set of codimension one $\operatorname{GL}_{\beta}$-orbits. We require the
condition that $\mathbb{K}[\mathcal{C}]$ is a UFD. Under this assumption, our
methods allow us to describe the generators of
$\operatorname{SI}(Q,\mathcal{C})$. This leads us to one of our main results:

######  Theorem A.

Let $\mathcal{C}$ be an irreducible component of $\operatorname{{\rm
rep}}_{\beta}(Q,R)$ which is not an orbit closure but which contains a
codimension one orbit. Suppose further that the coordinate ring
$\mathbb{K}[\mathcal{C}]$ is a UFD. Then, the semi-invariant ring
$\operatorname{SI}(Q,\mathcal{C})$ is a complete intersection whose
codimension is equal to $\\#\\{\text{monomials of weight $\chi$}\\}-2$, where
$\chi$ is the unique minimal weight such that
$\mathrm{dim}_{\mathbb{K}}\,\operatorname{SI}(Q,\mathcal{C})_{\chi}=2$.

The above theorem is stated in the context of representations of quivers with
relations. Our techniques can be applied in the following more general
setting. Let $G$ be a connected reductive algebraic group acting regularly on
a factorial variety $\mathcal{C}$. We can define the ring of semi-invariants
as $\mathbb{K}[\mathcal{C}]^{[G,G]}$ where $[G,G]$ is the derived subgroup of
$G$. As in the quiver case this ring has a weight space decomposition given by
the multiplicative characters of $G$. Theorem A applies when $\mathcal{C}$ has
a single closed orbit and its maximal orbits are of co-dimension 1.

For hereditary algebras, the variety $\operatorname{{\rm rep}}_{\beta}(Q)$ is
irreducible and its structure depends on the so-called canonical decomposition
of $\beta$, which is a decomposition of $\beta$ into a sum of special
dimension vectors known as Schur roots. These roots fall into three types, the
real, isotropic and imaginary Schur roots. A description of the semi-invariant
ring is known when the decomposition involves only real Schur roots or a
single isotropic Schur root.

If the canonical decomposition of the dimension vector $\beta$ includes real
Schur roots and exactly one isotropic Schur root, then we call such dimension
vector almost prehomogenous. We study some properties of these dimension
vectors in Proposition 5.0.2. By applying Theorem A to a quiver without
relations, we obtain a description of the semi-invariant ring
$\operatorname{SI}(Q,\beta)$, where $\beta$ is almost prehomogeneous. This
yields the second main result of this article and corroborates, albeit in a
slightly weakened form, the result outlined in [PW18, Corollary 6.3].

######  Theorem B.

Let $Q$ be an acyclic quiver without any relations and let $\beta$ be a
dimension vector. Suppose that the canonical decomposition of $\beta$ contains
exactly one isotropic Schur root and that all the other roots occurring in the
decomposition are real Schur roots. Then the semi-invariant ring
$\operatorname{SI}(Q,\beta)$ is a complete intersection.

Finally, we consider a more general case where the orbits of maximal dimension
are not necessarily of codimension one:

######  Theorem C.

If there is an irreducible component $\mathcal{C}\subseteq\operatorname{{\rm
rep}}_{\beta}(Q,R)$ such that $\mathcal{C}$ is not an orbit closure and the
coordinate ring $\mathbb{K}[\mathcal{C}]$ is a UFD, then the semi-invariant
ring $\operatorname{SI}(Q,\mathcal{C})$ is not multiplicity free.

A nice consequence of $\operatorname{SI}(Q,\mathcal{C})$ failing to be
multiplicity free is that there exists a dimension vector $\beta^{\prime}$
such that $\operatorname{{\rm rep}}_{\beta^{\prime}}(Q,R)$ contains infinitely
many non-isomorphic bricks $\\{B_{\ell}\\}_{\ell\in\Lambda}$, i.e., infinitely
many non-isomorphic modules $B_{\ell}$ with
$\mathrm{End}_{A}(B_{\ell})\cong\mathbb{K}$.

##  2\. Preliminaries

In this section, we recall some key notions on quiver representations and
their invariant theory. For more details, the reader is referred to [DW17].

A quiver $Q=(Q_{0},Q_{1})$ is a directed graph with $Q_{0}$ being the set of
vertices, and $Q_{1}$ being the set of arrows. Such a quiver comes equipped
with two maps $t,h:Q_{1}\rightarrow Q_{0}$ with $ta\coloneqq t(a)$, the tail
of the arrow $a$, and $ha\coloneqq h(a)$, the head of the arrow.

A path $p$ in $Q$ is a sequence of arrows $p=a_{r}\cdots a_{1}$ such that
$ha_{i}=ta_{i+1}$ for $1\leq i\leq r-1$. The path $p$ is said to have length
$r$. We can extend the functions $h,t$ to paths by defining $hp=ha_{r}$ and
$tp=ta_{1}$. For each $x\in Q_{0}$, we also define a trivial path $e_{x}$ of
length $0$ with $he_{x}=te_{x}=x$.

The path algebra $\mathbb{K}Q$ of the quiver $Q$ is constructed as follows. As
a $\operatorname{\mathbb{K}}$-vector space, it has a basis given by all paths
in $Q$ (including the trivial paths). The multiplication of two paths $p,q$ is
given by their concatenation, written as $qp$, if $h(p)=t(q)$, and is zero
otherwise.

Recall that an ideal $R$ of $\mathbb{K}Q$ is _admissible_ if there exists a
positive integer $t$ such that $R$ contains all paths of length $t$ and $R$
can be generated by elements that can be written as linear combination of
paths of length at least two. The importance in admissible ideals lies in the
well known fact that any basic finite dimensional algebra over
$\operatorname{\mathbb{K}}$ can be realized as an admissible quotient of a
path algebra of a finite quiver over $\operatorname{\mathbb{K}}$.

An element $u$ in $\mathbb{K}Q$ is said to be _uniform_ if it is a linear
combination of parallel paths, that is, paths sharing the same head and the
same tail. We note that an admissible ideal can always be generated by uniform
elements. Recall that a representation $V$ of $Q$ over
$\operatorname{\mathbb{K}}$ is given by $V=(\\{V_{x}\\}_{x\in
Q_{0}},\\{V(a)\\}_{a\in Q_{1}})$ where the $V_{x}$ are finite dimensional
$\operatorname{\mathbb{K}}$-vector spaces and $V(a):V_{ta}\to V_{ha}$ is a
$\operatorname{\mathbb{K}}$-linear map for each arrow $a\in Q_{1}$. For a path
$p=a_{r}\cdots a_{1}$ in $Q$ and a representation $V$, we define $V(p)$ to be
the composition $V(p)=V(a_{r})\cdots V(a_{1})$, which is a map from $V_{tp}$
to $V_{hp}$. If $u=\sum_{i=1}^{t}\lambda_{i}p_{i}$ is a uniform element, then
we define $V(u)=\sum_{i=1}^{t}\lambda_{i}V(p_{i})$. A representation $V$ is
said to be annihilated by $R$ if $V(u)=0$ for every uniform element $u$ in
$R$.

In this paper, we assume that our algebra $A$ is given by $A=\mathbb{K}Q/R$
where $Q$ is a finite quiver and $R$ is an admissible ideal of $\mathbb{K}Q$.
We denote by ${\rm mod}\,A$ the category of finite dimensional left
$A$-modules. It is well known that the category ${\rm mod}\,A$ is equivalent
to the category ${\operatorname{{\rm rep}}}(Q,R)$ of finite dimensional
$\operatorname{\mathbb{K}}$-representations of $Q$ annihilated by $R$.

The dimension vector $\beta\in\mathbb{N}^{Q_{0}}$ of a representation
$V\in\operatorname{{\rm rep}}(Q,R)$ is defined by
$\beta(x)=\dim_{\mathbb{K}}V_{x}$. For each $x\in Q_{0}$ we fix a basis for
the $\beta(x)$ dimensional vector space $V_{x}$, and define

$\operatorname{{\rm rep}}_{\beta}(Q,R)\coloneqq\left\\{(V(a))_{a\in
Q_{1}}\in\prod_{a\in
Q_{1}}\operatorname{Hom}_{\mathbb{K}}(V_{ta},V_{ha})\;\bigg{|}\;V(u)=0,\text{
for every uniform }u\in R\right\\}.$

We have a regular action of the algebraic group

$\operatorname{GL}_{\beta}\coloneqq\prod_{x\in
Q_{0}}\operatorname{GL}_{\beta(x)}$

on $\operatorname{{\rm rep}}_{\beta}(Q,R)$ given by $g\cdot
V=(g_{ha}V(a)g_{ta}^{-1})_{a\in Q_{1}}$ where $(V(a))_{a\in
Q_{1}}\in\operatorname{{\rm rep}}_{\beta}(Q,R)$ and $(g_{x})_{x\in
Q_{0}}\in\operatorname{GL}_{\beta}$. The orbits correspond to isomorphism
classes of $\beta$-dimensional representations of $\mathbb{K}Q/R$ modules.

The representation space $\operatorname{{\rm rep}}_{\beta}(Q,R)$ is an affine
variety which is not necessarily irreducible. Consider one of its irreducible
components $\mathcal{C}$. The action of $\operatorname{GL}_{\beta}$ on
$\operatorname{{\rm rep}}_{\beta}(Q,R)$ induces an action on the elements of
the coordinate ring $\mathbb{K}[\mathcal{C}]$ as follows: for every
$g\in\operatorname{GL}_{\beta},f\in\mathbb{K}[\mathcal{C}]$ and
$x\in\mathcal{C}$, we have $g\cdot f(x)\coloneqq f(g^{-1}x)$. Then,

$\operatorname{SI}(Q,\mathcal{C})\coloneqq\mathbb{K}[\mathcal{C}]^{\operatorname{SL}_{\beta}},\text{
where }\operatorname{SL}_{\beta}\coloneqq\prod_{x\in
Q_{0}}\operatorname{SL}_{\beta(x)}$

is called the ring of semi-invariant functions on $\mathcal{C}$.

For every weight $\sigma\in\mathbb{Z}^{Q_{0}}$, we can define a multiplicative
character of $\operatorname{GL}_{\beta}$ as follows:

$\begin{array}[]{ccccc}\chi_{\sigma}&:&\operatorname{GL}_{\beta}&\longrightarrow&\mathbb{K}^{*}\\\
&&(g_{x})_{x\in Q_{0}}&\longmapsto&\prod_{x\in
Q_{0}}\det(g_{x})^{\sigma(x)}\end{array}$

The semi-invariant ring $\operatorname{SI}(Q,\mathcal{C})$ has a decomposition
into weight spaces,

$\operatorname{SI}(Q,\mathcal{C})=\bigoplus_{\sigma}\operatorname{SI}(Q,\mathcal{C})_{\sigma}$

where

$\operatorname{SI}(Q,\mathcal{C})_{\sigma}\coloneqq\\{f\in\mathbb{K}[\mathcal{C}]\mid
g\cdot f=\chi_{\sigma}(g)\,f,\text{ for every
}g\in\operatorname{GL}_{\beta}\\}$

is called the space of semi-invariants of weight $\sigma$. A semi-invariant in
a given weight space is said to be _homogeneous_.

##  3\. Codimension one orbits and UFD

Let $\beta$ be a dimension vector, and let $\mathcal{C}\subseteq{\rm
rep}_{\beta}(Q,R)$ be an irreducible component.

######  Lemma 3.0.1.

The units of the coordinate ring $\mathbb{K}[\mathcal{C}]$ are the elements of
$\mathbb{K}^{*}$.

###### Proof.

Clearly, any element of $\mathbb{K}^{*}$ is a unit in
$\mathbb{K}[\mathcal{C}]$.

Let $u\in\mathbb{K}[\mathcal{C}]$ be a unit. Therefore, there exists a unit
$v\in\mathbb{K}[\mathcal{C}]$ such that $uv=1$, i.e., $u(x)v(x)=1$, for every
$x\in\mathcal{C}$. By [SR10, Lemma 5.4, Chap. 6], we know that
$u\in\operatorname{SI}(Q,\mathcal{C})_{\chi}$ for some character $\chi$.

Thus $u(g^{-1}x)=\chi(g)u(x)$, for every $x\in\mathcal{C}$ and every
$g\in\operatorname{GL}_{\beta}$. Multiplying by $v(x)$, we get,

$u(g^{-1}x)v(x)=\chi(g)u(x)v(x)=\chi(g)$

Take $x=0$, i.e., the point corresponding to the semi-simple representation.
Therefore, $\chi(g)=u(0)v(0)=1$ for all $g\in\operatorname{GL}_{\beta}$ and
$\chi$ is the trivial character. In particular, $u$ is an invariant function.
Hence, $u\in\mathbb{K}^{*}$. ∎

The next proposition gives a characterisation of the irreducible elements of
$\operatorname{SI}(Q,\mathcal{C})$ when the coordinate ring
$\mathbb{K}[\mathcal{C}]$ is a UFD. Recall an affine subvariety $W$ of
$\mathcal{C}$ is _pure_ if all of its irreducible components have the same
dimension.

######  Proposition 3.0.2.

Assume that $\mathbb{K}[\mathcal{C}]$ is a UFD and let $W\subseteq\mathcal{C}$
be a affine subvariety. Then $W$ is a pure codimension one
$\operatorname{GL}_{\beta}$-stable subvariety if and only if $W$ is the zero
set of a homogeneous semi-invariant, i.e., $W=\mathcal{V}(f)$, for some
homogeneous $f\in\operatorname{SI}(Q,\mathcal{C})$.

###### Proof.

The sufficiency is trivial. For the other direction, since
$\operatorname{GL}_{\beta}$ is connected, it is sufficient to consider the
case where $W$ is irreducible. Since $\mathbb{K}[\mathcal{C}]$ is a UFD and
$W$ is of codimension one and irreducible, we conclude that
$W=\mathcal{V}(f)$, for some irreducible element
$f\in\mathbb{K}[\mathcal{C}]$. Since $W$ is
$\operatorname{GL}_{\beta}$-stable, we have $\mathcal{V}(g\cdot
f)=\mathcal{V}(f)$, for every $g\in\operatorname{GL}_{\beta}$. By Hilbert’s
Nullstellensatz, we get, $\sqrt{(g\cdot f)}=\sqrt{(f)}$. Using the fact that
$\mathbb{K}[\mathcal{C}]$ is a UFD, and that $g\cdot f$ and $f$ are
irreducible, and hence prime, we see that $g\cdot f$ and $f$ are associates,
i.e., $g\cdot f=u_{g}f$, for some unit $u_{g}\in\mathbb{K}[\mathcal{C}]$.

By Lemma 3.0.1, we know that $u_{g}\in\mathbb{K}^{*}$. It follows that
$\chi:\operatorname{GL}_{\beta}\rightarrow\mathbb{K}^{*}$, defined by
$\chi(g)\coloneqq u_{g}$ is a multiplicative character of
$\operatorname{GL}_{\beta}$, and hence, $f$ is a homogeneous semi-invariant. ∎

Let $\mathcal{C}$ be an irreducible component of $\operatorname{{\rm
rep}}_{\beta}(Q,R)$. For the rest of this section we assume that
$\mathbb{K}[\mathcal{C}]$ is a UFD and that $\mathcal{C}$ is not an orbit
closure and $\mathcal{C}$ contains a codimension one orbit.

By upper semicontinuity of orbit dimension, it follows that the union $U$ of
all codimension one orbits forms an open subset of $\mathcal{C}$. Let
$\mathcal{I}$ be a complete set of representatives of these orbits. Then
$U=\bigcup_{M\in\mathcal{I}}\mathcal{O}(M)$.

Our goal is to describe the semi-invariant ring
$\operatorname{SI}(Q,\mathcal{C})$ in the above setting. By Hilbert’s
finiteness theorem the ring $\operatorname{SI}(Q,\mathcal{C})$ is a finitely
generated $\mathbb{K}$-algebra (see [DW17, Corollary 9.2.8]). Therefore, we
first determine a set of generators, and then the algebraic relations among
these generators.

###  3.1. Generators

In this subsection, we describe the generators of the semi-invariant ring
$\operatorname{SI}(Q,\mathcal{C})$. Decompose the complement $U^{\mathsf{c}}$
of $U$ into irreducible components, namely

$U^{\mathsf{c}}=\mathcal{Z}_{1}\cup\dotsb\cup\mathcal{Z}_{r}\cup\mathcal{Z}_{r+1}\cup\dotsb\cup\mathcal{Z}_{t}$

where we assume that $\mathrm{codim}\,\mathcal{Z}_{i}=1$, for $1\leq i\leq r$,
and $\mathrm{codim}\,\mathcal{Z}_{i}>1$, for $i>r$. We note that $r$ may be
zero. With the above notation we are ready to prove the following key lemma.

######  Lemma 3.1.1.

There exists at least one weight $\nu$ such that
$\dim_{\mathbb{K}}\operatorname{SI}(Q,\mathcal{C})_{\nu}\geq 2$.

###### Proof.

Suppose, by way of contradiction, that there does not exist any weight
$\sigma$ such that
$\dim_{\mathbb{K}}\operatorname{SI}(Q,\mathcal{C})_{\sigma}\geq 2$. Since
$\mathbb{K}[\mathcal{C}]$ is a UFD and the orbits $\mathcal{O}(M)$, for
$M\in\mathcal{I}$, are of codimension one, therefore, for every
$M\in\mathcal{I}$, we can write
$\overline{\mathcal{O}(M)}=\mathcal{V}(f_{M})$, for some
$f_{M}\in\mathbb{K}[\mathcal{C}]$. By Proposition 3.0.2, we know that $f_{M}$
is an irreducible semi-invariant, for every $M\in\mathcal{I}$. By our
assumption, the weights of the $f_{M}$ are all distinct. By Hilbert’s
finiteness theorem, there exists a finite set of homogeneous semi-invariants
$x_{1},x_{2},\dotsc,x_{n}$ which generate $\operatorname{SI}(Q,\mathcal{C})$.
Clearly we may assume this set of generators is a linearly independent set.
Now, there exists some $M^{\prime}\in\mathcal{I}$ such that $f_{M^{\prime}}$
is not a scalar multiple of $x_{l}$, for any $l\in\\{1,2,\dotsc,n\\}$. Since
$f_{M^{\prime}}$ is irreducible, it cannot be a monomial in the $x_{i}$.
Therefore, $f_{M^{\prime}}$ is a linear combination of at least two distinct
monomials in the $x_{i}$. If $\nu$ denotes the weight of these two semi-
invariants, we see that
$\dim_{\mathbb{K}}\,\operatorname{SI}(Q,\mathcal{C})_{\nu}\geq 2$. ∎

For $1\leq i\leq r$, $\mathcal{Z}_{i}$ is an irreducible closed set of
codimension one in $\mathcal{C}$. Since $\mathbb{K}[\mathcal{C}]$ is a UFD, we
know there exists $f_{i}\in\mathbb{K}[\mathcal{C}]$ such that
$\mathcal{Z}_{i}=\mathcal{V}(f_{i})$, for $1\leq i\leq r$.

######  Lemma 3.1.2.

The irreducible elements $f_{1},f_{2},\dotsc,f_{r}$ are homogeneous elements
of $\operatorname{SI}(Q,\mathcal{C})$.

###### Proof.

Since $\operatorname{GL}_{\beta}$ is connected (hence irreducible), each orbit
(of $\mathcal{Z}_{1}\cup\cdots\cup\mathcal{Z}_{r}$) is irreducible. Hence, we
see that the elements of $\operatorname{GL}_{\beta}$ fix each irreducible
component $\mathcal{Z}_{i}$. Now, we apply Proposition 3.0.2. ∎

We define the cone of weights

$\Sigma(Q,\mathcal{C})\coloneqq\\{\sigma\in\mathbb{Z}^{Q_{0}}\mid\operatorname{SI}(Q,\mathcal{C})_{\sigma}\not=0\\}$

We have a partial ordering on the weights $\sigma\in\Sigma(Q,\mathcal{C})$.
For $\sigma_{1},\sigma_{2}\in\Sigma(Q,\mathcal{C})$, we say
$\sigma_{1}\succeq\sigma_{2}$, if there exists a semi-invariant in
$\operatorname{SI}(Q,\mathcal{C})_{\sigma_{1}}$ divisible by some semi-
invariant in $\operatorname{SI}(Q,\mathcal{C})_{\sigma_{2}}$. Now, it follows
from Lemma 3.1.1 that there exists a weight $\chi$ such that
$\operatorname{SI}(Q,\mathcal{C})_{\chi}$ is at least two dimensional and
$\chi$ is minimal with respect to this property. Let us fix such a weight
$\chi$.

######  Lemma 3.1.3.

By possibly reducing $U$ to a smaller open non-empty set, we may assume that
$U^{\mathsf{c}}$ has $r$ irreducible components of codimension one:
$\mathcal{V}(f_{1}),\mathcal{V}(f_{2}),\dotsc,\mathcal{V}(f_{r})$ where $r\geq
2$ and such that there are two linearly independent monomials in
$f_{1},f_{2},\dots,f_{r}$ having weight $\chi$.

###### Proof.

Starting with the given set $U$, suppose that no two linearly independent
monomials in the $f_{i}$ share same weight. Take two linearly independent
semi-invariants of weight $\chi$, say $y$ and $z$. Notice that the
$\operatorname{GL}_{\beta}$-stable irreducible codimension one closed subsets
in $\mathcal{C}$ are precisely the $\mathcal{V}(f_{i})$ and the orbit closures
of representations in $\mathcal{I}$. We consider the vanishing sets of $y$ and
$z$ and decompose them into their irreducible components as follows:

$\displaystyle\mathcal{V}(y)=\cup_{i\in
S}\mathcal{V}(f_{i})\,\bigcup\,\cup_{M\in\mathcal{I}_{1}}\overline{\mathcal{O}(M)}$
$\displaystyle\mathcal{V}(z)=\cup_{i\in
T}\mathcal{V}(f_{i})\,\bigcup\,\cup_{M\in\mathcal{I}_{2}}\overline{\mathcal{O}(M)}$

where $S,T\subseteq\\{1,2,\dotsc,r\\}$ and
$\mathcal{I}_{1},\mathcal{I}_{2}\subseteq\mathcal{I}$ are finite. By our
assumption, we may assume that $y$ is not a scalar multiple of a monomial in
the $f_{i}$, hence, $\mathcal{I}_{1}$ is non-empty.

Now, we shrink the open set $U$ by removing the orbit closures
$\overline{\mathcal{O}(M)}$, for $M\in\mathcal{I}_{1}\cup\mathcal{I}_{2}$,
from $U$. Therefore, we get a new open set in $\mathcal{C}$ which is a union
of orbits of codimension one, say
$U^{\prime}=\bigcup_{M\in\mathcal{I}^{\prime}}\mathcal{O}(M)$, and a new set
of semi-invariants $f_{j}$ with the desired property that there are at least
two monomials in the $f_{j}$ which are semi-invariants of the same weight. In
particular, this shrinking procedure allows us to assume that $r\geq 2$. ∎

From now on, we assume that the open set $U$ satisfies Lemma 3.1.3 and that
$\mathcal{I}$ denotes a complete set of representatives of the orbits in $U$.
Furthermore, the $\mathcal{V}(f_{i})$ for $1\leq i\leq r$ are the codimension
one irreducible components of $U^{\mathsf{c}}$. Observe that if
$\chi\succ\sigma$, then the weight space
$\operatorname{SI}(Q,\mathcal{C})_{\sigma}$ is spanned by a single monomial in
$f_{i}$.

######  Definition 3.1.1.

Two linearly independent monomials $p,q$ in the $f_{i}$, of equal weight form
a monomial pair, denoted by $(p,q)$.

######  Lemma 3.1.4.

Let $(p,q)$ be a monomial pair. Then for every $M\in\mathcal{I}$, we have that
$p(M)\neq 0$ and $q(M)\neq 0$.

###### Proof.

Recall that $p$ and $q$ are monomials in the $f_{i}$. Since the $f_{i}$ are
semi-invariants and $\mathcal{O}(M)\subseteq\overline{\mathcal{O}(M)}$ is
dense and irreducible, if $p$ vanishes on some $\mathcal{O}(M)$, and hence on
$\overline{\mathcal{O}(M)}$, then there exists $j$ such that $f_{j}$ divides
$p$ and $f_{j}(\overline{\mathcal{O}(M)})=0$. This cannot happen by the choice
of the $f_{i}$. Therefore, the statement follows. ∎

We fix a monomial pair $(p,q)$, and for $\alpha\in\mathbb{K}$, we define
$h_{\alpha}\coloneqq\alpha p-q$. Then, $h_{\alpha}$ is a non-constant semi-
invariant of weight $\chi$.

######  Remark 3.1.5.

By Lemma 3.1.4, we know $p$ and $q$ do not vanish anywhere on $U$. Therefore,
for every $M\in\mathcal{I}$, we can find $\alpha$ such that $h_{\alpha}$
vanishes on $\overline{\mathcal{O}(M)}$. In particular, if
$0=h_{\alpha}(M)=\alpha p(M)-q(M)$, then we get,

$\alpha=\frac{q(M)}{p(M)}.$

By Lemma 3.1.4, we know that $\alpha\not=0$. Therefore, we get a map

$\begin{array}[]{cccc}\phi:&\mathcal{I}&\longrightarrow&\mathbb{K}^{*}\\\
\textbf{ }&M&\longmapsto&\alpha=\frac{q(M)}{p(M)}\end{array}$

######  Proposition 3.1.6.

If $\alpha\in\operatorname{Im}(\phi)$, then $h_{\alpha}$ is irreducible and
there exists $M\in\mathcal{I}$ such that
$\mathcal{V}(h_{\alpha})=\overline{\mathcal{O}(M)}$.

###### Proof.

Note that $h_{\alpha}\in\operatorname{SI}(Q,\mathcal{C})_{\chi}$. Since
$\alpha\in\operatorname{Im}(\phi)$, we have $\alpha=\frac{q(M)}{p(M)}$, for
some $M\in\mathcal{I}$. Therefore,
$\mathcal{V}(h_{\alpha})\supseteq\overline{\mathcal{O}(M)}$. Since both
$\mathcal{V}(h_{\alpha})$ and $\overline{\mathcal{O}(M)}$ are of codimension
one in $\mathcal{C}$, we conclude,
$\mathcal{V}(h_{\alpha})=\overline{\mathcal{O}(M)}$, and hence, $h_{\alpha}$
is irreducible. ∎

By combining Proposition 3.1.6 with the above remark, we get the following
corollary.

######  Corollary 3.1.7.

For every $M\in\mathcal{I}$,
$\overline{\mathcal{O}(M)}=\mathcal{V}(h_{\alpha})$, where
$\alpha=\frac{q(M)}{p(M)}$, i.e. $\alpha\in\operatorname{Im}(\phi)$.

######  Remark 3.1.8.

From the above, every irreducible homogeneous semi-invariant is an associate
of some $f_{i}$ with $1\leq i\leq r$ or of some $h_{\alpha}$ with
$\alpha\in\operatorname{Im}(\phi)$.

######  Corollary 3.1.9.

For a monomial pair $(p,q)$, the monomials $p$ and $q$ are coprime.

###### Proof.

Both $p$ and $q$ are monomials in the $f_{i}$. If they are not coprime then
there exists $f_{j}$ dividing both. But then $p/f_{j}$ and $q/f_{j}$ would be
linearly independent monomials of the same weight. This violates the
minimality of $\chi$. ∎

######  Proposition 3.1.10.

The irreducible elements $f_{1},f_{2},\dotsc,f_{r}$ generate
$\operatorname{SI}(Q,\mathcal{C})$ as a $\operatorname{\mathbb{K}}$-algebra.

###### Proof.

Let $h$ be a non-constant irreducible homogeneous semi-invariant. Consider the
vanishing set $\mathcal{V}(h)$. Then, $\mathcal{V}(h)$ is an irreducible
closed set of codimension one. Therefore, either
$\mathcal{V}(h)=\mathcal{V}(f_{i})$, for some $1\leq i\leq r$, or
$\mathcal{V}(h)=\overline{\mathcal{O}(M)}=\mathcal{V}(h_{\alpha})$, for some
$M\in\mathcal{I}$. Using the fact that $h,f_{i}$ and $h_{\alpha}$, for
$\alpha\in\operatorname{Im}(\phi)$, are irreducible and hence prime (since
$\mathbb{K}[\mathcal{C}]$ is a UFD), we conclude that in the former case, $h$
and $f_{i}$ are associates and in the latter case, $h$ and $h_{\alpha}$ are
associates, for some $\alpha\in\operatorname{Im}(\phi)$. Thus either $h$ is a
scalar multiple of some $f_{i}$ or is a linear combination of two monomials in
the $f_{i}$. Therefore, we conclude $f_{1},f_{2},\dotsc,f_{r}$ generate
$\operatorname{SI}(Q,\mathcal{C})$. ∎

###  3.2. Weight Spaces and Relations

In this section, we describe the structure of the non-trivial weight spaces
and determine the algebraic relations that arise in different weight spaces.

######  Proposition 3.2.1.

The minimal weight $\chi$ is the unique minimal weight, with respect to the
partial ordering defined on the weights in $\Sigma(Q,\mathcal{C})$, such that
$\dim_{\mathbb{K}}\operatorname{SI}(Q,\mathcal{C})_{\chi}\geq 2$.

###### Proof.

Let $\chi^{\prime}$ be another such minimal weight such that
$\dim_{\mathbb{K}}\operatorname{SI}(Q,\mathcal{C})_{\chi^{\prime}}\geq 2$.
Choose a monomial pair $(p^{\prime},q^{\prime})$ of weight $\chi^{\prime}$.
For $\alpha^{\prime}\in\mathbb{K}$, we define,

$h^{\prime}_{\alpha^{\prime}}=\alpha^{\prime}p^{\prime}-q^{\prime}$

and a map

$\begin{array}[]{cccc}\phi^{\prime}:&\mathcal{I}&\longrightarrow&\mathbb{K}^{*}\\\
\textbf{
}&M&\longmapsto&\alpha^{\prime}=\frac{q^{\prime}(M)}{p^{\prime}(M)}\end{array}$

Arguing as above, we see that if
$\alpha^{\prime}\in\operatorname{Im}(\phi^{\prime})$, then
$h^{\prime}_{\alpha^{\prime}}$ is irreducible and the vanishing set
$\mathcal{V}(h^{\prime}_{\alpha^{\prime}})=\overline{\mathcal{O}(M)}$, for
some $M\in\mathcal{I}$.

Using Corollary 3.1.7, we get,
$\mathcal{V}(h^{\prime}_{\alpha^{\prime}})=\mathcal{V}(h_{\alpha})$, for some
$\alpha\in\operatorname{Im}(\phi)$. Therefore, by using the fact that
$\mathbb{K}[\mathcal{C}]$ is a UFD and the fact that $h_{\alpha}$ and
$h^{\prime}_{\alpha^{\prime}}$ are irreducible, for
$\alpha\in\operatorname{Im}(\phi)$ and
$\alpha^{\prime}\in\operatorname{Im}(\phi^{\prime})$, respectively, and hence
prime, we conclude that $h_{\alpha}$ and $h^{\prime}_{\alpha^{\prime}}$ are
associates and hence, $\chi^{\prime}=\chi$. ∎

In fact, we will prove that
$\dim_{\mathbb{K}}\operatorname{SI}(Q,\mathcal{C})_{\chi}=2$, but first we
will need some technical lemmas. The first lemma is a well-known observation.

######  Lemma 3.2.2.

Let $Z\in\mathbb{K}[\mathcal{C}]$ with vanishing set

$\mathcal{V}(Z)=\bigcup_{i=1}^{m}\mathcal{V}(Z_{i})$

where the $Z_{i}\in\mathbb{K}[\mathcal{C}]$ are non-associate primes. Then
$Z=\prod_{i=1}^{m}Z_{i}^{a_{i}}$, where $a_{i}\in\mathbb{Z}_{>0}$.

For the rest of this section we fix a monomial pair $(p,q)$ of weight $\chi$.
For any weight $\sigma\in\Sigma(Q,\mathcal{C})$, we can write $\sigma$ as
$\sigma=n\chi+\delta$, for some maximal $n\in\mathbb{Z}_{\geq 0}$ and for some
weight $\delta$, and $\sigma\succeq n\chi$. We will refer to $\delta$ as the
_remainder_ of $\sigma$ and denote it as ${\operatorname{rem}(\sigma)}$. It
follows from Proposition 3.2.1 that
$\dim_{\mathbb{K}}\operatorname{SI}(Q,\mathcal{C})_{{\operatorname{rem}(\sigma)}}\leq
1$.

######  Lemma 3.2.3.

Let $\sigma\in\Sigma(Q,\mathcal{C})$ be a weight and write
$\sigma=n\chi+{\operatorname{rem}(\sigma)}$. Suppose
$\operatorname{SI}(Q,\mathcal{C})_{{\operatorname{rem}(\sigma)}}$ is generated
by the monomial $X$. Observe that $\\{p^{i}q^{j}X\\}_{i+j=n}$ are distinct
monomials in the $f_{i}$ in $\operatorname{SI}(Q,\mathcal{C})_{\sigma}$.
Assume that $W$ is another monomial in the $f_{i}$ with
$W\in\operatorname{SI}(Q,\mathcal{C})_{\sigma}$ that is not an associate to
any of the elements in $\\{p^{i}q^{j}X\\}_{i+j=n}$. For
$\mu\in\mathbb{K}^{*}$, define

$g_{\mu}\coloneqq\mu p^{n}X-W.$

Then, there exists $\gamma\in\mathbb{K}^{*}$ such that $h_{\alpha}$ divides
$g_{\gamma}$ for some $\alpha\in\operatorname{Im}(\phi)$.

###### Proof.

By Remark 3.1.8, it suffices to find an element $\gamma\in\mathbb{K}^{*}$ such
that there exists an irreducible factor $z$ of $g_{\gamma}$ such that $z$ is
not an associate of any $f_{i}$.

If not, then there exists a monomial in the $f_{i}$, say
$f_{i_{1}}^{s_{1}}f_{i_{2}}^{s_{2}}\dotsb f_{i_{t}}^{s_{t}}$ where
$i_{k}\in\\{1,2,\dotsc,r\\}$, which is an associate of $g_{\mu}$ for
infinitely many $\mu\in\mathbb{K}^{*}$. Now choose two such elements
$\mu_{1},\mu_{2}\in\mathbb{K}^{*}$. Therefore, we have

$\mu_{1}p^{n}X-W=\lambda_{1}f_{i_{1}}^{s_{1}}f_{i_{2}}^{s_{2}}\dotsb
f_{i_{t}}^{s_{t}}\;\text{and}\;\mu_{2}p^{n}X-W=\lambda_{2}f_{i_{1}}^{s_{1}}f_{i_{2}}^{s_{2}}\dotsb
f_{i_{t}}^{s_{t}}$

After multiplying the first equation by $\lambda_{2}$ and the second equation
by $\lambda_{1}$, and subtracting them, we get
$(\lambda_{1}-\lambda_{2})W=(\lambda_{1}\mu_{2}-\lambda_{2}\mu_{1})p^{n}X$
showing that $W$ and $p^{n}X$ are associates, which contradicts our
assumption. ∎

In the next proposition we calculate the dimension of the weight space
$\operatorname{SI}(Q,\mathcal{C})_{\chi}$.

######  Proposition 3.2.4.

The weight space $\operatorname{SI}(Q,\mathcal{C})_{\chi}$ is $2$ dimensional.

###### Proof.

Let $\\{p,q,X_{1},X_{2},\dotsc,X_{t}\\}$ be all the monomials in the $f_{i}$
contained in the weight space $\operatorname{SI}(Q,\mathcal{C})_{\chi}$.

Fix a monomial $X_{i}$, where $i\in\\{1,2,\dotsc,t\\}$. For
$\mu\in\mathbb{K}^{*}$, define

$g_{\mu}\coloneqq\mu p-X_{i}.$

By Lemma 3.2.3, we know that there exists $\gamma\in\mathbb{K}^{*}$ and
$\alpha\in\operatorname{Im}(\phi)$ such that $h_{\alpha}$ divides
$g_{\gamma}$. Since $g_{\gamma}$ and $h_{\alpha}$ are semi-invariants of the
same weight $\chi$, it follows that $g_{\gamma}$ and $h_{\alpha}$ are
associates. Recall that $h_{\alpha}$ is a linear combination of $p$ and $q$.
Hence, it follows that $X_{i}$ can be written as a linear combination of $p$
and $q$. By definition, $p$ and $q$ are linearly independent. The statement
follows. ∎

######  Lemma 3.2.5.

The monomials $p$ and $q$ are algebraically independent.

###### Proof.

Assume that there is some algebraic dependence relation among $p$ and $q$,
which we may assume to be homogeneous. Let this relation be

$c_{p}p^{m}+\sum_{i+j=m}c_{ij}p^{i}q^{j}+c_{q}q^{m}=0,$

where $c_{p},c_{q}\in\mathbb{K}^{*}$, $c_{ij}\in\operatorname{\mathbb{K}}$ and
$l,i,j,m\in\mathbb{Z}_{\geq 0}$. Then we get,

$c_{p}p^{m}+\sum_{i,j}c_{ij}p^{i}q^{j}=-c_{q}q^{m}$

which contradicts Corollary 3.1.9. ∎

######  Proposition 3.2.6.

Let $\sigma\in\Sigma(Q,\mathcal{C})$ be a weight and write
$\sigma=n\chi+{\operatorname{rem}(\sigma)}$. Then,

$\dim_{\mathbb{K}}\,\operatorname{SI}(Q,\mathcal{C})_{\sigma}=\dim_{\mathbb{K}}\,\operatorname{SI}(Q,\mathcal{C})_{n\chi}=n+1.$

Moreover, $\\{p^{i}q^{j}X\\}_{i+j=n}$ where
$X\in\operatorname{SI}(Q,\mathcal{C})_{{\operatorname{rem}(\sigma)}}$, forms a
basis of $\operatorname{SI}(Q,\mathcal{C})_{\sigma}$.

###### Proof.

Let the one dimensional space
$\operatorname{SI}(Q,\mathcal{C})_{{\operatorname{rem}(\sigma)}}$ be generated
by $X$. Assume that the statement of the proposition does not hold, and let
$n$ be minimal such that
$\dim_{\mathbb{K}}\,\operatorname{SI}(Q,\mathcal{C})_{\sigma}>n+1$. Clearly,
$n>0$.

Observe that $\\{p^{j}q^{n-j}X\mid 0\leq j\leq n\\}$ are linearly independent
monomials in the $f_{i}$ contained in
$\operatorname{SI}(Q,\mathcal{C})_{\sigma}$. Let $W$ be another monomial in
the $f_{i}$ in $\operatorname{SI}(Q,\mathcal{C})_{\sigma}$. For
$\mu\in\mathbb{K}^{*}$, define

$g_{\mu}\coloneqq\mu p^{n}X-W.$

Using Lemma 3.2.3, we can find some $\gamma\in\mathbb{K}^{*}$ such that
$g_{\gamma}=h_{\alpha}X^{\prime}$, for some $\alpha\in\operatorname{Im}(\phi)$
and
$X^{\prime}\in\operatorname{SI}(Q,\mathcal{C})_{(n-1)\chi+{\operatorname{rem}(\sigma)}}$.
By minimality of $n$, it follows that $X^{\prime}$ is a linear combination of
the $\\{p^{i}q^{n-1-i}X\\}$. This, together with the fact that $h_{\alpha}$ is
a linear combination of $p,q$ yields that $W$ is a linear combination of the
$\\{p^{i}q^{n-i}X\\}$. This proves the proposition. ∎

######  Proposition 3.2.7.

Suppose the map $\phi$ is not surjective. If
$\gamma\in\mathbb{K}^{*}\setminus\operatorname{Im}(\phi)$, then $h_{\gamma}$
gives rise to a non-trivial linear dependence relation between three distinct
pairwise coprime monomials in the $f_{i}$ of weight $\chi$.

###### Proof.

Assume first that $h_{\gamma}$ is divisible by some $h_{\alpha}$ where
$\alpha\in\operatorname{Im}{\phi}$. Since $h_{\gamma},h_{\alpha}$ have the
same weight, that means that they are associate. Hence, there exists
$\lambda\in\operatorname{\mathbb{K}}$ such that $h_{\gamma}=\lambda
h_{\alpha}$ which gives a linear dependence relation between $p,q$, since
$\alpha\neq\gamma$. This is a contradiction. It follows from Remark 3.1.8 that
$h_{\gamma}=\lambda Z$ for some monomial $Z$ in the $f_{i}$, which cannot be
in $\\{p,q\\}$. We get the relation $\lambda Z-\gamma p+q=0$. By Corollary
3.1.9, it follows that $p,q,Z$ are pairwise coprime. This proves the
proposition. ∎

Notice that $\operatorname{Im}(\phi)$ is co-finite in
$\operatorname{\mathbb{K}}$. To see this note that $\phi$ can be naturally
defined as a morphism $\phi:U\to\mathbb{P}^{1}$ with $\phi(M)=[p(M):q(M)]$
and, since $U$ is irreducible, so is $\operatorname{Im}(\phi)$. Since,
clearly, the latter is not a single point, this implies the claim. In
particular, the number of relations provided by Proposition 3.2.7 is finite.
We let
$\mathbb{K}^{*}\setminus\operatorname{Im}(\phi)=\\{\gamma_{1},\ldots,\gamma_{m}\\}$.
For each $1\leq i\leq m$, we let $H_{i}$ denote the relation of the form
$\lambda_{i}Z_{i}-\gamma_{i}p+q$ obtained in the proof of Proposition 3.2.7,
where $\lambda_{i}$ is some scalar and $Z_{i}$ some monomial of weight $\chi$
in the $f_{i}$. Note that $\\{p,q,Z_{1},\ldots,Z_{m}\\}$ are pairwise coprime.
In particular, $m+2\leq r$.

######  Proposition 3.2.8.

The ideal of relations among the $f_{i}$ is generated by the $m$ relations
$H_{1},\ldots,H_{m}$.

###### Proof.

Consider the decomposition of $\sigma$ as
$\sigma=n\chi+{\operatorname{rem}(\sigma)}$. Recall that
$\operatorname{SI}(Q,\mathcal{C})_{\sigma}$ has a basis given by
$\\{p^{i}q^{j}X\\}_{i+j=n}$. Let $Y$ be any monomial in the $f_{i}$ of weight
$\sigma$ which is not in this basis. It follows from Proposition 3.2.6 that
$Y=XY^{\prime}$ where $Y^{\prime}$ is some homogeneous polynomial $P(p,q)$ of
degree $n$ in $p,q$. Then $P$ factors as $P=\lambda p^{a}\prod_{\mu\in
A}h_{\mu}$ for some $a$ and some scalar $\lambda$.

Writing $Y^{\prime}$ as $Y^{\prime}=\prod_{i=1}^{r}f_{i}^{a_{i}}$ we have
$\prod_{i=1}^{r}f_{i}^{a_{i}}=\lambda p^{a}\prod_{\mu\in A}h_{\mu}$. By the
unique factorization, this implies that $h_{\mu}$ is, up to a scalar, a
monomial in the $f_{i}$ for each $\mu\in A$. Every such $\mu$ cannot lie in
$\operatorname{Im}(\phi)$ since none of the $f_{i}$ vanish anywhere on $U$. ∎

###  3.3. Main Result

Now that we have determined the generators, the structure of the non-trivial
weight spaces and the type of algebraic relations that arise, we are ready to
prove the main result.

######  Theorem 3.3.1.

The semi-invariant ring $\operatorname{SI}(Q,\mathcal{C})$ is a complete
intersection of codimension $\\#\\{\text{monomials of weight }\chi\\}-2$.

###### Proof.

Since the semi-invariant ring $\operatorname{SI}(Q,\mathcal{C})$ is a sub-ring
of $\mathbb{K}[\mathcal{C}]$, which is an integral domain, it follows that
$\operatorname{SI}(Q,\mathcal{C})$ is an integral domain. In particular, the
ideal of relations, call it $J$, is a prime ideal.

From above discussion and Proposition 3.2.8, we know that the relations
$H_{1},\dots,H_{m}$ generate $J$.

Consider the Jacobian $\mathcal{J}=\left(\frac{\partial
H_{i}}{\partial{f_{j}}}\right)_{\begin{subarray}{c}1\leq i\leq m\\\ 1\leq
j\leq r\end{subarray}}$ which is of size $m\times r$. Since
$\\{p,q,X_{1},\ldots,X_{m}\\}$ are pairwise coprime, we see that this matrix
has rank $m$. Hence, we conclude that $\operatorname{SI}(Q,\mathcal{C})$ is a
complete intersection of codimension $\\#\\{\text{monomials of weight
}\chi\\}-2$. ∎

##  4\. Examples

In the previous section, we saw that under the UFD assumption and when the
maximal orbits have codimension one, the semi-invariant ring
$\operatorname{SI}(Q,\mathcal{C})$ is a complete intersection and is not
multiplicity free. However, just having an irreducible component with maximal
orbits of codimension one might not give us this. To illustrate this, we
present the following example.

###  4.1. Example 1:

Consider the algebra given by the following quiver $Q$,

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}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
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with $R=\langle x^{2},y^{2},xy,yx\rangle$, and consider the dimension vector
$\beta=(2)$. The algebra $A=\operatorname{\mathbb{K}}Q/R$ is a string algebra.
Notice that $\operatorname{{\rm rep}}_{\beta}(Q,R)$ has an irreducible
component $\mathcal{C}$ containing the following family of non-isomorphic band
modules $\\{M_{\lambda}\\}_{\lambda\not=0}$

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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
9.59444pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{} {{}{{}{}}{}}{{}{{}{}}{}}{{}{}}{{}}
{{}{{}{}}{}}{{{}}{{}}}{{}}{{}{{}{}}{}}{{{}}{{}}}{ {}{}{}}{}{{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{}{}{}}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.76804pt}{4.51888pt}\pgfsys@curveto{-31.58878pt}{49.23442pt}{-29.48842pt}{-56.55385pt}{-3.87134pt}{-12.18466pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.50002}{0.86601}{-0.86601}{0.50002}{-3.77135pt}{-12.01149pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.32553pt}{-9.22914pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\left[\begin{matrix}0&1\\\ 0&0\end{matrix}\right]}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}
{{}{{}{}}{}}{{}{{}{}}{}}{{}{}}{{}}
{{}{{}{}}{}}{{{}}{{}}}{{}}{{}{{}{}}{}}{{{}}{{}}}{ {}{}{}}{}{{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{}{}{}}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{5.76804pt}{4.51888pt}\pgfsys@curveto{31.58878pt}{49.23442pt}{29.48842pt}{-56.55385pt}{3.87134pt}{-12.18466pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.50002}{0.86601}{-0.86601}{-0.50002}{3.77135pt}{-12.01149pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.43661pt}{-9.47914pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\left[\begin{matrix}0&\lambda\\\
0&0\end{matrix}\right]}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

It is easy to check that the coordinate ring $\mathbb{K}[\mathcal{C}]$ is not
a UFD (or we can also use Remark 6.0.4 (2)). Note that string algebras are
tame. Therefore, it follows that $\mathrm{codim}\,\mathcal{O}(M_{\lambda})=1$
for $\lambda\not=0$.

Since the algebra given by the above quiver with relations is a local algebra,
by using King’s criterion (see [Kin94] or [DW17, Proposition 9.8.3]) we
conclude that $\mathcal{C}$ cannot have any $\theta$-semi-stable point for any
non-trivial stability parameter $\theta\in\mathbb{Z}^{Q_{0}}$. Therefore, the
semi-invariant functions on $\mathcal{C}$ are the constant functions, i.e.,
the semi-invariant ring $\operatorname{SI}(Q,\mathcal{C})$ is trivial.

The subsequent two examples showcase quivers with relations that not only
fulfill the hypothesis of our main result but also yield semi-invariant rings
that are complete intersection but not polynomial rings. Moreover, the
codimension of the complete intersection can be arbitrarily large.

###  4.2. Example 2:

Consider the algebra $A$ given by the following quiver $Q$,

$Q:\leavevmode\hbox to88.83pt{\vbox
to77.29pt{\pgfpicture\makeatletter\hbox{\hskip 44.4166pt\lower-38.64583pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-44.4166pt}{-38.64583pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
5.13885pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.83331pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\,}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
5.13885pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${2}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
6.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${1}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${3}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
6.80554pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${5}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
5.13885pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.83331pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\,}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
5.13885pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${4}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{7.00554pt}{21.46603pt}\pgfsys@lineto{30.29997pt}{1.79091pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.76396}{-0.64526}{0.64526}{0.76396}{30.45273pt}{1.66188pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.1583pt}{15.14108pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{x_{2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-30.60551pt}{1.53401pt}\pgfsys@lineto{-7.31108pt}{21.20914pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.76396}{0.64526}{-0.64526}{0.76396}{-7.15833pt}{21.33817pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-27.15897pt}{15.14226pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{x_{1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-30.60551pt}{-4.38193pt}\pgfsys@lineto{-7.4055pt}{-4.38193pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.20552pt}{-4.38193pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.80586pt}{-0.74028pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{x_{3}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-30.60551pt}{-10.29787pt}\pgfsys@lineto{-7.31108pt}{-29.97299pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.76396}{-0.64526}{0.64526}{0.76396}{-7.15833pt}{-30.10202pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-27.15897pt}{-25.63112pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{x_{5}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{7.00554pt}{-4.38193pt}\pgfsys@lineto{30.20555pt}{-4.38193pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.40553pt}{-4.38193pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.80519pt}{-0.74028pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{x_{4}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{7.00554pt}{-30.22989pt}\pgfsys@lineto{30.29997pt}{-10.55476pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.76396}{0.64526}{-0.64526}{0.76396}{30.45273pt}{-10.42574pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.1583pt}{-25.62994pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{x_{6}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

with the relation $x_{1}x_{2}+x_{3}x_{4}+x_{5}x_{6}=0$ and consider the
dimension vector $\beta=(1,1,1,1,1)$. The following is a family of pairwise
non-isomorphic representations $\\{M_{\lambda}\\}_{\lambda\not=0,-1}$

$M_{\lambda}:\leavevmode\hbox to97.17pt{\vbox
to78.46pt{\pgfpicture\makeatletter\hbox{\hskip 48.5833pt\lower-39.22916pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-48.5833pt}{-39.22916pt}\pgfsys@invoke{
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5.13885pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.83331pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\,}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
5.13885pt\hfil&\hfil\hskip 32.19441pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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8.19444pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
8.19444pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.19444pt\hfil&\hfil\hskip 32.19441pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.19444pt\hfil&\hfil\hskip 32.19441pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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8.19444pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
5.13885pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
5.13885pt\hfil&\hfil\hskip 32.19441pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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8.19444pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{7.98544pt}{21.2167pt}\pgfsys@lineto{31.6815pt}{2.35269pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{1}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-31.99442pt}{2.10385pt}\pgfsys@lineto{-8.30057pt}{20.96759pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.78233}{0.62285}{-0.62285}{0.78233}{-8.14413pt}{21.09215pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{1}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-31.99442pt}{-4.57637pt}\pgfsys@lineto{-8.7944pt}{-4.57637pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.59442pt}{-4.57637pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.94443pt}{-2.2236pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{1}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-32.40341pt}{-10.93608pt}\pgfsys@lineto{-8.70735pt}{-29.8001pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@transformcm{0.78236}{-0.62282}{0.62282}{0.78236}{-8.55089pt}{-29.92465pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.2517pt}{-27.3565pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{1}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{8.39444pt}{-4.57637pt}\pgfsys@lineto{31.59445pt}{-4.57637pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.79443pt}{-4.57637pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{18.15276pt}{-2.2236pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\lambda}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{8.39444pt}{-30.04893pt}\pgfsys@lineto{32.08829pt}{-11.1852pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.78233}{0.62285}{-0.62285}{0.78233}{32.24472pt}{-11.06064pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.75058pt}{-27.70636pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{-1-\lambda}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

We have

$\mathbb{K}[\operatorname{{\rm
rep}}_{\beta}(Q,R)]\cong\mathbb{K}[x_{1},x_{2},\dotsc,x_{6}]/(x_{1}x_{2}+x_{3}x_{4}+x_{5}x_{6}).$

In order to understand the latter, let us consider the quiver $Q^{\prime}$ of
type $\widetilde{\mathbb{D}}_{4}$ with the dimension vector
$\mathbf{d}=(1,1,1,1,2)$ as follows,

$Q^{\prime}:\leavevmode\hbox to88.83pt{\vbox
to77.29pt{\pgfpicture\makeatletter\hbox{\hskip 44.4166pt\lower-38.64583pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-44.4166pt}{-38.64583pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
6.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${1}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
6.80554pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${3}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${5}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
6.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${2}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${4}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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By the calculation in [DW17, Subsection 10.10.2], we know that
$\mathbb{K}[\operatorname{{\rm
rep}}_{\beta}(Q,R)]\cong\operatorname{SI}(Q^{\prime},\mathbf{d})$. Therefore,
$\mathbb{K}[\operatorname{{\rm rep}}_{\beta}(Q,R)]$ is a UFD and hence
${\mathcal{C}}:=\operatorname{{\rm
rep}}_{\beta}(Q,R)\cong\mathcal{V}(x_{1}x_{2}+x_{3}x_{4}+x_{5}x_{6})$ is an
irreducible hypersurface in $\mathbb{A}^{6}$.

Notice that each $M_{\lambda}$ is a brick (i.e., has a one dimensional
endomorphism algebra), hence
$\dim\mathcal{O}(M_{\lambda})=\dim\operatorname{GL}_{\beta}-1=4$. These
$M_{\lambda}$ lie on orbits of maximal dimension in ${\mathcal{C}}$. Clearly,
${\mathcal{C}}$ has dimension $5$, so ${\mathcal{C}}$ has an open set
consisting of infinitely many codimension one orbits.

Since the dimension vector for $Q$ is chosen to be $\beta=(1,1,1,1,1)$, we get

$\operatorname{SI}(Q,{\mathcal{C}})=\mathbb{K}[\operatorname{{\rm
rep}}_{\beta}(Q,R)]^{\operatorname{SL}_{\beta}}=\mathbb{K}[\operatorname{{\rm
rep}}_{\beta}(Q,R)]=\mathbb{K}[x_{1},x_{2},\dotsc,x_{6}]/(x_{1}x_{2}+x_{3}x_{4}+x_{5}x_{6})$

Therefore, $\operatorname{SI}(Q,{\mathcal{C}})$ is a hypersurface, and in
particular a complete intersection.

Next, we would like to generalize the last example to show that the number of
relations occurring in the weight space can be arbitrary. The next example
provides an example where the ring of semi-invariants is a complete
intersection but not a hypersurface.

###  4.3. Example 3:

Consider the following quiver $Q_{n}$, $n\geq 2$,

${1}$${2}$${{Q_{n}:}}$${0}$${\vdots}$${{n+3}}$${{n+1}}$${{n+2}}$$\scriptstyle{x_{1}}$$\scriptstyle{x_{2}}$$\scriptstyle{x_{n+1}}$$\scriptstyle{x_{n+2}}$$\scriptstyle{y_{1}}$$\scriptstyle{y_{2}}$$\scriptstyle{y_{n+1}}$$\scriptstyle{y_{n+2}}$

with the relations $\\{x_{1}y_{1}+kx_{2}y_{2}+x_{k+2}y_{k+2}\\}_{k=1}^{n}$ and
dimension vector $\beta=(1,1,\dotsc,1)$. We have a following family of non-
isomorphic representations, $\\{M_{\lambda}\\}_{\lambda\in\mathcal{I}}$, where

$\mathcal{I}=\\{\lambda\in\mathbb{K}^{*}\mid\lambda\not=-\frac{1}{k},1\leq
k\leq n\\},$
${{\mathbb{K}}}$${{\mathbb{K}}}$${{M_{\lambda}:}}$${{\mathbb{K}}}$${\vdots}$${{\mathbb{K}}}$${{\mathbb{K}}}$${{\mathbb{K}}}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{\lambda}$$\scriptstyle{-1-(n-1)\lambda}$$\scriptstyle{-1-n\lambda}$

Consider the polynomial ring $\mathbb{K}[x_{i},y_{i}\mid 1\leq i\leq n+2]$ and
the ideal of this ring generated by the above given relations,

$I_{n}=(x_{1}y_{1}+kx_{2}y_{2}+x_{k+2}y_{k+2}\mid 1\leq k\leq n)$

Then, we have,

$\mathbb{K}[\operatorname{{\rm
rep}}_{\beta}(Q_{n},R)]=\mathbb{K}[x_{i},y_{i}\mid 1\leq i\leq n+2]/I_{n}$

######  Proposition 4.3.1.

The ring $T_{n}\coloneqq\mathbb{K}[x_{i},y_{i}\mid 1\leq i\leq n+2]/I_{n}$ is
an integral domain.

###### Proof.

Make a change of coordinates as follows:

$\displaystyle x_{i}=z_{2i-1}-z_{2i}$ $\displaystyle y_{i}=z_{2i-1}+z_{2i}$

for $1\leq i\leq(n+2)$. Then, for $1\leq k\leq n$,

$x_{1}y_{1}+kx_{2}y_{2}+x_{k+2}y_{k+2}=z_{1}^{2}-z_{2}^{2}+kz_{3}^{2}-kz_{4}^{2}+z_{2k+3}^{2}-z_{2k+4}^{2}$

and let

$p_{k}\coloneqq z_{2}^{2}-z_{1}^{2}+kz_{4}^{2}-kz_{3}^{2}+z_{2k+4}^{2}.$

Therefore,
$I_{n}=(z_{5}^{2}-p_{1},z_{7}^{2}-p_{2},\dotsc,z_{2n+3}^{2}-p_{n})$, and
hence, $T_{n}=\mathbb{K}[z_{1},z_{2},\dotsc,z_{2n+4}]/I_{n}$. We first note
that the $p_{k}$ are irreducible as quadratic polynomials; this may be
confirmed by direct computations. It follows from this that each $p_{k}$ and
each ratio $\frac{p_{k}}{p_{k+1}}$ is a non-square in
$L\coloneqq\mathbb{K}(z_{1},z_{2},\dotsc,z_{2n+4})$.

By induction, it is not hard to prove that $\sqrt{p_{i}}\notin
L(\sqrt{p_{1}},\sqrt{p_{2}},\dotsc,\sqrt{p_{i-1}})$, which implies that

$[L(\sqrt{p_{1}},\dotsc,\sqrt{p_{n}}):L]=2^{n}$

Therefore, $\left\\{\sqrt{p_{1}^{i_{1}}p_{2}^{i_{2}}\dotsb p_{n}^{i_{n}}}\mid
i_{l}\in\\{0,1\\}\right\\}$ is an $L-$basis for
$L(\sqrt{p_{1}},\dotsc,\sqrt{p_{n}})$.

Let
$S\coloneqq\mathbb{K}[z_{1},z_{2},z_{3},z_{4},z_{6},z_{8},\dotsc,z_{2n+4}][z_{5},z_{7},\dotsc,z_{2n+3}]$.
Now, define

$\begin{array}[]{cccc}\phi:&S&\rightarrow&L(\sqrt{p_{1}},\sqrt{p_{2}},\dotsc,\sqrt{p_{n}})\\\
&f&\mapsto&f(z_{1},z_{2},z_{3},z_{4},\sqrt{p_{1}},z_{6},\sqrt{p_{2}},z_{8},\dotsc,\sqrt{p_{n}},z_{2(n+2)})\end{array}$

Clearly, $I_{n}\subseteq\mathrm{ker}(\phi)$. Let $g\in\mathrm{ker}(\phi)$,
then successively dividing $g$ by $z_{2k+3}^{2}-p_{k}$, for $1\leq k\leq n$,
we can write $g$ as,

$g=\sum_{0\leq i_{1},i_{2},\dotsc,i_{n}\leq 1}Q_{i_{1}i_{2}\dotsc
i_{n}}z_{5}^{i_{1}}z_{7}^{i_{2}}\dotsb z_{2n+3}^{i_{n}}+h,$

where $h\in I_{n}$ and $Q_{i_{1}i_{2}\dotsc
i_{n}}\in\mathbb{K}[z_{1},z_{2},z_{3},z_{4},z_{6},z_{8},\dotsc,z_{2(n+2)}]$.

Now,

$\phi(g)=\sum_{0\leq i_{1},i_{2},\dotsc,i_{n}\leq 1}Q_{i_{1}i_{2}\dotsc
i_{n}}\sqrt{p_{1}^{i_{1}}p_{2}^{i_{2}}\dotsb p_{n}^{i_{n}}}=0$

Therefore, each $Q_{i_{1}i_{2}\dotsc i_{n}}=0$ and hence, $g\in I_{n}$. So,
$\mathrm{ker}(\phi)=I_{n}$.

Therefore, we have an injective ring homomorphism,

$T_{n}\hookrightarrow L(\sqrt{p_{1}},\sqrt{p_{2}},\dotsc,\sqrt{p_{n}})$

Hence, $T_{n}$ is an integral domain. ∎

######  Lemma 4.3.2.

The element $\overline{x}_{1}\in T_{n}$ is a prime element.

###### Proof.

Let $J_{n}\coloneqq(kx_{2}y_{2}+x_{k+2}y_{k+2}\mid 1\leq k\leq n)$. Consider
the ideal $(\overline{x}_{1})\subseteq T_{n}$ generated by $\overline{x}_{1}$.

$\displaystyle T_{n}/(\overline{x}_{1})$
$\displaystyle\cong\mathbb{K}[x_{1},y_{1},x_{2},\dotsc,y_{n+2}]/(I_{n},x_{1})$
$\displaystyle\cong\mathbb{K}[x_{2},y_{2},x_{3},\dotsc,y_{n+2},y_{1}]/J_{n}$

Let $S\coloneqq\mathbb{K}[x_{2},y_{2},x_{3},\dotsc,y_{n+2}]/J_{n}$. Using a
similar argument as in the proof of Proposition 4.3.1, we conclude $S$ is an
integral domain. Therefore,

$T_{n}/(\overline{x}_{1})\cong S[y_{1}]$

is an integral domain, and hence, $\overline{x}_{1}\in T_{n}$ is a prime
element. ∎

######  Proposition 4.3.3.

The ring $T_{n}$ is a UFD.

###### Proof.

We will use induction on $n$ and Nagata’s criterion for UFD (refer to [Nag57,
Lemma 2]). From Example 4.1, we know that the statement is true for $n=1$.

Assume the statement is true for all $k<n$. Now, we’ll prove the statement for
$k=n$.

Subtracting the first relation from any other defining relation in $I_{n}$
gives us $kx_{2}y_{2}-x_{3}y_{3}+x_{k+3}y_{k+3}$ for $1\leq k\leq n-1$. Now,
let $A_{n-1}\coloneqq(kx_{2}y_{2}-x_{3}y_{3}+x_{k+3}y_{k+3}\mid 1\leq k\leq
n-1)$. By Proposition 4.3.1, we know that $T_{n}$ is an integral domain and by
Lemma 4.3.2, we know that $\overline{x}_{1}\in T_{n}$ is a prime element. Now,
localize $T_{n}$ at $\overline{x}_{1}$. Then,

$\displaystyle(T_{n})_{\overline{x}_{1}}\cong\mathbb{K}[x_{2},y_{2},x_{3}\dotsc,y_{n+2},x_{1},x_{1}^{-1}]/A_{n-1}\cong
T[x_{1},x_{1}^{-1}]$

where $T\coloneqq\mathbb{K}[x_{2},y_{2},x_{3},\dotsc,y_{n+2}]/A_{n-1}$. Notice
that $T\cong T_{n-1}$ which by the induction hypothesis is a UFD. Thus the
localization of $T_{n}$ at the prime element $\overline{x}_{1}$ is a UFD. By
Nagata’s criterion, we conclude that $T_{n}$ is a UFD. ∎

By Proposition 4.3.1, we get,

$\operatorname{{\rm rep}}_{\beta}(Q_{n},R)=\mathcal{V}(I_{n})$

is irreducible in $\mathbb{A}^{2n+4}$. Since the dimension vector for $Q_{n}$
is chosen to be $\beta=(1,1,\dotsc,1)$, we get,

$\operatorname{SI}(Q_{n},\beta)=\mathbb{K}[\operatorname{{\rm
rep}}_{\beta}(Q_{n},R)]^{\operatorname{SL}_{\beta}}=T_{n}$

Therefore, by the main result, we know that $\operatorname{SI}(Q_{n},\beta)$
is a complete intersection of codimension $n$.

######  Remark 4.3.4.

Notice that the coefficients $k$ in front of $x_{2}y_{2}$ in the defining
generators of $I_{n}$ in the above example are needed for the ring $T_{n}$ to
be a UFD, for $n\geq 2$. If we replace these coefficients by $1$, the ideal
$A_{n-1}$ in the above proof will be generated by binomials and the induction
will not be valid as $T_{2}$ would be
$\mathbb{K}[x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4}]/(x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3},x_{4}y_{4}-x_{3}y_{3})$
which is not a UFD.

##  5\. The Case of Hereditary Algebras

In this section we examine the situation in the context of hereditary
algebras. Let $Q=(Q_{0},Q_{1})$ be an acyclic quiver and $\beta$ be a
dimension vector. Notice that the representation space $\operatorname{{\rm
rep}}_{\beta}(Q)$ is an affine space and hence, irreducible. Also, the
coordinate ring $\mathbb{K}[\operatorname{{\rm rep}}_{\beta}(Q)]$ is a
polynomial ring, hence is a UFD. We will need few definitions before we get to
the main result of this section. We will denote the dimension vector of a
representation $V$ of $Q$ by $\underline{\dim}V$.

######  Definition 5.0.1.

A representation $V\in\operatorname{{\rm rep}}_{\beta}(Q)$ is said to be a
general representation (with respect to a property $\mathcal{P}$) if there
exists an open (dense) subset $U\subseteq\operatorname{{\rm rep}}_{\beta}(Q)$
such that $V\in U$ and every representation in $U$ satisfies the property
$\mathcal{P}$.

######  Definition 5.0.2.

A representation $V\in\operatorname{{\rm rep}}_{\beta}(Q)$ is called a Schur
representation (or a brick) if $\mathrm{End}(V)$ is a division algebra (or
equivalently, coincides with the scalar multiples of the identity, since
$\mathbb{K}$ is assumed to be algebraically closed). A dimension vector
$\beta$ is called a Schur root if a general representation in
$\operatorname{{\rm rep}}_{\beta}(Q)$ is a brick, i.e. there is an open non-
empty set $U\subseteq\operatorname{{\rm rep}}_{\beta}(Q)$ such that any
representation in $U$ is a brick.

For a quiver $Q=(Q_{0},Q_{1})$, we define the Euler form
$\langle\cdot,\cdot\rangle$ on the Grothendieck group $K_{0}(\mathbb{K}Q)$,
which can be identified with $\mathbb{Z}^{Q_{0}}$, in the following way:

$\langle\alpha,\beta\rangle\coloneqq\sum_{x\in
Q_{0}}\alpha(x)\beta(x)-\sum_{a\in
Q_{1}}\alpha(ta)\beta(ha),\;\text{where}\;\alpha,\beta\in\mathbb{Z}^{Q_{0}}$

The Euler characteristics of two representations $V,W$ of $Q$ is defined to be

$\chi(V,W)\coloneqq\dim\operatorname{Hom}(V,W)-\dim\operatorname{Ext}^{1}(V,W)$

By [DW17, Proposition 2.5.2], we know that the above Euler form coincides with
the Euler characteristics in the following way: if $V,W$ are two
representations of $Q$ then

$\langle\underline{\dim}V,\underline{\dim}W\rangle=\dim\operatorname{Hom}(V,W)-\dim\operatorname{Ext}^{1}(V,W)=\chi(V,W)$

Recall that if $\beta$ is a Schur root, then we call $\beta$ a real Schur root
if $\langle\beta,\beta\rangle=1$, we call $\beta$ an isotropic Schur root if
$\langle\beta,\beta\rangle=0$, and we call $\beta$ an imaginary Schur root if
$\langle\beta,\beta\rangle<0$.

We can also study the decomposition of a representation of dimension $\beta$
into its indecomposable representations. V. Kac defined the canonical
decomposition of a dimension vector and showed that if $V\in\operatorname{{\rm
rep}}_{\beta}(Q)$ is within an open (dense) set, then the dimension vector of
its indecomposable direct summands are independent of $V$ [Kac82, see page
150].

######  Definition 5.0.3.

Let $\beta$ be a dimension vector. If
$\beta=\beta_{1}+\beta_{2}+\dotsc+\beta_{t}$ is such that all $\beta_{i}$ are
Schur roots and a general representation $V$ of dimension $\beta$ decomposes
into its indecomposable direct summands as $V=V_{1}\oplus
V_{2}\oplus\dotsc\oplus V_{t}$ where $\underline{\dim}V_{i}=\beta_{i}$, then
we write

$\beta=\beta_{1}\oplus\beta_{2}\oplus\dotsc\oplus\beta_{t}$

and is called the _canonical decomposition_ of $\beta$.

######  Remark 5.0.1.

The canonical decomposition of a dimension vector always exists and is unique
(up to rearrangements), see [Kac82, page 150].

Let $\mathrm{ext}(\beta_{i},\beta_{j})$ denote the generic value of
$\operatorname{Ext}^{1}(V_{i},V_{j})$ on $\operatorname{{\rm
rep}}_{\beta_{i}}(Q)\times\operatorname{{\rm rep}}_{\beta_{j}}(Q)$. For a
dimension vector $\beta$, the decomposition
$\beta=\beta_{1}+\beta_{2}+\dotsc+\beta_{t}$ yields the canonical
decomposition if and only if all $\beta_{i}$ are Schur roots and
$\mathrm{ext}(\beta_{i},\beta_{j})=0$ for all $i\not=j$ (refer to [Kac82,
Proposition 3] or [DW17, Theorem 11.2.1]).

Recall that a dimension vector $\beta$ is called prehomogeneous if
$\operatorname{{\rm rep}}_{\beta}(Q)$ contains an open (dense)
$\operatorname{GL}_{\beta}$-orbit. In terms of canonical decomposition this
means that a dimension vector is prehomogeneous if and only if its canonical
decomposition only contains real Schur roots.

######  Definition 5.0.4.

We call a dimension vector $\beta$ almost prehomogeneous if
$\operatorname{{\rm rep}}_{\beta}(Q)$ has an open non-empty set which is a
union of codimension one $\operatorname{GL}_{\beta}$-orbits.

Recall that if $V\in\operatorname{{\rm rep}}_{\beta}(Q)$, then the
$\operatorname{GL}_{\beta}$-orbit of $V$ has codimension equal to
$\dim\operatorname{Ext}^{1}(V,V)$. Hence, we see that $\beta$ is almost
prehomogeneous exactly when a general representation $V$ of dimension vector
$\beta$ has $\operatorname{Ext}^{1}(V,V)$ one dimensional.

The proposition below gives a characterisation of almost prehomogeneous
dimension vectors and gives an interpretation of such a dimension vector in
terms of canonical decomposition.

######  Proposition 5.0.2.

Let $Q$ be an acyclic quiver and $\beta$ be a dimension vector. The following
statements are equivalent:

  1. (1)

A general module $M$ decomposes as $M=M_{1}\oplus M_{2}$, where

$\operatorname{Ext}^{1}(M,M)=\mathbb{K}=\operatorname{Ext}^{1}(M_{2},M_{2})$

  2. (2)

The representation space $\operatorname{{\rm rep}}_{\beta}(Q)$ consists of an
open set which is an union of codimension one
$\operatorname{GL}_{\beta}$-orbits.

  3. (3)

The dimension vector $\beta$ is almost prehomogeneous.

  4. (4)

If $\beta=\beta_{1}\oplus\beta_{2}\oplus\dotsb\oplus\beta_{t}$ is the
canonical decomposition of $\beta$, then for exactly one $j_{0}$,
$\beta_{j_{0}}$ is an isotropic Schur root and $\beta_{i}$ is a real Schur
root, for all $i\neq j_{0}$.

###### Proof.

$(\emph{1.}\Longrightarrow\emph{2.})$ Let $U$ be the union of all codimension
1 orbits. Then, $U$ is non-empty by the statement 1. Let

$\begin{array}[]{cccc}\Theta:&\operatorname{{\rm
rep}}_{\beta}(Q)&\longrightarrow&\mathbb{N}\\\
&M&\longmapsto&\mathrm{codim}\,\mathcal{O}(M)\end{array}$

We know that the map $\Theta$ is lower semi-continuous. Hence,
$\Theta^{-1}([0,1])$ is an open set in $\operatorname{{\rm rep}}_{\beta}(Q)$.
Notice that $\Theta^{-1}(0)=\emptyset$. Therefore, we conclude,
$\Theta^{-1}(1)=U$ is an open set.

$(\emph{2.}\Longrightarrow\emph{3.})$ This follows from the definition.

$(\emph{3.}\Longrightarrow\emph{4.})$ Since $\beta$ is almost prehomogeneous,
there exists an open set, $U$, which is a union of codimension 1
$\operatorname{GL}_{\beta}$-orbits. Let

$\beta=\beta_{1}\oplus\beta_{2}\oplus\dotsb\oplus\beta_{t}$

be the canonical decomposition of $\beta$. Any general representation $N$ in
this open set will decompose as

$N=N_{1}\oplus N_{2}\oplus\dotsb\oplus N_{t}$

where $N_{i}$ is an indecomposable representation with dimension vector
$\beta_{i}$ and $\operatorname{Ext}^{1}(N_{j},N_{k})=0$, for $j\not=k$. Since
$\mathrm{codim}\,\mathcal{O}(N)=1$, we have,
$\operatorname{Ext}^{1}(N,N)=\mathbb{K}$. Using the bilinearity of
$\operatorname{Ext}^{1}$, we conclude that for exactly one $1\leq j\leq t$,
$\operatorname{Ext}^{1}(N_{j},N_{j})=\mathbb{K}$, and
$\operatorname{Ext}^{1}(N_{i},N_{i})=0$, for all $1\leq i\leq t$ such that
$i\not=j$. Since each $N_{i}$ is indecomposable, we conclude that exactly one
$\beta_{j}$, for $1\leq j\leq t$ is an isotropic Schur root and all other
$\beta_{i}$ are real Schur root, for $1\leq i\leq t$ such that $i\not=j$.

$(\emph{4.}\Longrightarrow\emph{1.})$ This follows from the properties of
canonical decomposition and [DW17, Proposition 11.3.7]. ∎

######  Remark 5.0.3.

If in the statement $(1)$ of above proposition, $M_{2}$ is chosen to be
minimal with respect to the property
$\operatorname{Ext}^{1}(M_{2},M_{2})=\mathbb{K}$, then $M_{2}$ is a brick.
This is due to the following reason– using [DW17, Proposition 11.3.7] and the
fact that $M_{2}$ is minimal with respect to the property
$\operatorname{Ext}^{1}(M_{2},M_{2})=\mathbb{K}$, we conclude that the
canonical decomposition of $\underline{\dim}M_{2}$ is a single isotropic Schur
root. Therefore, by using the Euler form, we get

$0=\langle\underline{\dim}M_{2},\underline{\dim}M_{2}\rangle=\dim\operatorname{Hom}(M_{2},M_{2})-\dim\operatorname{Ext}^{1}(M_{2},M_{2})=\dim\mathrm{End}(M_{2})-1$

Hence, we get that $\mathrm{End}(M_{2})\cong\mathbb{K}$, i.e., $M_{2}$ is a
brick.

The following theorem was first proven in [PW18] for the case of an isotropic
Schur root. Specifically, when the canonical decomposition involves both real
Schur roots and a single isotropic Schur root, it is possible to simplify the
problem by considering a smaller quiver. This reduction can be achieved by
applying Schofield’s reduction theorem (refer to [DW17, Theorem 11.4.9]),
which allows one to remove the real Schur roots from the beginning and from
the end of the canonical decomposition. Consequently, the problem transforms
into computing the ring of semi-invariants for a single isotropic Schur root
over a smaller quiver.

In [PW18], it was proven that the ring of semi-invariants is a hypersurface.
However, considering that our techniques extend beyond the hereditary case and
building upon the examples presented in the previous section, the most
favorable outcome we can obtain is a complete intersection.

######  Theorem 5.0.4.

Let $Q$ be an acyclic quiver and $\beta$ be a dimension vector whose canonical
decomposition contains exactly one isotropic Schur root and all others are
real Schur root. Then, the semi-invariant ring $\operatorname{SI}(Q,\beta)$ is
a complete intersection.

###### Proof.

This follows from Proposition 5.0.2 and Theorem 3.3.1. ∎

##  6\. The General Case

In the previous sections, we have dealt with the case where the irreducible
component $\mathcal{C}$ has orbits of codimension at most one, under the UFD
assumption. In the case of an open orbit, we get a polynomial ring for the
ring of semi-invariants; and in case where there is no open orbit but
codimension one orbits, we get a complete intersection for the ring of semi-
invariants. When the maximal orbit has codimension at least two, it seems
difficult to get structural results for the ring of semi-invariants.
Nevertheless, under the UFD assumption, we can still draw some conclusion
about the semi-invariant ring.

As before, we assume that $A$ denotes a finite dimensional (associative and
unital) algebra over an algebraically closed field $\mathbb{K}$ of
characteristics $0$. Let $(Q,R)$ be a quiver with relations associated to the
algebra $A$, where $R\subseteq\mathbb{K}Q$ is an admissible ideal, and we let
$\beta$ be a dimension vector. In order to get to the main result of this
section, we will need a proposition, which is well known to experts.

######  Proposition 6.0.1.

Let $\mathcal{C}\subseteq\operatorname{{\rm rep}}_{\beta}(Q,R)$ be an
irreducible component such that the coordinate ring $\mathbb{K}[\mathcal{C}]$
is a UFD. If there exists a non-constant $\operatorname{GL}_{\beta}$-invariant
rational function $r$ on $\mathcal{C}$, then there exists homogeneous
$p,q\in\operatorname{SI}(Q,\mathcal{C})$ of the same weight such that $r=p/q$.

###### Proof.

We let $r=p/q$ where we may assume that $p$ and $q$ are coprime. For any
element $g\in\operatorname{GL}_{\beta}$, we have

(2) $(g\cdot p)q=(g\cdot q)p.$

Therefore, we get $p\mid g\cdot p$ and $q\mid g\cdot q$. Hence, there exists
$f_{1},f_{2}\in\mathbb{K}[\mathcal{C}]$ such that

$g\cdot p=pf_{1}\text{ and }g\cdot q=qf_{2}.$

By Equation 2, we conclude that $f_{1}=f_{2}$. Define $f_{g}\coloneqq
f_{1}=f_{2}$. By similar reasoning as above, there exists
$f_{g^{-1}}\in\mathbb{K}[\mathcal{C}]$ such that $g^{-1}\cdot p=pf_{g^{-1}}$
and $g^{-1}\cdot q=qf_{g^{-1}}$. Therefore, we get

$p=e\cdot p=(gg^{-1})\cdot p=g\cdot(g^{-1}\cdot
p)=g\cdot(pf_{g^{-1}})=pf_{g}(g\cdot f_{g^{-1}})$

Hence, we get $f_{g}(g\cdot f_{g^{-1}})=1$, i.e., $f_{g}$ is a unit. By Lemma
3.0.1, we know that $f_{g}\in\mathbb{K}^{*}$. Therefore, we conclude $p$ and
$g\cdot p$ are associates for every $g\in\operatorname{GL}_{\beta}$. Now, by
similar argument as in Proposition 3.0.2, we conclude that $p$ is a
homogeneous semi-invariant. Similarly, we prove that $q$ is a homogeneous
semi-invariant. Clearly, since $p/q$ is an invariant function, the semi-
invariants $p,q$ need to have the same weight. ∎

######  Definition 6.0.1.

  1. (1)

The semi-invariant ring $\operatorname{SI}(Q,\mathcal{C})$ is said to be
multiplicity free if every weight space has dimension at most one, i.e.,
$\dim_{\mathbb{K}}\operatorname{SI}(Q,\mathcal{C})_{\theta}\leq 1$ for every
weight $\theta\in\mathbb{Z}^{Q_{0}}$; otherwise it is said to be not
multiplicity free.

  2. (2)

The algebra $A$ is said to be multiplicity free if for every dimension vector
$\beta$ and every irreducible component
$\mathcal{C}\subseteq\operatorname{{\rm rep}}_{\beta}(Q,R)$, the semi-
invariant ring $\operatorname{SI}(Q,\mathcal{C})$ is multiplicity free;
otherwise it is said to be not multiplicity free (refer to [CKW13, Definition
2.3.1]).

######  Remark 6.0.2.

By (1), the algebra $A$ being multiplicity free is equivalent to the fact that
$\dim\mathcal{M}({\mathcal{C}})^{\theta-ss}=0$, for every irreducible
component $\mathcal{C}$ and every weight $\theta$.

######  Theorem 6.0.3.

If there is an irreducible component $\mathcal{C}\subseteq\operatorname{{\rm
rep}}_{\beta}(Q,R)$ which is not an orbit closure and the coordinate ring
$\mathbb{K}[\mathcal{C}]$ is a UFD, then the semi-invariant ring
$\operatorname{SI}(Q,\mathcal{C})$ is not multiplicity free.

###### Proof.

By Rosenlicht’s result (refer to [Ros63, Theorem]) we know that there exists a
$\operatorname{GL}_{\beta}$-stable, open (dense) subset
$U\subseteq\mathcal{C}$ such that there is a geometric quotient of $U$ by
$\operatorname{GL}_{\beta}$, say $\Phi:U\rightarrow W$. By [Fog83, Corollary,
Section 3] and [MFK94, Remark 2, Section 2, Chapter 0], we know that $W$ is an
integral scheme of finite type over $\mathbb{K}$. Since $U$ is a union of
infinitely many orbits, all of which are closed in $U$, we conclude that $\dim
W\geq 1$, and hence the transcendence degree of fraction field of $W$ is at
least 1. By Rosenlicht’s result (refer to [Ros63, Theorem]), we know that the
fraction field of $W$ can be identified with the
$\operatorname{GL}_{\beta}$-invariant rational functions on $U$, i.e.,
$\mathbb{K}(U)^{\operatorname{GL}_{\beta}}$.

By the argument in the previous paragraph we know that there exists a non-
constant $\operatorname{GL}_{\beta}$-invariant rational function in
$\mathbb{K}(U)^{\operatorname{GL}_{\beta}}$, say $p/q$. By Proposition 6.0.1,
we know that $p$ and $q$ are homogeneous semi-invariants of same weight, say
$\theta$. Also, $p$ and $q$ are linearly independent, since the
$\operatorname{GL}_{\beta}$-invariant rational function $p/q$ is a non-
constant function. Hence, we conclude that
$\dim_{\mathbb{K}}\operatorname{SI}(Q,\mathcal{C})_{\theta}\geq 2$. ∎

######  Remark 6.0.4.

Some immediate consequences of the above result are as follows:

  1. (1)

It follows from the contrapositive statement of [CKW13, Proposition 9] that if
the algebra $A$ is not multiplicity free, then there exists a dimension vector
$\alpha$ such that $\operatorname{{\rm rep}}_{\alpha}(Q,R)$ has infinitely
many non-isomorphic bricks. In particular, it follows from [DIJ17, Theorem
4.2] that algebra $A$ is $\tau$-tilting infinite.

  2. (2)

If we know that the algebra $A$ is $\tau$-tilting finite and there is an
irreducible component $\mathcal{C}$ which is not an orbit closure, then the
coordinate ring $\mathbb{K[\mathcal{C}]}$ is not a UFD.

Acknowledgement. C.P., and D.W. were partially supported by the National
Sciences and Engineering Research Council of Canada and by the Canadian
Defence Academy Research Programme.

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