# Quantum Grassmannians and their Associated Quantum Schubert Varieties at
roots of unity

Jason P. Bell111The first-named author’s research was partially supported by
NSERC grant RGPIN-2022-02951., Stéphane Launois222The second-named author’s
research was partly supported by EPSRC grant EP/R009279/1., and Alexandra
Rogers

###### Abstract

We study the PI degree of various quantum algebras at roots of unity,
including quantum Grassmannians, quantum Schubert varieties, partition
subalgebras, and their associated quantum affine spaces. By a theorem of De
Concini and Procesi, the PI degree of partition subalgebras and their
associated quantum affine spaces is controlled by skew-symmetric integral
matrices associated to (Cauchon-Le) diagrams. We prove that the invariant
factors of these matrices are always powers of 2. This allows us to compute
explicitly the PI degree of partition subalgebras.

Our results also apply to certain completely prime (homogeneous) quotients of
partition subalgebras. In particular, our results allow us to extend results
of Jakobsen and Jøndrup regarding the PI degree of quantum determinantal rings
at roots of unity [JJ01] and we present a method to construct an irreducible
representation of maximal dimension for quantum determinantal ideals.

Building on these results, we use the strong connection between partition
subalgebras and quantum Schubert varieties through noncommutative
dehomogenisation [LR08] to obtain expressions for the PI degree of quantum
Schubert varieties. In particular, we compute the PI degree of quantum
Grassmannians.

##  1 Introduction

The PI degree of a PI algebra is an invariant from which various properties of
the algebra can be deduced. For example, Brown and Yakimov [BY18] showed that,
under some mild conditions on a prime PI algebra, knowing its Azumaya locus
(an invariant which is linked to the PI degree of the algebra and the PI
degree of quotients by maximal ideals) provides valuable information about its
discriminant ideal (another invariant with applications in the study of
automorphism groups of PI algebras ([CPWZ16] and [CPWZ15])). The PI degree
also plays an important role in the representation theory of prime affine PI
algebras, giving an upper bound on the dimension of irreducible
representations ([BG02, Theorem I.13.5]). Our present aim is to provide
explicit computation of the PI degree of various quantum algebras at roots of
unity including quantum Grassmannians, quantum Schubert varieties, partition
subalgebras, and associated quantum affine spaces.

Let $\mathbb{K}$ be an arbitrary field and $1\neq q\in\mathbb{K}^{*}$ be a
field element. We will specify where we require $q$ to be a primitive
$\ell^{\text{th}}$ root of unity, with $\ell>2$. We start with the standard
(single parameter) quantised coordinate ring of $m\times n$ matrices over
$\mathbb{K}$, denoted $\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$. This is a
$\mathbb{K}$-algebra generated by $mn$ indeterminates
$(X_{i,j})_{i,j=1}^{m,n}$ subject to some commutation relations which will be
recapped in Section 2. For $\lambda=(\lambda_{1}\geq\dots\geq\lambda_{m})$ a
partition with $n\geq\lambda_{1}$ and with associated Young diagram denoted by
$Y_{\lambda}$, the partition subalgebra $\mathcal{O}_{q}(Y_{\lambda})$ of
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ is the subalgebra of
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ generated by the generators $X_{i,j}$
fitting into $Y_{\lambda}$. In particular,
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ is itself a partition subalgebra
associated to partition $(n,\dots,n)=n^{m}$. These algebras have appeared
naturally in the study of so-called quantum Grassmannians (see for instance
[LR08, LLR08, LLN]).

By taking the $\mathbb{K}$-algebra generated by all maximal quantum minors of
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ we obtain a quantum analogue of the
homogeneous coordinate ring of the Grassmannian of the $m$-dimensional
subspaces of $\mathbb{K}^{n}$. This is called the quantum Grassmannian and is
denoted $\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$. The maximal quantum minors of
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ are referred to as quantum Plücker
coordinates, and the set of quantum Plücker coordinates of
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$ is denoted by $\Pi_{m,n}$. There is a
natural partial order on $\Pi_{m,n}$ (see (1)) which allows to attach to each
quantum Plücker coordinate $\gamma$ the so-called quantum Schubert variety
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}$ associated to $\gamma$. This
is the quotient of $\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$ by the Schubert
ideal associated to $\gamma$, that is, the quotient by the ideal generated by
$\Pi_{\gamma}:=\\{\alpha\in\Pi_{m,n}|~{}\alpha\ngeq\gamma\\}$. This algebra is
a noncommutative deformation of the homogeneous coordinate ring of the
Schubert variety associated to the (classical) Plücker coordinate $\gamma$.
Note that $\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$ is itself a quantum Schubert
variety (associated to $\gamma=[1,\dots,m]$).

All these algebras are related by noncommutative dehomogenisation [LR08,
Theorem 3.1.6]: the quantum Schubert variety associated to $\gamma$ is a
noetherian domain and $\overline{\gamma}$ is a nonzero normal element of this
algebra. This allows us to form the localised ring
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}[\overline{\gamma}^{-1}]$. The
dehomogenisation isomorphism of Lenagan and Rigal then shows that this
localisation is a skew-Laurent extension of a partition algebra. More
precisely, we have:

$\mathcal{O}_{q^{-1}}(Y_{\lambda})[y^{\pm
1};\sigma]\cong\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}[\overline{\gamma}^{-1}],$

where $\sigma$ denotes the automorphism of $\mathcal{O}_{q^{-1}}(Y_{\lambda})$
defined by $\sigma(X_{i,j})=qX_{i,j}$ for all $(i,j)$ in the Young diagram of
$\lambda$, where $\lambda$ is the partition naturally associated to $\gamma$
(see Subsection 2.3 for more details).

When $q$ is a root of unity, all these algebras are PI, and this
dehomogenisation isomorphism provides a way to compute the PI degree of
quantum Schubert varieties. First, we should compute the PI-degree of quantum
partition subalgebras and then study the effect of a skew-Laurent extension on
the PI degree. This is roughly the strategy we will be using in this paper.
Before we explain our results, let us introduce more details.

Partition subalgebras can be presented as iterated Ore extensions over
$\mathbb{K}$ with the indeterminates $X_{i,j}$ fitting into the Young diagram
of $\lambda$ added in the lexicographic order:

$\mathcal{O}_{q}(Y_{\lambda})=\mathbb{K}[X_{1,1}]\cdots[X_{1,\lambda_{1}};\sigma_{1,\lambda_{1}},\delta_{1,\lambda_{1}}]\dots[X_{m,\lambda_{m}};\sigma_{m,\lambda_{m}},\delta_{m,\lambda_{m}}],$

for suitable automorphisms $\sigma_{i,j}$ and $\sigma_{i,j}$-skew derivations
$\delta_{i,j}$. We note that each automorphism $\sigma_{i,j}$ acts by
multiplication by powers of $q$ on the generators $X_{u,v}$ with
$(u,v)<_{\mathrm{lex}}(i,j)$. The behaviour of these algebras depend very much
on the choice of the parameter $q$. Indeed, it follows from [Hay08, Corollary
4.7] that $\mathcal{O}_{q}(Y_{\lambda})$ is PI if and only if $q$ is a root of
unity.

For the rest of this introduction, we assume that $q$ is a primitive
$\ell^{\text{th}}$ root of unity, with $\ell>2$, so that
$\mathcal{O}_{q}(Y_{\lambda})$ is a PI algebra. In this case, the PI degree of
the iterated Ore extension above does not depend on the skew-derivations
$\delta_{i,j}$ by [Hay08, Corollary 4.7]. In other words,

${\rm PI}\text{-}{\rm deg}(\mathcal{O}_{q}(Y_{\lambda}))={\rm PI}\text{-}{\rm
deg}(\mathbb{K}[T_{1,1}]\cdots[T_{1,\lambda_{1}};\sigma_{1,\lambda_{1}}]\dots[T_{m,\lambda_{m}};\sigma_{m,\lambda_{m}}]),$

with slightly abusive notation. We denote the iterated Ore extension on the
right-hand side by $\overline{\mathcal{O}_{q}(Y_{\lambda})}$. This algebra
$\overline{\mathcal{O}_{q}(Y_{\lambda})}$ is a (uniparameter) quantum affine
space, that is, it is the $\mathbb{K}$-algebra generated by the $T_{i,j}$
subject to $T_{i,j}T_{u,v}=q^{a_{(i,j),(u,v)}}T_{u,v}T_{i,j}$ for all
$(i,j),(u,v)$, where the $a_{(i,j),(u,v)}$ are integers coming from the
automorphisms $\sigma_{i,j}$. We denote by $M_{\lambda}$ the integral skew-
symmetric matrix associated to this quantum affine space, that is,
$M_{\lambda}=(a_{(i,j),(u,v)})\in M_{N}(\mathbb{Z})$, where the rows and
columns of $M_{\lambda}$ are indexed by $(i,j)\in Y_{\lambda}$ ordered in the
lexicographic order.

The above discussion shows that the PI degree of
$\mathcal{O}_{q}(Y_{\lambda})$ is equal to the PI degree of the (uniparameter)
quantum affine space associated to the matrix $M_{\lambda}$. The PI degree of
quantum affine spaces at roots of unity has been studied in [CP93], where it
was shown that this PI degree is controlled by the invariant factors of the
skew-symmetric integral matrix $M_{\lambda}$. A priori, computing invariant
factors can prove a difficult task. In the present situation, matrix
$M_{\lambda}$ can be obtained from a combinatorial gadget called (Cauchon-Le)
diagrams. These objects have naturally appeared in the study of the primitive
spectrum of the quantum Grassmannian on one hand [LLR08] and the cell
decomposition of the totally nonnegative Grassmannian on the other hand
[Pos06]. Our first main result asserts that the invariant factors of any
matrix associated to a diagram (in the sense of Section 3.2) are always powers
of $2$. Combining with results from [BCL12] where the dimension of the kernel
of $M_{\lambda}$ was computed (in terms of the so-called toric permutation
associated to a given diagram–see Section 3.3), we obtain a closed formula for
the PI degree of $\mathcal{O}_{q}(Y_{\lambda})$ as follows:

######  Theorem 1.

Assume that $q$ is a primitive $\ell^{\mathrm{th}}$ root of unity with $\ell$
odd. Then

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(Y_{\lambda}))=\ell^{\frac{N-r}{2}},$

where $N=\sum_{i=1}^{m}\lambda_{i}$ is the number of boxes in $Y_{\lambda}$
and $r$ is the number of odd cycles in the disjoint cycle decomposition of the
toric permutation associated to $\lambda$.

In the case of the partition $\lambda=n^{n}$, we retrieve that the PI degree
of $\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ is equal to
$\ell^{\frac{n(n-1)}{2}}$. This result was first obtained by Jakobsen and
Zhang in [JZ97] with the assumption that $\mathrm{char}(\mathbb{K})=0$, and
then reproved in [Jon00] and [JJ01].

We note that our results are not only applicable to partition subalgebras, but
also to their completely prime quotients by Cauchon ideals in the sense of
[LLR, Definition 3.20]. We illustrate this in the specific case of quantum
determinantal rings, that is, the quotient of
$\mathcal{O}_{q}(M_{n}(\mathbb{K}))$ by the ideal generated by all quantum
minors of a given size. More precisely, our results allow us to compute the PI
degree of quantum determinantal ideals, thus extending [JJ01, Proposition 6.2]
which was established with the assumption that the base field has
characteristic $0$. We also construct an explicit irreducible representation
of maximal dimension (equal to the PI degree) for all quantum determinantal
rings.

Now that we have a good grasp on the PI degree of partition subalgebras, we
turn our attention quantum Schubert varieties. Quantum Schubert varieties are
PI precisely when $q$ is a root of unity thanks to [Hay08], and so we continue
to assume that $q$ is a primitive $\ell^{\text{th}}$ root of unity, with
$\ell>2$. In view of the dehomogenisation isomorphism above and [Hay08,
Corollary 4.7], the computation of the PI degree of
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}$ boils down to computing the
dimension of the kernel and the invariant factors of the skew-symmetric matrix
$M_{\gamma}=\left(\begin{array}[]{cc}M_{\lambda}&{\bf 1}\\\
{\bf-1}^{T}&0\end{array}\right)$ with ${\bf
1}\in\mathbb{Z}^{\lambda_{1}+\dots+\lambda_{m}}$ being a column vector
consisting of $1$ in each entry and with
${\bf-1}\in\mathbb{Z}^{\lambda_{1}+\dots+\lambda_{m}}$ being a row vector
consisting of $-1$ in each entry. This naturally leads us to compare the
invariant factors of an integral skew-symmetric matrix $M$ with the invariant
factors of the matrix $M^{E}=\left(\begin{array}[]{cc}M&{\bf 1}\\\
{\bf-1}^{T}&0\end{array}\right)$. We refer to the matrix $M^{E}$ as the
extended matrix of $M$. Interestingly, computing the invariant factors of
$M_{\lambda}^{E}$ is trickier than computing those of $M_{\lambda}$. For
instance, invariant factors of $M^{E}$ do not need to be powers of $2$
anymore. However, by a careful combinatorial study, we prove the following
result:

######  Theorem 2.

Let $q$ be a primitive $\ell^{\mathrm{th}}$ root of unity such that the
smallest prime factor $\ell_{0}$ of $\ell$ satisfies
$\ell_{0}>\min\\{m,n,2\\}$.

Let $\gamma=[\gamma_{1}<\dots<\gamma_{m}]$ be a quantum Plücker coordinate of
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$ and let $\lambda$ be the partition
associated to $\gamma$. We denote by $N$ the number of boxes in $Y_{\lambda}$
and by $r$ the number of odd cycles for the toric permutation associated to
$\lambda$.

Then the PI degree of the quantum Schubert variety
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}$ is given by

${\rm PI}\text{-}{\rm
deg}\left(\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}\right)=\begin{cases}\ell^{\frac{N-r}{2}}&\text{
if $\ker(M_{\lambda})\subseteq\mathbf{1}^{\bot}$};\\\
\ell^{\frac{N-r}{2}+1}&\text{ if
$\ker(M_{\lambda})\nsubseteq\mathbf{1}^{\bot}$}.\end{cases}$

It would be interesting to characterise when
$\ker(M_{\lambda})\subseteq\mathbf{1}^{\bot}$ via $\gamma$ or $\lambda$. We
are able to do so in the case of the quantum Grassmannian. The result suggests
that this might indeed be an interesting combinatorial question.

If $i$ is a positive integer greater than or equal to $2$, we denote by
$\mu_{2}(i)$ the 2-adic valuation of $i$; that is, the largest integer $j$
such that $2^{j}$ divides $i$. Then we have:

######  Theorem 3.

Let $q$ be a primitive $\ell^{\mathrm{th}}$ root of unity such that the
smallest prime factor $\ell_{0}$ of $\ell$ satisfies
$\ell_{0}>\min\\{m,n,2\\}$. Then the PI degree of the quantum Schubert variety
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$ is given by

${\rm PI}\text{-}{\rm
deg}\left(\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))\right)=\begin{cases}\ell^{\frac{m(n-m)}{2}}&\text{
if }\mu_{2}(m)\neq\mu_{2}(n);\\\ \ell^{\frac{m(n-m)-\gcd(m,n)}{2}+1}&\text{
otherwise}.\end{cases}$

The paper is organised as follows. In Section 2, we introduce all the algebras
involved and we use results of Haynal [Hay08] to reduce the computation of
their PI degree to the computation of the PI degree of a (uniparameter)
quantum affine space. In Section 3, we explain De Concini-Procesi’s strategy
to compute the PI degree of a quantum affine space through the invariant
factors of their associated integral skew-symmetric matrix [CP93] and proceed
to prove that these invariant factors are always powers of $2$ for integral
skew-symmetric matrices coming from a diagram. This allows us to compute the
PI degree of partition subalgebras (see Theorem 3.8) and quantum determinantal
rings (see Theorem 3.13). Section 4 is devoted to the construction of an
irreducible representation of maximal dimension for quantum determinantal
rings. Finally, in Section 5, we compare invariant factors of an integral
skew-symmetric matrix and its associated extended matrix. This allows us to
prove Theorems 2 and 3.

Acknowledgment: In [LLR], the second-named author, third-named author and
Samuel Lopes have generalised the computation of the PI degree and the
construction of irreducible representations to completely prime quotients of
Quantum Nilpotent Algebras at roots of unity. The authors of the present
article would like to thank Samuel Lopes for the useful discussions that have
led to Section 4 and for allowing us to include some of the material of [LLR]
here.

We also thank H. P. Jakobsen for pointing out that the PI degree of quantum
determinantal rings had already been computed in [JJ01] when the
characteristic of the base field is $0$.

The material presented in this article is part of the PhD thesis of the third-
named author [Rog19]. She would like to thank the School of Mathematics,
Statistics and Actuarial Science at the University of Kent for their financial
support.

##  2 Quantum Schubert Varieties

###  2.1 Quantum matrices and partition subalgebras

We denote the quantised coordinate ring of $m\times n$ matrices over
$\mathbb{K}$ by $\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$. This is the
$\mathbb{K}$-algebra generated by $m\times n$ indeterminates
$\\{X_{i,j}\\}_{i,j=1}^{m,n}$ subject to the following relations: for
$(1,1)\leq(i,j)<(s,t)\leq(m,n)$ (in lexicographic ordering), we have

$\displaystyle X_{i,j}X_{s,t}=\begin{cases}X_{s,t}X_{i,j}&i<s,\,j>t;\\\
qX_{s,t}X_{i,j}&(i=s,\,j<t)\text{ or }(i<s,\,j=t);\\\
X_{s,t}X_{i,j}+(q-q^{-1})X_{i,t}X_{s,j}&i<s,\,j<t.\end{cases}$

The _quantum determinant_ $D_{q}$ of $\mathcal{O}_{q}(M_{n}(\mathbb{K}))$ can
be expressed as

$D_{q}=\sum_{\pi\in S_{n}}(-q)^{\ell(\pi)}X_{\pi(1),1}\cdots X_{\pi(n),n},$

where $\ell(\pi)$ gives the length of the permutation $\pi$. For a pair of
subsets $(I,J)\in\llbracket 1,m\rrbracket\times\llbracket 1,n\rrbracket$, with
$|I|=|J|=s\leq\min(m,n)$, called an _index pair_ , the _quantum minor_
$[I|J]\in\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ is defined to be the quantum
determinant of the subalgebra $\mathcal{O}_{q}(M_{s}(\mathbb{K}))$ generated
by $\\{X_{i,j}\\}_{i\in I,j\in J}$. We denote by $\Delta_{m,n}$ the set of
index pairs.

We finish this section with the definition of partition subalgebras:

######  Definition 2.1.

Let $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{m})$ be a
partition with associated Young diagram $Y_{\lambda}$. In
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$, with $n\geq\lambda_{1}$, look at the
subring $\mathcal{O}_{q}(Y_{\lambda})$ generated over $\mathbb{K}$ by those
$X_{ij}$ that fit into the Young diagram for $\lambda$. We call this
subalgebra the partition subalgebra of $\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$
associated with the partition $\lambda$.

For instance, for the partition $\lambda=(4,3,3,1)$,
$\mathcal{O}_{q}(Y_{\lambda})$ is generated by $X_{1,1}$, $X_{1,2}$,
$X_{1,3}$, $X_{1,4}$, $X_{2,1}$, $X_{2,2}$, $X_{2,3}$, $X_{3,1}$, $X_{3,2}$,
$X_{3,3}$ and $X_{4,1}$.

It is well known that $\mathcal{O}_{q}(Y_{\lambda})$ is an iterated Ore
extension over $\mathbb{K}$, with the indeterminates $X_{i,j}$ fitting into
the Young diagram of $\lambda$ added in the lexicographic order.

###  2.2 Quantum Grassmannians and quantum Schubert varieties

The _quantum Grassmannian_ , $\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$, $m<n$, is
defined to be the subalgebra of $\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$
generated by all $m\times m$ maximal quantum minors. As for quantum matrices,
a maximal quantum minor
$\gamma:=[\\{1,\dots,m\\}|\\{\gamma_{1}<\dots<\gamma_{m}\\}]$ is uniquely
defined by its _index set_
$\gamma:=\\{\gamma_{1}<\gamma_{2}<\cdots<\gamma_{m}\\}\subseteq\llbracket
1,n\rrbracket$. Such maximal quantum minors $[\gamma_{1}\cdots\gamma_{m}]$ are
called the _quantum Plücker coordinates_ of
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$. Again, we identify the set of all
index sets with the set of elements in $\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))$
and label this $\Pi_{m,n}$. There is a natural partial order on $\Pi_{m,n}$
given by

$[\gamma_{1}<\cdots<\gamma_{m}]\leq[\gamma_{1}^{\prime}<\cdots<\gamma_{m}^{\prime}]\iff(\gamma_{i}\leq\gamma_{i}^{\prime}\mbox{
for all }i\in\llbracket 1,m\rrbracket).$ (1)

This partial ordering is used to define a quantum analogue of the homogeneous
coordinate ring of Schubert varieties in the Grassmannian.

######  Definition 2.2 ([LR08]).

Let $\gamma\in\Pi_{m,n}$ and set
$\Pi_{m,n}^{\gamma}:=\\{\alpha\in\Pi_{m,n}\mid\alpha\ngeq\gamma\\}$. The
_quantum Schubert variety_ associated to $\gamma$ is:

$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}:=\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))/\langle\Pi_{m,n}^{\gamma}\rangle.$

It follows from [LR08, Corollary 3.1.7] that
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}$ is a noetherian domain.
Moreover, this algebra is a quantum algebra with a straightening law [LR08,
Example 2.1.3], and it follows from [LR08, Remark 2.1.4] and [LR06, Lemma
1.2.1] that the coset $\overline{\gamma}$ of $\gamma$ in
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}$ is a normal element.

###  2.3 Quantum Schubert varieties and partition subalgebras

We now describe an isomorphism between a localisation of
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}$ and a skew-Laurent extension
of a partition subalgebra of $\mathcal{O}_{q^{-1}}(M_{m,n-m}(\mathbb{K}))$.
This isomorphism mainly follows from [LR08, Theorem 3.1.6]. It was established
under the assumption that $q$ is not a root of unity in [LLN]. However, the
existence of this isomorphism, see [LLN, Equation (55) in Section 9.5],
carries over when we just assume that $q\neq 0$.

Before we describe this isomorphism, we introduce the necessary ingredients.

######  Definition 2.3.

The _ladder_ associated to $\gamma$ is denoted by $\mathcal{L}_{\gamma}$ and
defined by

$\mathcal{L}_{\gamma}=\\{(i,j)\in\llbracket 1,m\rrbracket\times\llbracket
1,n\rrbracket\ |\ j>\gamma_{m+1-i}\mbox{ and }j\neq\gamma_{l}\ \mbox{ for all
}l\in\llbracket 1,m\rrbracket\\}.$

For $(i,j)\in\mathcal{L}_{\gamma}$, one defines
$m_{i,j}:=[\\{\gamma_{1},\dots,\gamma_{m}\\}\setminus\\{\gamma_{m+1-i}\\}\sqcup\\{j\\}]$
(which clearly belongs to $\Pi_{m,n}\setminus\Pi_{m,n}^{\gamma}$).

Notice that $\gamma_{i}-i=|\\{a\in\llbracket 1,n\rrbracket\setminus\gamma\ |\
a<\gamma_{i}\\}|$ for each $i\in\llbracket 1,m\rrbracket$. It follows easily
that if we define $\lambda_{i}=n-m-(\gamma_{i}-i)$ for each $i\in\llbracket
1,m\rrbracket$, then $(\lambda_{1},\ldots,\lambda_{m})$ is a partition with
$n-m\geq\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{m}\geq 0$. Let $c$ be
as large as possible such that $\lambda_{c}\neq 0$ and denote by $\lambda$ the
partition $(\lambda_{1},\ldots,\lambda_{c})$. We say that $\lambda$ is the
partition associated to $\gamma$.

Note that the south and east borders of $Y_{\lambda}$ give rise to path of
length $n$, from the north-east corner to the south-west corner of the
$m\times(n-m)$ rectangle. Label each of edges of this path with the numbers
$1$ through $n$ (starting from the north-east corner). Then the elements of
$\gamma$ coincide with the vertical steps in this numbering. The following
example illustrates this construction when $\gamma=[1347]$.

$1$$2$$3$$4$$5$$6$$7$$8$ (2)

We are now ready to state the following result that makes a link between
quantum Schubert varieties and partition subalgebras.

######  Theorem 2.4.

There exists an isomorphism

$\displaystyle\begin{split}\Phi:\mathcal{O}_{q^{-1}}(Y_{\lambda})[y^{\pm
1};\sigma]&\xrightarrow{\cong}(\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))/\langle\Pi_{\gamma}\rangle)[\overline{\gamma}^{-1}]\\\
X_{i,j}&\mapsto\overline{m_{m+1-i,a_{n-m+1-j}}}\cdot\overline{\gamma}^{-1}\hskip
19.91692pt((i,j)\in Y_{\lambda})\\\ y&\mapsto\overline{\gamma},\\\
\end{split}$ (3)

where $\sigma$ denotes the automorphism of $\mathcal{O}_{q^{-1}}(Y_{\lambda})$
defined by $\sigma(X_{i,j})=qX_{i,j}$ for all $(i,j)$ in the Young diagram of
$\lambda$, where we have set $\\{a_{1}<\cdots<a_{n-m}\\}:=\llbracket
1,n\rrbracket\setminus\gamma$ and where $\lambda$ is the partition associated
to $\gamma$.

###### Proof.

This is proved in a similar manner than [LLN, Equation (55) in Section 9.5]
thanks to [LR08, Theorem 3.1.6]. ∎

###  2.4 Partition subalgebras and quantum Schubert varieties at roots of
unity

Recall that $\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ can be written as an
iterated Ore extension of the form:

$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))=\mathbb{K}[X_{11}][X_{12};\sigma_{12},\delta_{12}]\cdots[X_{mn};\sigma_{mn},\delta_{mn}],$

where the indeterminates $X_{i,j}$ are added in the lexicographic order, and
where the $\sigma_{ij}$ are automorphisms and the $\delta_{ij}$ are left
$\sigma_{ij}$-derivations of the appropriate subalgebras.

With a slightly abusive notation, the partition subalgebra
$\mathcal{O}_{q}(Y_{\lambda})$ associated to the partition
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{m})$ with
$n\geq\lambda_{1}$ can also be written as an iterated Ore extention as
follows:

$\mathcal{O}_{q}(Y_{\lambda})=\mathbb{K}[X_{11}][X_{12};\sigma_{12},\delta_{12}]\cdots[X_{mn};\sigma_{mn},\delta_{mn}],$

where the indeterminates $X_{i,j}$ with $(i,j)$ fitting in the Young diagram
$Y_{\lambda}$ are added in the lexicographic order. Note that these
automorphisms $\sigma_{i,j}$ and skew-derivations $\delta_{i,j}$ are in
general distinct from the ones above. However, we kept the same notation to
emphasize the fact that they are defined on the previous generators in the
same way in both cases.

Assume $q$ is a primitive $\ell^{\mbox{th}}$ root of unity. Under this
assumption, it follows from [Hay08, Theorem 1.2] that
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ is a PI algebra. The fact that
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ satisfies the required assumptions to
apply [Hay08, Theorem 1.2] follows from [Hay08, Theorem 2.8]. More generally,
the same arguments can be used to prove that $\mathcal{O}_{q}(Y_{\lambda})$ is
a PI algebra.

It then easily follows from Theorem 2.4 that quantum Schubert varieties are
also PI algebra. Note that quantum Schubert algebras are actually PI only
under the assumption that $q$ is a root of unity.

Moreover, it follows from Theorem 2.4 that:

${\rm PI}\text{-}{\rm deg}(\mathcal{O}_{q^{-1}}(Y_{\lambda})[y;\sigma])={\rm
PI}\text{-}{\rm deg}(\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}).$

In this article, we are interested in computing the PI degree of
$\mathcal{O}_{q}(Y_{\lambda})$ and
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}$. It follows from [Hay08,
Theorem 1.2] that

${\rm PI}\text{-}{\rm deg}(\mathcal{O}_{q}(Y_{\lambda}))={\rm PI}\text{-}{\rm
deg}(\mathbb{K}[T_{11}][T_{12};\sigma_{12}]\cdots[T_{mn};\sigma_{mn}]),$

where the indeterminates $T_{i,j}$ with $(i,j)$ fitting in the Young diagram
$Y_{\lambda}$ are added in the lexicographic order. We denote this algebra
$\mathbb{K}[T_{11}][T_{12};\sigma_{12}]\cdots[T_{mn};\sigma_{mn}]$ by
$\overline{\mathcal{O}_{q}(Y_{\lambda})}$. Note that once again we have used
slightly abusive notation for the automorphisms $\sigma_{i,j}$. These are
defined by

$\sigma_{i,j}(T_{s,t})=\begin{cases}T_{s,t}&i<s,\,j>t;\\\
qT_{s,t}&(i=s,\,j<t)\text{ or }(i<s,\,j=t);\\\ T_{s,t}&i<s,\,j<t,\end{cases}$

for all $(s,t)<_{\mbox{lex}}(i,j)$ such that $(s,t),(i,j)\in Y_{\lambda}$.

We note that the algebra
$\overline{\mathcal{O}_{q}(Y_{\lambda})}=\mathbb{K}[T_{11}][T_{12};\sigma_{12}]\cdots[T_{mn};\sigma_{mn}]$
is a quantum affine space. Recall that the (uniparameter) quantum affine space
$\mathcal{O}_{q^{M}}(\mathbb{K}^{N})$ associated to the integral skew-
symmetric matrix $M=(a_{i,j})\in M_{N}(\mathbb{Z})$ is the
$\mathbb{K}$-algebra generated $T_{1},\dots,T_{N}$ subject to
$T_{i}T_{j}=q^{a_{i,j}}T_{j}T_{i}$ for all $1\leq i,j\leq N$.

The above discussion shows that

${\rm PI}\text{-}{\rm deg}(\mathcal{O}_{q}(Y_{\lambda}))={\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M_{\lambda}}}[T_{1},\dots,T_{N}]),$

where $N=\lambda_{1}+\dots+\lambda_{m}$ is the number of boxes in the Young
diagram $Y_{\lambda}$ and $M_{\lambda}\in M_{N}(\mathbb{Z})$ is the skew-
symmetric integral matrix defined as follows: Label the boxes of $Y_{\lambda}$
from $1$ to $N$ in such a way that the labels increase along the rows (from
left to right) and down the columns (from top to bottom). Given such a
labelling, we construct $M_{\lambda}$ according the rule

$M_{\lambda}[i,j]=\begin{cases}1&\text{ if square }i\text{ is strictly below
or strictly to the right of square }j;\\\ -1&\text{ if square }i\text{ is
strictly above or strictly to the left of square }j;\\\ 0&\text{
otherwise}.\end{cases}$

We are now ready to summarise the main result of this section.

######  Proposition 2.5.

Assume $q$ is a primitive $\ell^{\mbox{th}}$ root of unity.

  1. 1.

Let $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{m})$ be a
partition. Then $\mathcal{O}_{q}(Y_{\lambda})$ is a PI algebra and its PI
degree is given by:

${\rm PI}\text{-}{\rm deg}(\mathcal{O}_{q}(Y_{\lambda}))={\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M_{\lambda}}}(\mathbb{K}^{N})),$

where $M_{\lambda}$ is the matrix deduced from $M$ by deleting rows and
columns indexed by $(i,j)$ not fitting in $Y_{\lambda}$.

  2. 2.

Let $\gamma$ be an indexed set. Then the quantum Schubert variety
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\lambda}$ is a PI algebra and

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\lambda})={\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M_{\gamma}}}(\mathbb{K}^{N})),$

where $M_{\gamma}=\left(\begin{array}[]{cc}M_{\lambda}&{\bf 1}\\\
{\bf-1}^{T}&0\end{array}\right)$ with ${\bf
1}\in\mathbb{Z}^{\lambda_{1}+\dots+\lambda_{m}}$ being a column vector
consisting of $1$ in each entry and with
${\bf-1}\in\mathbb{Z}^{\lambda_{1}+\dots+\lambda_{m}}$ being a row vector
consisting of $-1$ in each entry.

In the next section, we develop new results to compute the PI degree of
quantum affine spaces attached to matrices coming from (Cauchon-Le) diagrams.
This will allow us to give an explicit formula for ${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(Y_{\lambda}))$ thanks to Proposition 2.5.

##  3 PI degree of quantum affine spaces associated to diagrams

###  3.1 The PI Setting

First, we present a general result relating the PI degree of an arbitrary
quantum affine space $\mathcal{O}_{q^{M}}(\mathbb{K}^{N})$ to properties of
its skew-symmetric commutation matrix $M\in M_{N}(\mathbb{Z})$. It is a well-
known result that $M$ is congruent (in the sense of [New72, Chapter IV]) to
its skew-normal form. We denote the skew normal form by $S$ or
$\mathrm{Sk}(M)$ if it is not clear to which matrix $M$ the skew-normal form
$S$ is congruent, and we write $M\sim_{C}S$. This is a block diagonal matrix
of the form

$S=\left(\begin{smallmatrix}0&h_{1}&&&&&&&\\\ -h_{1}&0&&&&&&&\\\
&&0&h_{2}&&&&&\\\ &&-h_{2}&0&&&&&\\\ &&&&\ddots&&&\\\ &&&&&0&h_{s}&\\\
&&&&&-h_{s}&0&\\\ &&&&&&&\mathbf{0}\end{smallmatrix}\right),$

where $\mathbf{0}$ is a square matrix of zeros of dimension $\dim(\ker(M))$ so
that $2s=N-\dim(\ker(M))$, and
$h_{1},\,h_{1},\,h_{2},\,h_{2},\,\ldots,\,h_{s},\,h_{s}\in\mathbb{Z}\backslash\\{0\\}$
are called the _invariant factors_ of $M$. As they always come in pairs, from
now on we will avoid repetition and list the invariant factors simply as
$h_{1},\,h_{2},\,\ldots,\,h_{s}$. These have the property that $h_{i}|h_{i+1}$
for all $i\in\llbracket 1,s\rrbracket$.

######  Lemma 3.1.

Take $q$ to be a primitive $\ell^{\mathrm{th}}$ root of unity. For some $N>1$,
let $M\in M_{N}(\mathbb{Z})$ be a skew-symmetric integral matrix with
invariant factors $h_{1},\ldots,h_{s}$. Then the PI degree of
$\mathcal{O}_{q^{M}}(\mathbb{K}^{N})$ is given as

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M}}(\mathbb{K}^{N}))=\prod_{i=1}^{\frac{N-\dim(\ker(M))}{2}}\frac{\ell}{\gcd(h_{i},\ell)}.$

###### Proof.

By [CP93, Proposition 7.1], the PI degree of this quantum affine space is
$\sqrt{h}$, where $h$ is the cardinality of the image of the homomorphism

${\mathbb{Z}^{N}}$${\mathbb{Z}^{N}}$${(\mathbb{Z}/\ell\mathbb{Z})^{N},}$$\scriptstyle{M}$$\scriptstyle{\pi}$

where $\pi$ denotes the canonical epimorphism. Let $S$ be the skew-normal form
of $M$. Then $S$ is congruent to $M$ and, by [Pan95, Lemma 2.4], the two
quantum tori $\mathcal{O}_{q^{M}}((\mathbb{K}^{*})^{N})$ and
$\mathcal{O}_{q^{S}}((\mathbb{K}^{*})^{N})$ are isomorphic. Also, since the PI
degree of a prime, noetherian ring does not change upon localisation [BG02,
Corollary I.13.3], the PI degree of a quantum affine space is equal to the PI
degree of the associated quantum torus, and thus

$\displaystyle{\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M}}(\mathbb{K}^{N}))={\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M}}((\mathbb{K}^{*})^{N})={\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{S}}((\mathbb{K}^{*})^{N})={\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{S}}(\mathbb{K}^{N})).$

It is therefore enough to compute the cardinality, $h$, of the image of the
homomorphism

${\mathbb{Z}^{N}}$${\mathbb{Z}^{N}}$${(\mathbb{Z}/\ell\mathbb{Z})^{N}.}$$\scriptstyle{S}$$\scriptstyle{\pi}$
(4)

Applying this map to a general element
$\underline{z}=(z_{1},\ldots,z_{N})^{T}\in\mathbb{Z}^{N}$, we obtain the
following:

$(\pi\circ
S)(\underline{z})=(\overline{h_{1}z_{2}},\overline{-h_{1}z_{1}},\overline{h_{2}z_{4}},\overline{-h_{2}z_{3}},\ldots,\overline{h_{s}z_{2s}},\overline{-h_{s}z_{2s-1}},\mathbf{\overline{0}})^{T},$

where $\overline{h_{i}z_{j}}$ denotes the canonical image of $h_{i}z_{j}$ in
$\mathbb{Z}/\ell\mathbb{Z}$. Since $\dim(\ker(M))=\dim(\ker(S))$, the
dimension of the zero matrix $\mathbf{0}$ in $S$ is equal to $\dim(\ker(M))$
and hence $2s=N-\dim(\ker(M))$.

We now turn our attention to the entries of $(\pi\circ S)(\underline{z})$.
Each nonzero entry is of the form $\pm\overline{h_{i}}\overline{z_{j}}$ for
some $\overline{z_{j}}\in\mathbb{Z}/\ell\mathbb{Z}$. Consider then, for each
invariant factor $h_{i}$, the map

$\displaystyle f_{i}:~{}\mathbb{Z}$
$\displaystyle{}~{}\longrightarrow~{}\mathbb{Z}/\ell\mathbb{Z}$ $\displaystyle
z_{j}$ $\displaystyle{}~{}\longmapsto~{}\overline{h_{i}z_{j}}.$

The image of $f_{i}$ is

$\operatorname{Im}(f_{i})=\\{\overline{h}_{i}\overline{z}_{j}\mid\overline{z}_{j}\in\mathbb{Z}/\ell\mathbb{Z}\\}\subseteq\mathbb{Z}/\ell\mathbb{Z}$

and this forms an additive subgroup of $(\mathbb{Z}/\ell\mathbb{Z},+)$.
Furthermore, the image of $f_{i}$ is the cyclic subgroup generated by
$\overline{h}_{i}$ and denoted by $\langle\overline{h}_{i}\rangle$. Since
$(\mathbb{Z}/\ell\mathbb{Z},+)$ is the cyclic group generated by $1$
containing $\ell$ elements then the order of
$\overline{h}_{i}\in(\mathbb{Z}/\ell\mathbb{Z},+)$ is $\ell/\gcd(h_{i},\ell)$.
Hence

$|\operatorname{Im}(f_{i})|=|\langle\overline{h}_{i}\rangle|=\frac{\ell}{\gcd(h_{i},\ell)}.$

The image of $(\pi\circ S)$ in $(\mathbb{Z}/\ell\mathbb{Z})^{N}$ consists of
copies of the subgroup $\operatorname{Im}(f_{i})\in\mathbb{Z}/\ell\mathbb{Z}$
in positions $2i$ and $2i-1$, for each $i\in\llbracket 1,s\rrbracket$, and
zeros in positions $2s+1,\ldots,N$. Therefore, for each $i\in\llbracket
1,s\rrbracket$ the image of $f_{i}$ describes two entries in the image of
$(\pi\circ S)$. The cardinality of the whole image of $(\pi\circ S)$ is
therefore given as

$h=\prod_{i=1}^{s}|\operatorname{Im}(f_{i})|^{2}=\prod_{i=1}^{s}|\langle\overline{h}_{i}\rangle|^{2}=\prod_{i=1}^{s}\left(\frac{\ell}{\gcd(h_{i},\ell)}\right)^{2}.$

Substituting $2s=N-\dim(\ker(M))$, the PI degree of
$\mathcal{O}_{q^{M}}(\mathbb{K}^{N})$ becomes

$\sqrt{h}=\sqrt{\prod_{i=1}^{\frac{N-\dim(\ker(M))}{2}}\left(\frac{\ell}{\gcd(h_{i},\ell)}\right)^{2}}=\prod_{i=1}^{\frac{N-\dim(\ker(M))}{2}}\frac{\ell}{\gcd(h_{i},\ell)},$
(5)

as desired. ∎

Lemma 3.1 tells us we need to find out the invariant factors and the kernel of
the matrix $M$ in order to obtain the PI degree of the associated quantum
affine space. For an arbitrary matrix $M$ this may need to be calculated
specifically. However, we will see in the next section that for matrices
associated to a (Cauchon-Le) diagram we can compute explicitly the PI degree
of the corresponding quantum affine space.

###  3.2 Properties of Matrices associated to (Cauchon-Le) Diagrams and PI
degree of their associated quantum affine spaces

An _$m\times n$ diagram_ is an $m\times n$ grid with each square coloured
either black or white. If the diagram satisfies the rule that if a square is
black then all squares directly above it, or all squares directly to the left
of it, are also black then we say that the diagram is a _Cauchon-Le diagram_.
This name recognises the fact that these diagrams were discovered
independently by both Cauchon [Cau03], in relation to torus-invariant prime
ideals of generic quantum matrices, and Postnikov [Pos06], in relation to the
totally nonnegative Grassmannian where they are called “Le” diagrams.

Given an $m\times n$ diagram $D$ we may compute its _toric permutation_
$\tau$, as defined in [BCL12, Section 4.1], by laying pipes over the squares
such that we place a “cross” on each black square and a “hyperbola” on each
white square. We label the sides of the diagram with the numbers
$1,\ldots,m+n$ such that each pair of opposite sides share the same labels in
the same order. The permutation, $\tau$, may then be read off this diagram by
defining $\tau(i)$ to be the label (on the left or top side of $D$) reached by
following the pipe starting at label $i$ (on the right or bottom side of $D$).
See Figure 1 for an example of a diagram with $\tau=(17)(26384)$.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$~{}~{}~{}~{}\longleftrightarrow~{}~{}~{}$ | $1$$2$$3$$1$$2$$3$$4$$5$$6$$7$$8$$4$$5$$6$$7$$8$
---|---
Figure 1: A labelled $3\times 5$ (Cauchon-Le) diagram (left) with pipe dream
construction (right).

For each $m\times n$ diagram, $D$, with $N$ white squares, we may construct a
skew-symmetric integral matrix, $M(D)\in M_{N}(\mathbb{Z})$, as follows: Label
the white squares from $1$ to $N$ in such a way that the labels increase along
the rows (from left to right) and down the columns (from top to bottom). Given
such a labelling, we construct $M(D)$ according the rule

$M(D)[i,j]=\begin{cases}1&\text{ if square }i\text{ is strictly below or
strictly to the right of square }j;\\\ -1&\text{ if square }i\text{ is
strictly above or strictly to the left of square }j;\\\ 0&\text{
otherwise}.\end{cases}$

The matrix associated to the diagram $D$ in Figure 1 is:

$M(D)=\left(\hbox{}\,\vbox{\Let@\restore@math@cr\default@tag\halign{\hfil$\m@th\scriptstyle#$&&\thickspace\hfil$\m@th\scriptstyle#$\cr
0&1&1&1&0&0&0&0&0\\\ -1&0&1&0&1&0&0&0&0\\\ -1&-1&0&0&0&0&1&0&1\\\
-1&0&0&0&1&1&1&0&0\\\ 0&-1&0&-1&0&1&1&0&0\\\ 0&0&0&-1&-1&0&1&1&0\\\
0&0&-1&-1&-1&-1&0&0&1\\\ 0&0&0&0&0&-1&0&0&1\\\
0&0&-1&0&0&0&-1&-1&0\crcr}}\,\right).$

Our main aim in this section is to show that the invariant factors of a matrix
$M(D)$ are always powers of $2$. The next lemma is a simplification of [BL10,
Proposition 4.6] where we consider diagrams without requiring them to have the
Cauchon-Le property. This is needed for the theorem that follows. The proof
follows that of the aforementioned proposition. We incorporate it for
completeness.

######  Lemma 3.2.

Let $D$ be an $m\times n$ diagram. Suppose that the kernel of $M(D)$ has
dimension $e\geq 1$. Then there is a diagram $D^{\prime}\supseteq D$ obtained
by adding exactly one black box to $D$ such that $M(D^{\prime})$ has an
$(e-1)$-dimensional kernel.

###### Proof.

Without loss of generality we may assume that $D$ has no all-black columns.
Make $D$ into a labelled diagram by assigning the white boxes labels
$\llbracket 1,N\rrbracket$, for some $N\geq 1$. Clearly if any one of these
boxes is coloured black, we are left with a diagram with $N-1$ white boxes.

For every $i\in\llbracket 1,N\rrbracket$, denote by $D_{i}$ the diagram
obtained by colouring the white block with label $i$ black. Assume for
contradiction, that $\dim(\ker(M(D_{i})))\geq e$ for all $i$. Due to parity
arguments, this means that $\dim(\ker(M(D_{i})))\geq e+1$ and we can take
$e+1$ linearly independent vectors of $\ker(M(D_{i}))$, which we denote
$\\{v_{1}^{(i)},\ldots,v_{e+1}^{(i)}\\}$.

We now construct $e+1$ linearly independent vectors of $\ker(M(D))$ as
follows. For $1\leq j\leq e+1$ and $1\leq i\leq N$, let $w_{j}^{(i)}$ be the
vector whose $\ell$-th coordinate is the $\ell$-th coordinate of $v_{j}^{(i)}$
if $\ell<i$, is $0$ if $\ell=i$, and is the $(\ell-1)$-th coordinate of
$v_{j}^{(i)}$ if $\ell>i$. By construction, the set
$\\{w_{j}^{(i)}\\}_{j=1}^{e+1}$ is orthogonal to every row of $M(D)$, except
possibly row $i$. Let $r$ denote row $i$ of $M(D)$ and define the following
map:

$\displaystyle\mathrm{Span}\\{w_{1}^{(i)},\ldots,w_{e+1}^{(i)}\\}$
$\displaystyle\longrightarrow$ $\displaystyle\mathbb{Q}$ $\displaystyle w$
$\displaystyle\longmapsto$ $\displaystyle r\cdot w$

This map must be surjective, since if not then it is the zero map, in which
case the linearly independent vectors $\\{w_{j}^{(i)}\\}_{i=1}^{e+1}$ are
contained within the kernel of $M(D)$. However, per the conditions of the
lemma, the kernel of $M(D)$ has dimension $e$ so this is not possible. The
kernel of this map is therefore an $e$-dimensional subspace of the span which
lies in the kernel of $M(D)$, itself having dimension $e$. From this we see
that every vector in $\ker(M(D))$ has $i$-th coordinate equal to zero.
Furthermore, this is true for all $i\in\llbracket 1,N\rrbracket$, thus
$\ker(M(D))=\\{\mathbf{0}\\}$. This contradicts the dimension condition again,
hence there must exist at least one $i\in\llbracket 1,N\rrbracket$ with
$\dim(\ker(M(D_{i})))\leq e-1$.

To show that $\dim(\ker(M(D_{i})))=e-1$ we simply observe that it is no loss
of generality to assume that in the basis
$\\{v_{1},\ldots,v_{e}\\}\subseteq\ker(M(D))$, the vectors
$v_{1},\ldots,v_{e-1}$ have $i$-th coordinate equal to $0$. Therefore,
deleting the $i$-th coordinate from each of these vectors to obtain
$v^{\prime}_{1},\ldots,v^{\prime}_{e-1}$ gives us a linearly independent set
of vectors of $\ker(M(D_{i}))$. The result follows. ∎

We are now ready to state our main result regarding the invariant factors of a
matrix $M(D)$.

######  Theorem 3.3.

Let $D$ be an $m\times n$ diagram and $M(D)$ be its associated matrix. Then
the invariant factors of $M(D)$ are all powers of $2$.

###### Proof.

We prove that all invariant factors of $M(D)$ are powers of $2$ using
increasing induction on the dimension, $d$, of the kernel of $M(D)$.

If $d=0$ then the matrix $M(D)$ is invertible and it follows, from [BLL10,
Theorem 2.2], that the determinant of $M(D)$ is a power of $4$. Let
$\det(M(D))=4^{a}$ for some $a\in\mathbb{N}$. The matrix $M(D)$ is congruent
to the block diagonal matrix

$\mathrm{Sk}(D):=\left(\begin{smallmatrix}0&h_{1}&&&&&\\\ -h_{1}&0&&&&&\\\
&&0&h_{2}&&&\\\ &&-h_{2}&0&&&\\\ &&&&\ddots&&\\\ &&&&&0&h_{s}\\\
&&&&&-h_{s}&0\\\ \end{smallmatrix}\right),$

where $\mathrm{rank}(M(D))=2s$ and $h_{i}\mid
h_{i+1}\in\mathbb{Z}\backslash\\{0\\}$, for $i\in\llbracket 1,s-1\rrbracket$.
The matrix $\mathrm{Sk}(D)$ is clearly equivalent (in the sense of [New72,
Chapter II]) to the Smith normal form of $M(D)$, which we denote by

$\mathrm{Sm}(D)=\mathrm{diag}(h_{1},h_{1},h_{2},h_{2},\ldots,h_{s},h_{s}).$

We see that $\det(\mathrm{Sk}(D))=\det(\mathrm{Sm}(D))$ since $\mathrm{Sm}(D)$
is obtained from $\mathrm{Sk}(D)$ by performing $s$ distinct row swaps and
multiplying $s$ rows by $-1$, thus changing the determinant of
$\mathrm{Sk}(D)$ by a factor of $(-1)^{2s}=1$. The congruence relation then
implies that

$\det(M(D))=\det(\mathrm{Sk}(D))=\det(\mathrm{Sm}(D)).$

Since the nonzero entries $h_{i}$ in $\mathrm{Sm}(D)$ are the invariant
factors of the matrix $M(D)$, we may use the observations above to deduce that

$4^{a}=\det(M(D))=\det(\mathrm{Sm}(D))=\prod_{i=1}^{s}h_{i}^{2}\quad\Longrightarrow\quad
2^{a}=\prod_{i=1}^{s}h_{i}.$

Therefore, each $h_{i}$ is a power of $2$.

Now let $d>0$ and assume the statement of the theorem holds for all diagrams
whose associated matrix has kernel of dimension less than $d$. That is, for
all diagrams $D^{\prime}$, with $\dim(\ker(M(D^{\prime})))<d$, the invariant
factors of $M(D^{\prime})$ are powers of $2$. Consider a diagram $D$ whose
associated matrix $M(D)$ has kernel of dimension $d$, and let $N$ be the
number of white squares in $D$ so that $M(D)$ is an $N\times N$ matrix. Then,
by Lemma 3.2, there exists a diagram $D^{\prime}$ obtained by adding exactly
one black square to $D$, with the property

$\dim(\ker(M(D^{\prime})))=d^{\prime}=d-1.$

Label the white squares of $D$ as $l_{1}<\ldots<l_{N}$ and suppose we obtain
$D^{\prime}$ by colouring the white square labelled $l_{i_{0}}$ black, for
some $i_{0}\in\llbracket 1,N\rrbracket$. Then $M(D^{\prime})$ is the
$(N-1)\times(N-1)$ submatrix of $M(D)$ obtained by removing the row and column
indexed by $i_{0}$. Noting that $d^{\prime}=d-1<d$, we apply the induction
hypothesis to $M(D^{\prime})$ to see that all its invariant factors, denoted
by $h^{\prime}_{1},\ldots,h^{\prime}_{s^{\prime}}$ for some
$s^{\prime}\in\mathbb{N}$, are powers of $2$. Let $h_{1},\ldots,h_{s}$ be the
invariant factors of $M(D)$ and note that

$s^{\prime}=N-1-(d-1)=N-d=s.$

We may therefore write the Smith normal forms of $M(D)$ and $M(D^{\prime})$,
respectively, as

$\displaystyle\mathrm{Sm}(D)$
$\displaystyle{}=\mathrm{diag}(h_{1},h_{1},\ldots,h_{s},h_{s},0,\ldots,0),$
(6) $\displaystyle\mathrm{Sm}(D^{\prime})$
$\displaystyle{}=\mathrm{diag}(h^{\prime}_{1},h^{\prime}_{1},\ldots,h^{\prime}_{s},h^{\prime}_{s},0,\ldots,0).$
(7)

A well-known result for the Smith normal form (see [New72, Chapter II
subsection 16]) applied to $M(D^{\prime})$ allows us to write

$\displaystyle D_{2i-1}(M(D^{\prime}))$ $\displaystyle{}=h^{\prime}_{i}\cdot
D_{2i-2}(M(D^{\prime})),$ (8) $\displaystyle D_{2i}(M(D^{\prime}))$
$\displaystyle{}=h^{\prime}_{i}\cdot D_{2i-1}(M(D^{\prime})),$ (9)

where $D_{j}(M(D^{\prime}))$ is the greatest common divisor of all $j\times j$
minors of the matrix $M(D^{\prime})$, for all $j\in\llbracket
1,N-1\rrbracket$, and $D_{0}(M(D^{\prime})):=1$. Clearly any minor of
$M(D^{\prime})$ of size $j$ is also a minor of $M(D)$ of size $j$, so
$D_{j}(M(D))$ divides $D_{j}(M(D^{\prime}))$. Therefore, if
$D_{j}(M(D^{\prime}))$ is a power of 2, for all $j\in\llbracket
1,2s\rrbracket$, then $D_{j}(M(D))$ is also a power of 2, for all
$j\in\llbracket 1,2s\rrbracket$. We now apply an induction argument on
$i\in\llbracket 1,s\rrbracket$ to show that $D_{2i-1}(M(D^{\prime}))$ and
$D_{2i}(M(D^{\prime}))$ are powers of 2 for all $i$.

The base case, when $i=1$, is easily seen to hold since $h^{\prime}_{1}$ is a
power of 2, by the inductive hypothesis on $d$, and (8) and (9) give

$\displaystyle D_{1}(M(D^{\prime}))$ $\displaystyle{}=h^{\prime}_{1}\cdot
D_{0}(M(D^{\prime}))=h^{\prime}_{1}\cdot 1,$ $\displaystyle
D_{2}(M(D^{\prime}))$ $\displaystyle{}=h^{\prime}_{1}\cdot
D_{1}(M(D^{\prime}))=h^{\prime}_{1}\cdot h^{\prime}_{1}.$

Assume now that $D_{2i-1}(M(D^{\prime}))$ and $D_{2i}(M(D^{\prime}))$ are
powers of 2, for some $i\in\llbracket 1,s-1\rrbracket$, and consider
$D_{2i+1}(M(D^{\prime}))$ and $D_{2i+2}(M(D^{\prime}))$. Using (8) and (9), we
write these as

$\displaystyle D_{2i+1}(M(D^{\prime}))$ $\displaystyle{}=h^{\prime}_{i+1}\cdot
D_{2i}(M(D^{\prime})),$ $\displaystyle D_{2i+2}(M(D^{\prime}))$
$\displaystyle{}=h^{\prime}_{i+1}\cdot D_{2i+1}(M(D^{\prime})),$

where $h^{\prime}_{i+1}$ is a power of $2$, by the induction hypothesis on
$d$, and $D_{2i}(M(D^{\prime}))$ is a power of $2$, by the induction
hypothesis on $i\in\llbracket 1,s\rrbracket$. Hence, using the equations
above, we may write

$\displaystyle D_{2i+1}(M(D^{\prime}))$ $\displaystyle{}=2^{a}\cdot
2^{b}=2^{a+b},$ $\displaystyle D_{2i+2}(M(D^{\prime}))$
$\displaystyle{}=2^{a}\cdot 2^{a+b}=2^{2a+b},$

for some $a,b\in\mathbb{N}$. This proves the inductive step for the induction
on $i$ and, hence, we conclude that $D_{j}(M(D))$ is a power of $2$, for all
$j\in\llbracket 1,2s\rrbracket$.

Finally, to conclude the main induction on $d$ we apply the identity (9) to
$M(D)$, and use the result of the previous induction argument, to show that,
for any $i\in\llbracket 1,s\rrbracket$,

$h_{i}=\frac{D_{2i}(M(D))}{D_{2i-1}(M(D))}=\frac{2^{e}}{2^{f}}=2^{e-f}\in\mathbb{Z},$

for some $e,f\in\mathbb{N}$ such that $e\geq f$. This proves that all the
invariant factors, $h_{i}$, of $M(D)$ are powers of 2, thus completing the
main proof by induction. ∎

Before we turn our attention back to partition subalgebras, we observe that
combining [BCL12, Theorem 4.6 and Lemma 4.3] gives the following result:

######  Proposition 3.4 ([BCL12]).

Let $D$ be an $m\times n$ diagram and $M(D)$ be its associated matrix. Then
the dimension of the kernel of $M(D)$ is the number of odd cycles in the
disjoint cycle decomposition of its toric permutation $\tau$.

From the two results above we see that if a matrix $M=M(D)$ is associated to
an $m\times n$ diagram with $N$ white boxes, then we have methods to obtain
the properties required to calculate the PI degree of
$\mathcal{O}_{q^{M}}(\mathbb{K}^{N})$, when $q$ is a root of unity. More
precisely, if $q$ is a primitive $\ell^{\mathrm{th}}$ root of unity, then

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M(D)}}(\mathbb{K}^{N}))=\begin{cases}\ell^{\frac{N-r}{2}}&\text{if
$\ell$ is odd};\\\
\prod_{i=1}^{\frac{N-r}{2}}\frac{\ell}{\gcd(h_{i},\ell)}&\text{if $\ell$ is
even},\end{cases}$

where $h_{1},\ldots,h_{(N-r)/2}\in\mathbb{Z}$ are the invariant factors of the
matrix $M(D)$, $N$ is the number of white squares in $D$, $r$ is the number of
odd cycles in the disjoint cycle decomposition of $\tau$.

In fact, our results work more generally as explained in the following remark.

######  Remark 3.5.

The results in this section dealing with diagrams can easily be stated for
Young diagrams upon noting that every diagram on a Young tableau can be
considered a diagram (in a rectangle) by embedding it in a suitably sized
rectangle and colouring all squares outside the Young tableau black. More
formally: Let $\lambda=(\lambda_{1},\ldots,\lambda_{m})$ be a partition with
$n\geq\lambda_{1}\geq\lambda_{2},\cdots\geq\lambda_{m}\geq 0$ and let
$Y_{\lambda}$ be the corresponding Young tableau. We can put this inside an
$m\times n$ diagram, colouring all boxes outside of $Y_{\lambda}$ black (see
Figure 2). Furthermore, any Young diagram on $Y_{\lambda}$ obtained by
colouring a set of white boxes black will, in turn, become another $m\times n$
diagram, $D^{\prime}$, by colouring the corresponding white boxes in $D$
black. Combinatorially, there is no difference between the Young diagram and
the corresponding diagram $D^{\prime}$; their associated matrices will have
the same invariant factors and dimension of kernel.

Therefore, we can apply the results above to any Young diagram and see that
all invariant factors of its associated matrix are powers of 2 and that the
kernel dimension can be calculated by counting the odd cycles in its toric
permutation. We note that (Cauchon-Le) diagrams on Young tableaux naturally
arise in the study of the prime and primitive spectra of the quantum
Grassmannian [LLR08].

| $\leftrightarrow\quad$
---|---
Figure 2: Diagram on a Young diagram (left) embedded into an $3\times 5$
diagram (right).

We are now ready to state our main result for this section.

######  Theorem 3.6.

Let $Y_{\lambda}$ be a Young tableau, $D$ be a diagram on $Y_{\lambda}$ with
$N$ white boxes, and $\tau$ be the toric permutation on $Y_{\lambda}$. If $q$
is a primitive $\ell^{\mathrm{th}}$ root of unity, then

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M(D)}}(\mathbb{K}^{N}))=\begin{cases}\ell^{\frac{N-r}{2}}&\text{if
$\ell$ is odd};\\\
\prod_{i=1}^{\frac{N-r}{2}}\frac{\ell}{\gcd(h_{i},\ell)}&\text{if $\ell$ is
even},\end{cases}$

where $h_{1},\ldots,h_{(N-r)/2}\in\mathbb{Z}$ are the invariant factors of the
matrix $M(D)$ and $r$ is the number of odd cycles in the disjoint cycle
decomposition of $\tau$.

###  3.3 PI degree of partition subalgebras

In view of Theorem 3.6 and Proposition 2.5, we know that the PI degree of
$\mathcal{O}_{q}(Y_{\lambda})$ depends only on the toric permutation
associated to $Y_{\lambda}$. More precisely, let
$\lambda=\\{n=\lambda_{1}\geq\dots\geq\lambda_{m}\\}$ be a partition with
associated Young diagram $Y_{\lambda}$. We can associate to $\lambda$ an
$m\times n$ diagram $D_{\lambda}$ by colouring black every box $(i,j)$ with
$j>\lambda_{i}$, see Figure 2. Then, we deduce from Proposition 2.5 that

${\rm PI}\text{-}{\rm deg}(\mathcal{O}_{q}(Y_{\lambda}))={\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M(D_{\lambda})}}(\mathbb{K}^{N})).$

Moreover, Theorem 3.6 shows that ${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{M(D_{\lambda})}}(\mathbb{K}^{N}))$ essentially depends on
the toric permutation associated to $D_{\lambda}$.

Our next aim is to identify this toric permutation $\tau_{\lambda}$. To
achieve this, we need another permutation attached to a diagram, which we call
the $w$-permutation. In the construction of toric permutations through pipe
dreams, we made a choice for the labelling of the sides of a diagram. By
making a different choice, we obtain a different permutation. More precisely,
the permutation $w$ attached to an $m\times n$ diagram $D$ is obtained by
laying pipes over the squares such that we place a “cross” on each black
square and a “hyperbola” on each white square. The difference with the toric
permutation associated to $D$ is in the way we label the sides of $D$: to
compute $w$, we label the sides of the diagram with the numbers $1,\ldots,m+n$
starting from the North East box. The permutation, $w$, may then be read off
this diagram by defining $w(i)$ to be the label (on the left or top side of
$D$) reached by following the pipe starting at label $i$ (on the right or
bottom side of $D$). See Figure 3 for an example of a diagram with
$w=(23)(4587)$.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$~{}~{}~{}~{}\longleftrightarrow~{}~{}~{}$ | $8$$7$$6$$3$$2$$1$$8$$7$$6$$5$$4$$5$$4$$3$$2$$1$
---|---
Figure 3: A labelled $3\times 5$ (Cauchon-Le) diagram (left) with pipe dream
construction (right).

######  Proposition 3.7.

Let $\lambda=\\{n=\lambda_{1}\geq\dots\geq\lambda_{m}\\}$ be a permutation.
Let $\tau_{\lambda}$ its associated toric permutation, $D_{\lambda}$ its
associated diagram and $w_{\lambda}$ its associated $w$-permutation. Similarly
to Section 2.3, we define $\gamma=\\{\gamma_{1}<\dots<\gamma_{m}\\}$ by
$\lambda_{i}+\gamma_{i}=n+i$ for all $1\leq i\leq m$.

  1. 1.

$\tau_{\lambda}=w_{0}w_{\lambda}w_{0}^{\prime}$, where $w_{0}$ is the longest
element of $S_{m+n}$ and $w_{0}^{\prime}$ is the longest element of
$S_{m}\times S_{n}$, that is, in 1-line notation, $w_{0}=[m+n~{}m+n-1~{}\dots
2~{}1]$ and $w_{0}^{\prime}=[m~{}m-1~{}\dots 2~{}1~{}m+n~{}m+n-1~{}\dots
m+2~{}m+1]$.

  2. 2.

Let $\gamma^{\prime}=\\{\gamma^{\prime}_{1}<\dots<\gamma^{\prime}_{n}\\}$ be
the complement of $\gamma$ in $\\{1,\dots,m+n\\}$. Then, in 1-line notation,
we have:

$w_{\lambda}=[\gamma_{1}~{}\gamma_{2}~{}\dots\gamma_{m}~{}\gamma^{\prime}_{1}~{}\gamma^{\prime}_{2}~{}\dots~{}\gamma^{\prime}_{n}].$

###### Proof.

The first part is trivial as the permutations $w_{0}$ and $w_{0}^{\prime}$ are
just ‘fixing’ the different ways of labelling the sides of a diagram in the
constructions of toric and $w$ permutations.

For the second part, we first consider the case where $i\leq m$. In this case,
the pipe starting at $i$ will first go across $n-\lambda_{i}$ black boxes
(starting from the right) on row $i$. After this, it will only encounter white
boxes. As a consequence, we see that:

$w_{\lambda}(i)=i+(n-\lambda_{i})=\gamma_{i}.$

To conclude we just observe that if $j>i>m$, then
$w_{\lambda}(j)>w_{\lambda}(i)$. ∎

Putting together the previous proposition and Theorem 3.6, we are now ready to
state our main result concerning partition subalgebras.

######  Theorem 3.8.

Let $Y_{\lambda}$ be a Young tableau and $\tau$ be the toric permutation on
$Y_{\lambda}$. Assume that $q$ is a primitive $\ell^{\mathrm{th}}$ root of
unity with $\ell$ odd.
Then

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(Y_{\lambda}))=\ell^{\frac{N-r}{2}},$

where $N=\sum_{i=1}^{m}\lambda_{i}$ and $r$ is the number of odd cycles in the
disjoint cycle decomposition of
$\tau_{\lambda}=w_{0}w_{\lambda}w_{0}^{\prime}$.

For instance, for the partition $\lambda=(5,3,2)$, and $\ell=5$, we obtain
$\tau_{\lambda}=(138764)(25)$ and so ${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(Y_{\lambda}))=5^{\frac{10-2}{2}}=5^{4}$.

###  3.4 PI degree of quantum determinantal rings

For $t\in\llbracket 1,n-1\rrbracket$, let $\mathcal{I}_{t}$ be the quantum
determinantal ideal generated by all $(t+1)\times(t+1)$ quantum minors of
$\mathcal{O}_{q}(M_{n}(\mathbb{K}))$ and denote the quantum determinantal ring
as $R_{t}(M_{n}):=\mathcal{O}_{q}(M_{n}(\mathbb{K}))/\mathcal{I}_{t}$. This is
a noetherian domain, see for instance [GL00].

We begin by stating the following lemma, in which the first isomorphism is
exactly [LR08, Lemma 4.4] and the second isomorphism follows from the map
defined in [PW91, Proposition 3.7.1(3)] which preserves the normality of
$\delta=[1,\ldots,t\mid 1,\ldots,t]$ when mapping it to
$\delta_{t}=[n+t-1,\ldots,n\mid n+t-1,\ldots,n]$:

######  Lemma 3.9.

Let $\delta:=[1,\ldots,t\mid 1,\ldots,t]$ be a $t\times t$ quantum minor in
$R=\mathcal{O}_{q}(M_{n}(\mathbb{K}))$, for some $q\in\mathbb{K}^{*}$. Let
$\bar{\delta}\in R_{t}(M_{n})$ be its canonical image and
$\delta_{t}:=[n+t-1,\ldots,n,\mid n+t-1,\ldots,n]$ be a $t\times t$ quantum
minor in $R^{\mathrm{op}}$. Then

$R_{t}(M_{n})[\bar{\delta}^{-1}]\cong A_{t}[\delta^{-1}]\cong
B_{t}[\delta_{t}^{-1}],$

where $A_{t}$ and $B_{t}$ are the following subalgebras:

$\displaystyle A_{t}$ $\displaystyle{}:=\langle X_{i,j}\in R\mid i\leq t\text{
or }j\leq t\rangle\subseteq R,$ $\displaystyle B_{t}$
$\displaystyle{}:=\langle X_{i,j}\in R^{\mathrm{op}}\mid i\geq n-t+1\text{ or
}j\geq n-t+1\rangle\subseteq R^{\mathrm{op}}.$

######  Remark 3.10.

Lemma 3.9 implies that $\mathrm{Frac}(R_{t}(M_{n}))\cong\mathrm{Frac}(B_{t})$.

We point out that $B_{t}$ may be expressed as an iterated Ore extension over
$\mathbb{K}$ by taking the nonzero indeterminates $X_{i,j}\in B_{t}$, for
$(i,j)\in\\{(1,n-t+1),(1,n-t+2),\ldots,(n,n)\\}$, and adjoining them in
lexicographic order to the base field, $\mathbb{K}$. Let $\alpha_{k}$ be the
$\mathbb{K}$-automorphism, and $\beta_{k}$ be the $\alpha_{k}$-derivation,
appearing in the $k^{\text{th}}$ Ore extension of $B_{t}$ (these are easily
computed using the commutation rules of $R$), and let $Y_{k,t}$ and $Y_{l,t}$
be the indeterminates in the $k^{\text{th}}$ and $l^{\text{th}}$ extensions of
$B_{t}$ respectively. That is,

$Y_{1,t}=X_{1,n-t+1},~{}~{}\ldots,~{}~{}Y_{t,t}=X_{1,n},~{}~{}Y_{t+1,t}=X_{2,n-t+1},~{}~{}\ldots,~{}~{}Y_{2nt-t^{2},t}=X_{n,n}.$

Then,

$\displaystyle B_{t}$
$\displaystyle{}=\mathbb{K}[X_{1,n-t+1}][X_{1,n-t+2};\alpha_{2},\beta_{2}]\cdots[X_{n,n};\alpha_{2nt-t^{2}},\beta_{2nt-t^{2}}]$
$\displaystyle{}=\mathbb{K}[Y_{1,t}][Y_{2,t};\alpha_{2},\beta_{2}]\cdots[Y_{2nt-t^{2},t};\alpha_{2nt-t^{2}},\beta_{2nt-t^{2}}]$

and, for all $1\leq l<k\leq 2nt-t^{2}$, we have

$\displaystyle Y_{k,t}\ast Y_{l,t}$ $\displaystyle{}=\alpha_{k}(Y_{l,t})\ast
Y_{k,t}+\beta_{k}(Y_{l,t})$ $\displaystyle{}=q^{m_{k,l}}Y_{l,t}\ast
Y_{k,t}+\beta_{k}(Y_{l,t}),$

for some $m_{k,l}\in\mathbb{Z}$. These $m_{k,l}\in\mathbb{Z}$ form a skew-
symmetric matrix $M_{n,t}=(m_{k,l})\in M_{2nt-t^{2}}(\mathbb{Z})$. Note that
we use $\ast$ above to denote the multiplication in the opposite algebra.

It was shown in [Hay08, Example 5.4] that $R$ satisfies the conditions of
[Hay08, Theorem 4.6]. As $B_{t}$ is isomorphic to a subalgebra of $R$, it too
must satisfy these conditions because the relevant properties of the iterated
Ore extension of $R$ are preserved in $B_{t}$. We may therefore apply [Hay08,
Corollary 4.7] to $B_{t}$ and observe the following results:

######  Lemma 3.11.

  1. 1.

$\mathrm{Frac}(R_{t}(M_{n}))\cong\mathrm{Frac}(B_{t})\cong\mathrm{Frac}(\mathcal{O}_{q^{M_{n,t}}}(\mathbb{K}^{2nt-t^{2}}))$.

  2. 2.

If $q$ is a root of unity, then $B_{t}$ is a PI algebra and

${\rm PI}\text{-}{\rm deg}(R_{t}(M_{n}))={\rm PI}\text{-}{\rm deg}(B_{t})={\rm
PI}\text{-}{\rm deg}(\mathcal{O}_{q^{M_{n,t}}}(\mathbb{K}^{2nt-t^{2}})).$

It can easily be verified that $-M_{n,t}=M(C)$, where $C$ is the $n\times n$
Cauchon-Le diagram whose last $t$ rows and $t$ columns are white (and all
other boxes are black). It follows from [Pan94, Corollary] that the PI degree
of $R_{t}(M_{n})$ depends only on the properties of this matrix $M(C)$. Since
these specific diagrams and matrices are important for the results that
follow, we define notation for them in the following corollary to Lemmas 3.11
and 3.1:

######  Corollary 3.12.

Let $C_{n,t}$ be the $n\times n$ Cauchon-Le diagram whose last $t$ rows and
$t$ columns are white (and all other boxes are black) and let
$\mathcal{M}_{n,t}:=M(C_{n,t})$ be its associated matrix with invariant
factors $\\{h_{i}\\}_{i}$. Then:

  1. 1.

$M_{n,t}\sim_{C}\mathcal{M}_{n,t}$ and thus
$\mathrm{Frac}(R_{t}(M_{n}))\cong\mathrm{Frac}(\mathcal{O}_{q^{M_{n,t}}}(\mathbb{K}^{2nt-t^{2}}))\cong\mathrm{Frac}(\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}}))$.

  2. 2.

If $q$ is a primitive $\ell^{\mathrm{th}}$ root of unity, then

${\rm PI}\text{-}{\rm deg}(R_{t}(M_{n}))={\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}}))=\prod_{i=1}^{\frac{2nt-t^{2}-\dim(\ker(\mathcal{M}_{n,t}))}{2}}\frac{\ell}{\gcd(h_{i},\ell)}.$

Having proved that all invariant factors of $\mathcal{M}_{n,t}$ are powers of
2 (Theorem 3.3), we now wish to find $\dim(\ker(\mathcal{M}_{n,t}))$. We
observe that [JJ01, Proposition 6.2] have computed the PI degree of
$R_{t}(M_{n})$ over a field of characteristic $0$. However, our above
corollary shows that ${\rm PI}\text{-}{\rm deg}(R_{t}(M_{n}))$ is independent
of the characteristic of the base field $\mathbb{K}$, and so we conclude from
[JJ01, Proposition 6.2] the following generalisation of their result.

######  Theorem 3.13.

Let $n\in\mathbb{N}_{>0}$ and $t\in\llbracket 1,n-1\rrbracket$ and take $q$ to
be a primitive $\ell^{\mathrm{th}}$ root of unity with $\ell>2$. Then

${\rm PI}\text{-}{\rm
deg}(R_{t}(M_{n}))=\begin{cases}\ell^{\frac{2nt-t^{2}-t}{2}}&\ell\text{ is
odd};\\\ \ell^{\frac{2nt-t^{2}-t}{2}}\cdot
2^{-nt+\frac{t^{2}+t}{2}+n-1}&\ell\text{ is even}.\end{cases}$

We note that the above result shows that the toric permutation associated to
$C_{n,t}$ has $t$ odd cycles in its disjoint cycle decomposition. While we
will not need to be more precise, for completeness, we state the following
result whose proof can be found in [Rog19, Proposition 6.10].

######  Proposition 3.14.

Fix some $n\in\mathbb{N}_{>0}$ and $t\in\llbracket 1,n-1\rrbracket$. Then the
toric permutation, $\tau$, associated to $C_{n,t}$ is the product of $t$
disjoint odd cycles $\tau=(\eta_{1}\cdots\eta_{t})$.

In particular, write $n=ut+r$, for some $u\in\mathbb{N}$ and $r\in\llbracket
0,t-1\rrbracket$. Then the cycles depend on $n-t$ and $t$, and are the
following:

$\bullet$ $0<n-t<t$:

$\eta_{i}:=\begin{cases}(i,i+t,2t+r+i,t+r+i),&1\leq i\leq r;\\\
(i,t+r+i),&r<i\leq t.\end{cases}$

$\bullet$ $n-t=t$:

$\eta_{i}:=(i,\,i+t,\,i+t+n,\,i+n),\quad 1\leq i\leq t.$

$\bullet$ $t<n-t$:

$\eta_{i}:=\begin{cases}(i,\,i+t,\,\ldots,\,i+ut,\,i+ut+n,\,i+(u-1)t+n,\,\ldots,\,i+n),&1\leq
i\leq r;\\\
(i,\,i+t,\,\ldots,\,i+(u-1)t,\,i+(u-1)t+n,\,i+(u-2)t+n,\,\ldots,\,i+n),&r<i\leq
t.\end{cases}$

##  4 Irreducible Representations of Quantum Determinantal Rings

In this section we construct an irreducible representation for $R_{t}(M_{n})$
at a root of unity, with dimension equal to the PI degree calculated above.

######  Definition 4.1.

An index pair $(I,J)\in\Delta_{n,n}$ corresponds to a _final quantum minor_ of
$\mathcal{O}_{q}(M_{n}(\mathbb{K}))$ of size $s\in\llbracket 1,n\rrbracket$ if
$I=\llbracket i,i+(s-1)\rrbracket$ and $J=\llbracket j,j+(s-1)\rrbracket$, and
either $i+(s-1)=n$ or $j+(s-1)=n$.

We denote the set of all final quantum minors of
$\mathcal{O}_{q}(M_{n}(\mathbb{K}))$ by $\Omega_{n,n}\subseteq\Delta_{n,n}$,
and the set of all final quantum minors of
$\mathcal{O}_{q}(M_{n}(\mathbb{K}))$ of size less than or equal to $t$ by
$\Omega_{n,n}^{t}\subseteq\Omega_{n,n}$. Note that all the quantum minors in
$\Omega_{n,n}^{t}$ “survive” in the quotient algebra $R_{t}(M_{n})$ and
generate a subalgebra, which we denote by
$\mathbb{K}\langle\Omega_{n,n}^{t}\rangle\subseteq R_{t}(M_{n})$. It may also
be verified that $|\Omega_{n,n}^{t}|=2nt-t^{2}$.

######  Lemma 4.2.

Let $A=\mathbb{K}\langle\Omega_{n,n}^{t}\rangle$ and $\Sigma\subset A$ be the
multiplicative set generated by $\Omega_{n,n}^{t}$. Then,

  1. (i)

The elements of $\Omega_{n,n}^{t}$ commute with each other up to powers of
$q$;

  2. (ii)

$A\subseteq R_{t}(M_{n})\subseteq A\Sigma^{-1}$;

  3. (iii)

$A$ is a quantum affine space
$\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})$, for some
$M^{\prime}\in M_{2nt-t^{2}}(\mathbb{Z})$, and $A\Sigma^{-1}$ is the quantum
torus associated to $A$;

  4. (iv)

$M^{\prime}\in M_{2nt-t^{2}}(\mathbb{Z})$ has rank $2nt-t^{2}-t$ and all its
invariant factors are powers of $2$.

###### Proof.

  1. (i)

This result follows from an analogous result for initial quantum minors of
$\mathcal{O}_{q}(M_{n}(\mathbb{K}))$: a quantum minor $[I|J]\in\Delta_{n,n}$
is called _initial_ if $I=\\{i,i+1,\ldots,i+t\\}$ and
$J=\\{j,j+1,\ldots,j+t\\}$, for some $t\in\llbracket 0,n\rrbracket$, and $1\in
I\cup J$. Noting that all pairs of initial quantum minors $[I|J],\,[M|N]$
satisfying $i=m=1$ are _weakly separated_ in the sense of Leclerc and
Zelevinsky [LZ98], we may apply their result [LZ98, Lemma 2.1] to see that the
$[I|J],\,[M|N]$ above quasi-commute (that is, commute up to a power of $q$).
Applying the transpose automorphism $\tau_{q}^{1}$ of [PW91, Proposition
3.7.1(1)] reveals that any pair of initial quantum minors
$[I^{\prime}|J^{\prime}],\,[M^{\prime}|N^{\prime}]$ satisfying
$j^{\prime}=n^{\prime}=1$ also quasi-commute, as is shown in [Goo06, (4-13)].
Finally, [Goo06, Corollary 6.5] shows that pairs of initial quantum minors
$[I|J],\,[M^{\prime}|N^{\prime}]$ satisfying $i=n^{\prime}=1$ quasi-commute
and hence all initial quantum minors of $\mathcal{O}_{q}(M_{n}(\mathbb{K}))$
quasi-commute with each other. This result passes to the final quantum minors
of $\mathcal{O}_{q}(M_{n}(\mathbb{K}))$, and hence to all elements of
$\Omega_{n,n}^{t}$, by use of the anti-automorphism $\tau_{q}^{2}$ of [PW91,
Proposition 3.7.1(2)] and the property found in [PW91, Lemma 4.3.1].

  2. (ii)

$A\subseteq R_{t}(M_{n})$ is immediate. We deduce from part (i) that the
elements of $\Omega_{n,n}^{t}$ are normal in $A$. They are also regular, since
$R_{t}(M_{n})$ is a noetherian domain. Therefore, $\Sigma\subseteq A$ is an
Ore set at which we can localise. We now show that each generator of
$R_{t}(M_{n})$ is contained in $A\Sigma^{-1}$:

Define sets $Q_{1}:=\llbracket n-t+1,n\rrbracket\times\llbracket
1,n\rrbracket,\;Q_{2}:=\llbracket 1,n\rrbracket\times\llbracket
n-t+1,n\rrbracket$, and $Q:=Q_{1}\cup Q_{2}$. To prove $R_{t}(M_{n})\subseteq
A\Sigma^{-1}$ we first use decreasing induction on $(i,j)\in Q$ (for the
lexicographic order) to show that $\bar{X}_{i,j}\in A\Sigma^{-1}$, where
$\bar{X}_{i,j}\in R_{t}(M_{n})$ is the canonical image of the generator
$X_{i,j}\in R$. We then show that $\bar{X}_{i,j}\in A\Sigma^{-1}$, for all
$(i,j)\in\llbracket 1,n-t\rrbracket\times\llbracket 1,n-t\rrbracket$. This
will prove the result for all $(i,j)\in\llbracket
1,n\rrbracket\times\llbracket 1,n\rrbracket$ since $\llbracket
1,n\rrbracket\times\llbracket 1,n\rrbracket=Q\cup(\llbracket
1,n-t\rrbracket\times\llbracket 1,n-t\rrbracket)$.

We start with the induction argument. Since $\bar{X}_{i,j}\in A$ for all
$(i,j)\in(\\{n\\}\times\llbracket 1,n\rrbracket)\cup(\llbracket
1,n\rrbracket\times\\{n\\})$, by definition, the first non-trivial $(i,j)\in
Q$ to prove in the induction is when $(i,j)=(n-1,n-1)$. We take this as our
base case.

By the definition of quantum minors, we may write

$[\\{n-1,n\\}|\\{n-1,n\\}]=\bar{X}_{n-1,n-1}\bar{X}_{n,n}-q\bar{X}_{n-1,n}\bar{X}_{n,n-1}.$

Rearranging this we obtain

$\bar{X}_{n-1,n-1}=([\\{n-1,n\\}|\\{n-1,n\\}]+q\bar{X}_{n-1,n}\bar{X}_{n,n-1})\bar{X}_{n,n}^{-1}\in
A\Sigma^{-1}.$

Since
$[\\{n-1,n\\}|\\{n-1,n\\}],\,\bar{X}_{n-1,n},\,\bar{X}_{n,n-1},\,\bar{X}_{n,n}\in
A$, and $\bar{X}_{n,n}^{-1}\in A\Sigma^{-1}$, this proves the base case.

For ease of reading we now fix some notation: for some $(i,j)\in Q$, denote by
$[I_{i,j}|J_{i,j}]\in\Delta_{n,n}$ the final quantum minor which has $i$ as
its first row index and $j$ as its first column index; that is,

$I_{i,j}:=\llbracket i,i+\min\\{n-i,n-j\\}\rrbracket,~{}~{}J_{i,j}:=\llbracket
j,j+\min\\{n-i,n-j\\}\rrbracket.$

Setting $s_{i,j}:=\min\\{n-i,n-j\\}$ we see that $[I_{i,j}|J_{i,j}]$ has size
$s_{i,j}+1$ and $s_{i,j}+1\in\llbracket 1,t\rrbracket$ because $(i,j)\in Q$.
Therefore, $[I_{i,j}|J_{i,j}]\in\Omega_{n,n}^{t}\subseteq A$ (note that to
avoid cumbersome notation we do not distinguished between a quantum minor and
its coset). One may also verify that $[I_{i,j}|J_{i,j}]$ is generated by sums
of monomials in $\bar{X}_{l,k}$, where $(l,k)\geq(i,j)$ and $(l,k)\in Q$.

For the inductive step, fix some $(i,j)\in Q$, with $(i,j)<(n-1,n-1)$, and
assume that $\bar{X}_{l,k}\in A\Sigma^{-1}$, for all $(i,j)<(l,k)\leq(n,n)$
and $(l,k)\in Q$. Consider $\bar{X}_{i,j}$. With the notation above we can
rewrite $[I_{i,j}|J_{i,j}]\in A$, using the quantum Laplace relations from
[NYM93, Proposition 1.1], as

$\displaystyle[I_{i,j}|J_{i,j}]=\bar{X}_{i,j}[I_{i+1,j+1}|J_{i+1,j+1}]+\sum_{r=1}^{s_{i,j}}(-q)^{r}\bar{X}_{i,j+r}[I_{i,j}\backslash\\{i\\}|J_{i,j}\backslash\\{j+r\\}].$
(10)

Since $(i+1,j+1)\in Q$ then $[I_{i+1,j+1}|J_{i+1,j+1}]\in\Omega_{n,n}^{t}$ and
is hence invertible in $A\Sigma^{-1}$. Using the inductive hypothesis we
deduce that $[I_{i,j}\backslash\\{i\\}|J_{i,j}\backslash\\{j+r\\}]\in
A\Sigma^{-1}$, for all $r\in\llbracket 0,s_{i,j}\rrbracket$, as it is the sum
of monomials in $\bar{X}_{l,k}$, where $(l,k)>(i,j)$ and $(l,k)\in Q$.
Similarly, $\bar{X}_{i,j+r}\in A\Sigma^{-1}$, for all $r\in\llbracket
1,s_{i,j}\rrbracket$, because $(i,j+r)\in Q$ and $(i,j+r)>(i,j)$. We may
therefore rearrange (10) to obtain

$\displaystyle\bar{X}_{i,j}=\left([I_{i,j}|J_{i,j}]-\sum_{r=1}^{s_{i,j}}(-q)^{r}\bar{X}_{i,j+r}[I_{i,j}\backslash\\{i\\}|J_{i,j}\backslash\\{j+r\\}]\right)[I_{i+1,j+1}|J_{i+1,j+1}]^{-1}\in
A\Sigma^{-1}.$

This proves the inductive step and we conclude that $\bar{X}_{i,j}\in
A\Sigma^{-1}$, for all $(i,j)\in Q$.

For any $(i,j)\in\llbracket 1,n-t\rrbracket\times\llbracket 1,n-t\rrbracket$,
define

$\hat{I}_{i,j}:=\\{i,n-t+1,n-t+2,\ldots,n\\},\quad\hat{J}_{i,j}:=\\{j,n-t+1,n-t+2,\ldots,n\\}$

so that $|\hat{I}_{i,j}|=|\hat{J}_{i,j}|=t+1$. Then
$[\hat{I}_{i,j}|\hat{J}_{i,j}]=0$ in $R_{t}(M_{n})$ and hence also in $A$. We
may write this as

$\displaystyle
0=\bar{X}_{i,j}[\hat{I}_{i,j}\backslash\\{i\\}|\hat{J}_{i,j}\backslash\\{j\\}]+\sum_{r=1}^{t}(-q)^{r}\bar{X}_{i,n-t+r}[\hat{I}_{i,j}\backslash\\{i\\}|\hat{J}_{i,j}\backslash\\{n-t+r\\}].$
(11)

Note that, for all $r\in\llbracket 1,t\rrbracket$, the quantum minor
$[\hat{I}_{i,j}\backslash\\{i\\}|\hat{J}_{i,j}\backslash\\{n-t+r\\}]$ is a sum
of monomials in $\bar{X}_{l,k}$, where $l\in\llbracket n-t+1,n\rrbracket$ so
that $(l,k)\in Q$. By the induction above we deduce that $\bar{X}_{l,k}\in
A\Sigma^{-1}$ and hence
$[\hat{I}_{i,j}\backslash\\{i\\}|\hat{J}_{i,j}\backslash\\{n-t+r\\}]\in
A\Sigma^{-1}$, for all $r\in\llbracket 1,t\rrbracket$. Furthermore,
$[\hat{I}_{i,j}\backslash\\{i\\}|\hat{J}_{i,j}\backslash\\{j\\}]=[I_{n-t+1,n-t+1},I_{n-t+1,n-t+1}]\in\Omega_{n,n}^{t}$
and is therefore invertible in $A\Sigma^{-1}$. Rearranging (11) we obtain

$\bar{X}_{i,j}=\left(-\sum_{r=1}^{t}(-q)^{r}\bar{X}_{i,n-t+r}[\hat{I}_{i,j}\backslash\\{i\\}|\hat{J}_{i,j}\backslash\\{n-t+r\\}]\right)[\hat{I}_{i,j}\backslash\\{i\\}|\hat{J}_{i,j}\backslash\\{j\\}]^{-1}\in
A\Sigma^{-1}.$

Hence $\bar{X}_{i,j}\in A\Sigma^{-1}$, for all $(i,j)\in\llbracket
1,n-t\rrbracket\times\llbracket 1,n-t\rrbracket$, and, together with the
induction proof above, this proves that $R_{t}(M_{n})\subseteq A\Sigma^{-1}$.

  3. (iii)

It was shown in part (i) that elements of $\Omega_{n,n}^{t}$ commute with each
other up to powers of $q$ as determined by a skew-symmetric matrix, which we
denote by $M^{\prime}\in M_{2nt-t^{2}}(\mathbb{Z})$. Hence, there is a
surjective homomorphism

$\displaystyle f:\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})$
$\displaystyle{}\longrightarrow A.$ (12)

We will use the GK-dimension to show that this surjection is, in fact, an
isomorphism.

Claim 1: $\mathrm{GKdim}(A)=\mathrm{GKdim}(R_{t}(M_{n}))=2nt-t^{2}$.

When $q$ is not a root of unity then [GLL18, Corollary 4.8] applied to
$R_{t}(M_{n})$, with $J_{w}=\mathcal{I}_{t}$ and $w=\llbracket
1,n-t\rrbracket\times\llbracket 1,n-t\rrbracket$, shows that $R_{t}(M_{n})$ is
Tdeg-stable in the generic case. When $q$ is a root of unity then
$R_{t}(M_{n})$ is a PI algebra which is a noetherian domain. Thus [Zha96,
Theorem 5.3] says that $R_{t}(M_{n})$ is also Tdeg-stable in the root of unity
case. Taking total rings of fractions of the algebras in part (ii) (possible
since all algebras involved are noetherian domains), we obtain

$\mathrm{Frac}(A)\subseteq\mathrm{Frac}(R_{t}(M_{n}))\subseteq\mathrm{Frac}(A\Sigma^{-1})=\mathrm{Frac}(A)\quad\implies\quad\mathrm{Frac}(R_{t}(M_{n}))=\mathrm{Frac}(A).$

The algebras $A$ and $R_{t}(M_{n})$ therefore satisfy the conditions of
[Zha96, Proposition 3.5(3)], which tells us that
$\mathrm{GKdim}(A)\geq\mathrm{GKdim}(R_{t}(M_{n}))$. The equality of GK-
dimension follows from the basic property that $A\subseteq R_{t}(M_{n})$
implies $\mathrm{GKdim}(A)\leq\mathrm{GKdim}(R_{t}(M_{n}))$.

Finally, we see that $\mathrm{GKdim}(R_{t}(M_{n}))=2nt-t^{2}$, using [LR08,
Remark 4.2 (iii) & (iv)], replacing the $t$ with $t+1$ to make the result in
the paper compatible with our definition of $R_{t}(M_{n})$.

Claim 2: The map, $f$, in (12) is an isomorphism.

From (12) and the First Isomorphism Theorem, we have
$\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})/\ker(f)\cong A$ and
hence

$\mathrm{GKdim}\left(\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})/\ker(f)\right)=\mathrm{GKdim}(A)=2nt-t^{2}.$

As a consequence of Goldie’s Theorem (see [GW04, Corollary 6.4]), every
nonzero ideal of $\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})$
contains a regular element. Therefore, if $\ker(f)$ were not trivial then, by
[KL00, Proposition 3.15],

$\displaystyle\mathrm{GKdim}\left(\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})/\ker(f)\right)+1$
$\displaystyle{}\leq\mathrm{GKdim}\left(\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})\right).$
(13)

However,
$\mathrm{GKdim}\left(\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})\right)=2nt-t^{2}$,
by [LMO88, Lemma 2], so the inequality in (13) becomes $2nt-t^{2}+1\leq
2nt-t^{2}$, which is clearly false. Therefore $\ker(f)$ must be trivial and
$\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}})\cong A$.

  4. (iv)

In Corollary 3.12 it was shown that
$\mathrm{Frac}(R_{t}(M_{n}))\cong\mathrm{Frac}(\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}}))$
and in Theorem 3.3 and Proposition 3.14 we deduced that $\mathcal{M}_{n,t}$
has rank $2nt-t^{2}-t$ and that all its invariant factors are powers of $2$.
Parts (ii) and (iii) of this lemma imply that
$\mathrm{Frac}(R_{t}(M_{n}))=\mathrm{Frac}(A)\cong\mathrm{Frac}(\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}}))$,
therefore

$\mathrm{Frac}(\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}}))\cong\mathrm{Frac}(R_{t}(M_{n}))\cong\mathrm{Frac}(\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}})).$

Specialising $q$ to be a non-root of unity allows us to apply [Pan95, Theorem
2.19] to this isomorphism, which states that

$\mathrm{Frac}(\mathcal{O}_{q^{M^{\prime}}}(\mathbb{K}^{2nt-t^{2}}))\cong\mathrm{Frac}(\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}}))\quad\iff\quad
M^{\prime}\sim_{C}\mathcal{M}_{n,t}.$

Hence, $M^{\prime}$ shares the same invariant factors and rank as
$\mathcal{M}_{n,t}$ thus proving part (iv) in the case when $q$ is not a root
of unity. Since the matrix $M^{\prime}$ also defines the commutation relations
on the quantum affine space $A$ when $q$ is a root of unity, this also proves
part (iv) for any $1\neq q\in\mathbb{K}^{*}$.

∎

For the rest of this section we take $\mathbb{K}$ to be algebraically closed
as this is required for the application of the Jacobson Density Theorem in
Proposition 4.4(ii). All results that follow in this section will take $q$ to
be a root of unity, as will be clearly stated.

We start by constructing an irreducible representation of a quantum affine
space at a root of unity, with dimension equal to its PI degree. This result
is not believed to be new, however, we present a proof of it here to convince
the reader that we do indeed get an irreducible representation, as this is
used heavily in the results that follow. The notation of the lemma has been
chosen to suit its application in the results to come, in which we will take
tensor products of a family of quantum affine planes indexed by $i$.

######  Lemma 4.3.

Let $q$ be a primitive $\ell^{\mathrm{th}}$ root of unity, for some $\ell>1$.
Let $\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$ denote the quantum affine plane with
relations $x_{i}y_{i}=q^{h_{i}}y_{i}x_{i}$, for some
$h_{i}\in\mathbb{Z}\backslash\\{0\\}$ satisfying $\gcd(h_{i},\ell)=1$. Let
$V_{i}$ be an $\ell$-dimensional $\mathbb{K}$-vector space with basis
$\\{v^{(i)}_{1},\ldots,v^{(i)}_{\ell}\\}\subseteq V_{i}$ and define the map

$\displaystyle\varphi_{i}:\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{End}_{\mathbb{K}}(V_{i})$ (14)
$\displaystyle x_{i}$ $\displaystyle{}~{}\longmapsto~{}\varphi_{i}(x_{i})$
$\displaystyle y_{i}$ $\displaystyle{}~{}\longmapsto~{}\varphi_{i}(y_{i}),$

where the endomorphisms $\varphi_{i}(x_{i}),\,\varphi_{i}(y_{i})$ act on the
basis vectors $v_{j}^{(i)}\in\\{v_{1}^{(i)},\ldots,v_{\ell}^{(i)}\\}$ in the
following way:

$\displaystyle\varphi_{i}(x_{i})\cdot v^{(i)}_{j}$
$\displaystyle{}=\lambda_{i}q^{(j-1)h_{i}}v^{(i)}_{j},$ (15)
$\displaystyle\varphi_{i}(y_{i})\cdot v^{(i)}_{j}$
$\displaystyle{}=\begin{cases}v^{(i)}_{j+1},&\text{ if }j\in\llbracket
1,\ell-1\rrbracket;\\\ v^{(i)}_{1},&\text{ if }j=\ell,\end{cases}$ (16)

for some $\lambda_{i}\in K^{*}$. In particular, we can write
$\varphi_{i}(x_{i}),\,\varphi_{i}(y_{i})$ as matrices in
$M_{\ell}(\mathbb{K})$ with respect to this basis:

$\varphi_{i}(x_{i})=\left(\begin{smallmatrix}\lambda_{i}&&&&\\\
&\lambda_{i}q^{h_{i}}&&&\\\ &&\lambda_{i}q^{2h_{i}}&&\\\ &&&\ddots&\\\
&&&&\lambda_{i}q^{(\ell-1)h_{i}}\end{smallmatrix}\right),\qquad\varphi_{i}(y_{i})=\left(\begin{smallmatrix}0&0&\ldots&0&1\\\
1&0&\ldots&0&0\\\ 0&1&\ldots&0&0\\\ \vdots&&\ddots&\vdots&\vdots\\\
0&\ldots&\ldots&1&0\end{smallmatrix}\right).$ (17)

Then $\varphi_{i}$ is a surjective algebra homomorphism which defines an
irreducible representation of $\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$ on $V$ of
dimension $\ell$ and satisfies the property
$\varphi_{i}(x_{i})^{\ell}=\lambda_{i}^{\ell}\operatorname{Id}_{V_{i}}$ and
$\varphi_{i}(y_{i})^{\ell}=\operatorname{Id}_{V_{i}}$.

###### Proof.

Using (15) and (16) it may be verified that
$\varphi_{i}(x_{i})\varphi_{i}(y_{i})=q^{h_{i}}\varphi_{i}(y_{i})\varphi_{i}(x_{i})$,
so that the relations between generators
$x_{i},y_{i}\in\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$ are preserved in the image
of $\varphi_{i}$. By the universal property of algebras with generators and
relations, $\varphi_{i}$ then becomes an algebra homomorphism and thus defines
a representation of $\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$ on $V_{i}$. This
gives $V_{i}$ a $\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$-module structure with
multiplication defined as $rv=\varphi_{i}(r)\cdot v$, for all
$r\in\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$ and $v\in V$.

To show that $(\varphi_{i},V_{i})$ defines an irreducible representation of
the quantum affine space, we suppose that there exists a nonzero subspace
$W\subseteq V_{i}$ which is a sub-representation of $V_{i}$ and we argue that
$W=V_{i}$. Assume (for contradiction) that $v_{j}^{(i)}\notin W$, for all
$j\in\llbracket 1,\ell\rrbracket$, and let
$w:=\sum_{j=1}^{\ell}\alpha_{j}v^{(i)}_{j}\in W$, with
$\alpha_{1},\ldots,\alpha_{\ell}\in\mathbb{K}$, be a nonzero element of $W$
which is minimal with respect to the number of nonzero summands. Note that the
assumption $v_{j}^{(i)}\notin W$ implies that at least two coefficients in
$\\{\alpha_{1},\ldots,\alpha_{\ell}\\}$ are nonzero. Since $W$ is a sub-
representation then $\varphi_{i}(y_{i})\cdot w\in W$ and
$\varphi_{i}(x_{i}y_{i})\cdot w\in W$, where

$\displaystyle\varphi_{i}(y_{i})\cdot w$
$\displaystyle{}=\sum_{j=1}^{\ell-1}\alpha_{j}v^{(i)}_{j+1}+\alpha_{\ell}v^{(i)}_{1},$
$\displaystyle\varphi_{i}(x_{i}y_{i})\cdot
w=\varphi_{i}(x_{i})\cdot(\varphi_{i}(y_{i})\cdot w)$
$\displaystyle{}=\sum_{j=1}^{\ell-1}\lambda_{i}q^{jh_{i}}\alpha_{j}v^{(i)}_{j+1}+\lambda_{i}\alpha_{\ell}v^{(i)}_{1}.$

We now use these elements of $W$ to arrive at a contradiction. If
$\alpha_{\ell}\neq 0$ then we consider
$\varphi_{i}(y_{i}-\lambda_{i}^{-1}x_{i}y_{i})\cdot w\in W$, where

$\varphi_{i}(y_{i}-\lambda_{i}^{-1}x_{i}y_{i})\cdot
w=\sum_{j=1}^{\ell-1}\alpha_{j}(1-q^{jh_{i}})v^{(i)}_{j+1}.$

This is nonzero, since $\ell\nmid jh_{i}$ for all $j\in\llbracket
1,\ell-1\rrbracket$, and it has fewer summands than $w$, thus contradicting
the minimality of $w$. If, instead, $\alpha_{\ell}=0$ and $\alpha_{k}\neq 0$,
for some $k\in\llbracket 1,\ell-1\rrbracket$, then we consider the element
$\varphi_{i}(y_{i}-q^{-kh_{i}}\lambda_{i}^{-1}x_{i}y_{i})\cdot w$ and argue in
a similar way to show that this also contradicts the minimality of $w$.
Therefore, our assumption is false and there must exist some $j\in\llbracket
1,\ell\rrbracket$ such that $v_{j}^{(i)}\in W$. But then $W=V_{i}$, since
$\varphi_{i}(y_{i})$ permutes all the basis vectors, as can be seen in (16).
We deduce from this that $(\varphi_{i},V_{i})$ is irreducible.

To verify surjectivity of $\varphi_{i}$, note that $V_{i}$ is a simple
$\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$-module, finite-dimensional over
$\mathbb{K}$, and $\mathbb{K}$ is algebraically closed. Thus, by Schur’s
Lemma,
$\mathrm{End}_{\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]}(V_{i})\cong\mathbb{K}$ and
we can apply the Jacobson Density Theorem to
$R=\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$, with
$R^{\prime}=\mathrm{End}_{\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]}(V_{i})$, to see
that the map
$\rho:\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]\rightarrow\mathrm{End}_{\mathbb{K}}(V_{i})$,
sending $r$ to $f_{r}$, is surjective. Recall that $f_{r}$ was defined as
$f_{r}(v)=rv$, for all $r\in\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$ and $v\in
V_{i}$, and this is precisely the definition of $\varphi_{i}(r)$ as stated
earlier. Hence $\varphi_{i}=\rho$ is surjective.

Finally, it is easily verified that
$\varphi_{i}(x_{i})^{\ell}=\lambda_{i}^{\ell}\operatorname{Id}_{V_{i}}$ and
$\varphi_{i}(y_{i})^{\ell}=\operatorname{Id}_{V_{i}}$. This proves the final
property in the statement. ∎

Consider a quantum affine space $\mathbb{K}_{q^{M}}[T_{1},\ldots,T_{N}]$ for
some matrix $M$ with a kernel of dimension $t\in\llbracket 0,N-1\rrbracket$
and invariant factors $\\{h_{i}\\}_{i=1}^{s}$, where $2s=N-t$. The skew normal
form of $M$ is $S=EME^{T}\in M_{N}(\mathbb{Z})$, where $E=(e_{i,j})_{i,j}\in
M_{N}(\mathbb{Z})$ is invertible. We denote the quantum affine space
associated to $S$ as

$D:=\mathbb{K}_{q^{S}}[x_{1},y_{1},x_{2},y_{2},\ldots,x_{s},y_{s},z_{1},\ldots,z_{t}].$

If $\gcd(h_{i},\ell)=1$, for all $i\in\llbracket 1,s\rrbracket$, then each
subalgebra $\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]\subseteq D$ is a quantum
affine plane satisfying the conditions of Lemma 4.3, hence it must have an
irreducible representation $(\varphi_{i},V_{i})$ of dimension $\ell$. Let
$V:=V_{1}\otimes\cdots\otimes V_{s}$ so that the dimension of $V$ is
$\ell^{s}$. Using this notation we state the following proposition:

######  Proposition 4.4.

Let $q$ be a primitive $\ell^{\mathrm{th}}$ root of unity, for some $\ell>1$,
and $M\in M_{N}(\mathbb{Z})$ be a skew-symmetric matrix with invariant factors
$h_{1},\ldots,h_{s}\in\mathbb{Z}\backslash\\{0\\}$ satisfying
$\gcd(h_{i},\ell)=1$, for all $i\in\llbracket 1,s\rrbracket$. The following
statements hold:

  1. (i)

${\rm PI}\text{-}{\rm deg}(\mathbb{K}_{q^{M}}[T_{1},\ldots,T_{N}])=\ell^{s}$.

  2. (ii)

There is an algebra homomorphism
$\varphi:D\rightarrow\mathrm{End}_{\mathbb{K}}(V)$ which defines an
irreducible representation of $D$ on $V$ of dimension $\ell^{s}$. It is
defined using the tensor product of the maps $\varphi_{i}$ found in Lemma 4.3
and it acts on the generators of $D$ as follows:

$\displaystyle\varphi(x_{i})$
$\displaystyle{}=\operatorname{Id}_{V_{1}}\otimes\cdots\otimes\operatorname{Id}_{V_{i-1}}\otimes\varphi_{i}(x_{i})\otimes\operatorname{Id}_{V_{i+1}}\otimes\cdots\otimes\operatorname{Id}_{V_{s}},$
$\displaystyle\varphi(y_{i})$
$\displaystyle{}=\operatorname{Id}_{V_{1}}\otimes\cdots\otimes\operatorname{Id}_{V_{i-1}}\otimes\varphi_{i}(y_{i})\otimes\operatorname{Id}_{V_{i+1}}\otimes\cdots\otimes\operatorname{Id}_{V_{s}},$
$\displaystyle\varphi(z_{j})$ $\displaystyle{}=\xi_{j}\operatorname{Id}_{V},$

where $\xi_{j}\in\mathbb{K}^{*}$ for all $j\in\llbracket 1,t\rrbracket$,
$\operatorname{Id}_{V_{i}}$ denotes the identity map on $V_{i}$ for all
$i\in\llbracket 1,s\rrbracket$, and
$\operatorname{Id}_{V}:=\operatorname{Id}_{V_{1}}\otimes\cdots\otimes\operatorname{Id}_{V_{s}}$
denotes the identity map on $V$. Moreover,
$\varphi(x_{i})^{-1}=\lambda_{i}^{-\ell}\varphi(x_{i})^{\ell-1}$ and
$\varphi(y_{i})^{-1}=\varphi(y_{i})^{\ell-1}$, for all $i\in\llbracket
1,s\rrbracket$.

  3. (iii)

$(\varphi,V)$ induces an irreducible representation, $(\phi,V)$, of
$\mathbb{K}_{q^{M}}[T_{1},\ldots,T_{N}]$ where, for all $i\in\llbracket
1,N\rrbracket$, we have

$\phi(T_{i}):=\varphi(x_{1})^{e^{\prime}_{i,1}}\varphi(y_{1})^{e^{\prime}_{i,2}}\cdots\varphi(x_{s})^{e^{\prime}_{i,2s-1}}\varphi(y_{s})^{e^{\prime}_{i,2s}}\varphi(z_{1})^{e^{\prime}_{i,2s+1}}\cdots\varphi(z_{t})^{e^{\prime}_{i,2s+t}},$

with $E^{-1}:=(e^{\prime}_{i,j})_{i,j}\in M_{N}(\mathbb{Z})$. Moreover, for
all $i\in\llbracket 1,2s+t\rrbracket$ there exists some
$\nu_{i}\in\mathbb{K}^{*}$ such that
$\phi(T_{i})^{-1}=\nu_{i}^{-1}\phi(T_{i})^{\ell-1}$.

###### Proof.

  1. (i)

Straightforward application of Lemma 3.1.

  2. (ii)

It is clear from the matrix $S$ that the elements $z_{1},\ldots,z_{t}\in Z(D)$
generate a polynomial ring $\mathbb{K}[z_{1},\ldots,z_{t}]\subseteq D$ and
that, for each $i\in\llbracket 1,s\rrbracket$, the pair $x_{i},\,y_{i}$
generates a quantum affine plane $\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]\subseteq
D$ such that $x_{i},\,y_{i}$ commute with $x_{j},\,y_{j}$, for all
$j\in\llbracket 1,s\rrbracket\backslash\\{i\\}$. The generators of $D$
therefore share the same commutation relations as the generators of
$\left(\bigotimes_{i=1}^{s}\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]\right)\otimes\mathbb{K}[z_{1},\ldots,z_{t}]$.
Using the universal property of Ore extensions we deduce that the following
map extends to an algebra homomorphism, and that this is in fact an
isomorphism:

$\displaystyle\iota:D$
$\displaystyle{}~{}\longrightarrow~{}\mathbb{K}_{q^{h_{1}}}[x_{1},y_{1}]\otimes\ldots\otimes\mathbb{K}_{q^{h_{s}}}[x_{s},y_{s}]\otimes\mathbb{K}[z_{1},\ldots,z_{t}]$
(18) $\displaystyle x_{i}$
$\displaystyle{}~{}\longmapsto~{}1_{1}\otimes\cdots\otimes 1_{i-1}\otimes
x_{i}\otimes 1_{i+1}\otimes\cdots\otimes 1_{s}\otimes 1$ $\displaystyle y_{i}$
$\displaystyle{}~{}\longmapsto~{}1_{1}\otimes\cdots\otimes 1_{i-1}\otimes
y_{i}\otimes 1_{i+1}\otimes\cdots\otimes 1_{s}\otimes 1$ $\displaystyle z_{j}$
$\displaystyle{}~{}\longmapsto~{}1_{1}\otimes\cdots\otimes 1_{s}\otimes
z_{j},$

where $1_{i}$ is the identity element in
$\mathbb{K}_{q^{h_{i}}}[x_{i},y_{i}]$, for all $i\in\llbracket 1,s\rrbracket$.

That is, for a general element
$d=\sum_{\underline{l}\in\mathbb{N}^{2s+t}}\alpha_{\underline{l}}x_{1}^{l_{1}}y_{1}^{l_{2}}\cdots
x_{s}^{l_{2s-1}}y_{s}^{l_{2s}}z_{1}^{l_{2s+1}}\cdots z_{t}^{l_{2s+t}}\in D$,

$\iota(d)=\sum_{\underline{l}\in\mathbb{N}^{2s+t}}\alpha_{\underline{l}}x_{1}^{l_{1}}y_{1}^{l_{2}}\otimes\cdots\otimes
x_{s}^{l_{2s-1}}y_{s}^{l_{2s}}\otimes z_{1}^{l_{2s+1}}\cdots
z_{t}^{l_{2s+t}}.$

We can define an irreducible representation $(\varphi^{\prime},V)$ of the
tensor product algebra on the right-hand side of (18) by applying the Jacobson
Density Theorem, which maintains irreducibility, and by noting that the image
of $\mathbb{K}[z_{1},\ldots,z_{t}]$ under $\varphi^{\prime}$ is contained in
$\mathbb{K}$ by Schur’s Lemma. Here,
$\varphi^{\prime}:=\varphi_{1}\otimes\cdots\varphi_{s}$ for the maps
$\varphi_{i}$ as defined in Lemma 4.3. We then pull this representation back
to an irreducible representation $(\varphi,V)$ of $D$ via the isomorphism
above by setting $\varphi:=\varphi^{\prime}\circ\iota$. This is defined on the
generators of $D$ as

$\displaystyle\varphi:D$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{End}_{\mathbb{K}}(V)$
$\displaystyle x_{i}$
$\displaystyle{}~{}\longmapsto~{}\operatorname{Id}_{V_{1}}\otimes\cdots\otimes\operatorname{Id}_{V_{i-1}}\otimes\varphi_{i}(x_{i})\otimes\operatorname{Id}_{V_{i+1}}\otimes\cdots\otimes\operatorname{Id}_{V_{s}}$
$\displaystyle y_{i}$
$\displaystyle{}~{}\longmapsto~{}\operatorname{Id}_{V_{1}}\otimes\cdots\otimes\operatorname{Id}_{V_{i-1}}\otimes\varphi_{i}(y_{i})\otimes\operatorname{Id}_{V_{i+1}}\otimes\cdots\otimes\operatorname{Id}_{V_{s}}$
$\displaystyle z_{j}$
$\displaystyle{}\longmapsto~{}\xi_{j}\operatorname{Id}_{V},$ (19)

where $\operatorname{Id}_{V_{i}}$ is the identity map on $V_{i}$ and
$\operatorname{Id}_{V}:=\operatorname{Id}_{V_{1}}\otimes\cdots\otimes\operatorname{Id}_{V_{s}}$
is the identity map on $V$.

We may form a basis $\\{\mu_{i_{1},i_{2},\ldots,i_{s}}\mid 1\leq
i_{1},i_{2},\ldots,i_{s}\leq\ell\\}$ of $V$ in terms of the basis of each
$V_{i}$ by setting:

$\mu_{i_{1},i_{2},\ldots,i_{s}}:=v_{i_{1}}^{(1)}\otimes
v_{i_{2}}^{(2)}\otimes\cdots\otimes v_{i_{s}}^{(s)}.$

Then, using (15) and (16), one may verify the following:

$\displaystyle\varphi(x_{j})\cdot\mu_{i_{1},\ldots,i_{s}}$
$\displaystyle{}=v_{i_{1}}^{(1)}\otimes\cdots\otimes
v_{i_{j-1}}^{(j-1)}\otimes\varphi_{j}(x_{j})\cdot v_{i_{j}}^{(j)}\otimes
v_{i_{j+1}}^{(j+1)}\otimes\cdots\otimes v_{i_{s}}^{(s)}$
$\displaystyle{}=v_{i_{1}}^{(1)}\otimes\cdots\otimes
v_{i_{j-1}}^{(j-1)}\otimes\lambda_{j}q^{(i_{j}-1)h_{j}}v_{i_{j}}^{(j)}\otimes
v_{i_{j+1}}^{(j+1)}\otimes\cdots\otimes v_{i_{s}}^{(s)}$
$\displaystyle{}=\lambda_{j}q^{(i_{j}-1)h_{j}}\mu_{i_{1},\ldots,i_{s}}$
$\displaystyle\varphi(y_{j})\cdot\mu_{i_{1},\ldots,i_{s}}$
$\displaystyle{}=v_{i_{1}}^{(1)}\otimes\cdots\otimes
v_{i_{j-1}}^{(j-1)}\otimes\varphi_{j}(y_{j})\cdot v_{i_{j}}^{(j)}\otimes
v_{i_{j+1}}^{(j+1)}\otimes\cdots\otimes v_{i_{s}}^{(s)}$
$\displaystyle{}=\begin{cases}v_{i_{1}}^{(1)}\otimes\cdots\otimes
v_{i_{j-1}}^{(j-1)}\otimes v_{1+i_{j}}^{(j)}\otimes
v_{i_{j+1}}^{(j+1)}\otimes\cdots\otimes v_{i_{s}}^{(s)},&\text{ if
}i_{j}\neq\ell;\\\ v_{i_{1}}^{(1)}\otimes\cdots\otimes
v_{i_{j-1}}^{(j-1)}\otimes v_{1}^{(j)}\otimes
v_{i_{j+1}}^{(j+1)}\otimes\cdots\otimes v_{i_{s}}^{(s)},&\text{ if
}i_{j}=\ell,\end{cases}$
$\displaystyle{}=\begin{cases}\mu_{i_{1},\ldots,i_{j-1},1+i_{j},i_{j+1},\ldots,i_{s}},&\text{
if }i_{j}\neq\ell;\\\
\mu_{i_{1},\ldots,i_{j-1},1,i_{j+1},\ldots,i_{s}},&\text{ if
}i_{j}=\ell,\end{cases}$
$\displaystyle\varphi(z_{j})\cdot\mu_{i_{1},\ldots,i_{s}}$
$\displaystyle{}=\xi_{j}\mu_{i_{1},\ldots,i_{s}}.$

Fixing this basis for $V$ allows us to represent the maps
$\varphi(x_{i}),\,\varphi(y_{i}),\,\varphi(z_{j})\in\mathrm{End}_{\mathbb{K}}(V)$
as matrices in $M_{\ell^{s}}(\mathbb{K})$ by first defining
$\varphi_{i}(x_{i}),\,\varphi_{i}(y_{i})$ as the matrices in (17), and then
taking the images in (ii) to be the Kronecker product of these matrices.

  3. (iii)

The algebra homomorphism $\varphi:D\rightarrow\mathrm{End}_{\mathbb{K}}(V)$
preserves the relations

$\varphi(x_{i})\varphi(y_{i})=q^{h_{i}}\varphi(y_{i})\varphi(x_{i}),$

for all $i\in\llbracket 1,s\rrbracket$, and, as was shown in Lemma 4.3,
$\varphi_{i}(x_{i})^{\ell}=\lambda_{i}^{\ell}\operatorname{Id}_{V_{i}}$ and
$\varphi_{i}(y_{i})^{\ell}=\operatorname{Id}_{V_{i}}$. Hence we obtain

$\left.\begin{aligned}
\varphi(x_{i})^{-1}&{}=\lambda_{i}^{-\ell}\varphi(x_{i})^{\ell-1}=\varphi(\lambda_{i}^{-\ell}x_{i}^{\ell-1})\\\
\varphi(y_{i})^{-1}&{}=\varphi(y_{i})^{\ell-1}=\varphi(y_{i}^{\ell-1})\\\
\varphi(z_{j})^{-1}&{}=\xi_{j}^{-1}\operatorname{Id}_{V}\end{aligned}\right\\}\in\varphi(D).$
(20)

These identities show that $\varphi$ may be extended to define a
representation of the quantum torus associated to $D$ and hence, any negative
powers of the generators $x_{i},\,y_{i},\,z_{j}$ in the argument of $\varphi$
may be replaced with positive powers, upon an appropriate multiplication by a
scalar.

From the identity $E^{-1}S(E^{-1})^{T}=M$ we obtain the equality

$m_{i,j}=\sum_{k=1}^{s}h_{k}(e^{\prime}_{i,2k-1}e^{\prime}_{j,2k}-e^{\prime}_{i,2k}e^{\prime}_{j,2k-1}).$
(21)

Any algebra homomorphism
$\phi:\mathbb{K}_{q^{M}}[T_{1},\ldots,T_{N}]\rightarrow\mathrm{End}_{\mathbb{K}}(V)$
must preserve the commutation rules between the $T_{i}$. That is, for all
$i,j\in\llbracket 1,N\rrbracket$,

$\phi(T_{i})\phi(T_{j})=q^{m_{i,j}}\phi(T_{j})\phi(T_{i})=q^{\sum_{k=1}^{s}h_{k}(e^{\prime}_{i,2k-1}e^{\prime}_{j,2k}-e^{\prime}_{i,2k}e^{\prime}_{j,2k-1})}\phi(T_{j})\phi(T_{i}).$

Let $E^{-1}_{i}$ denote the $i^{\text{th}}$ row of $E^{-1}$, for all
$i\in\llbracket 1,N\rrbracket$, and let $\underline{d}:=x_{1}y_{1}\ldots
x_{s}y_{s}z_{1}\ldots z_{t}\in D$ be the ordered monomial of all the
generators of $D$. We define the following elements of the quantum torus
$D\Sigma^{-1}$, where $\Sigma\subset D$ is the multiplicatively closed set
generated by the generators of $D$:

$\underline{d}^{E^{-1}_{i}}:=x_{1}^{e^{\prime}_{i,1}}y_{1}^{e^{\prime}_{i,2}}\cdots
x_{s}^{e^{\prime}_{i,2s-1}}y_{s}^{e^{\prime}_{i,2s}}z_{1}^{e^{\prime}_{i,2s+1}}\cdots
z_{t}^{e^{\prime}_{i,2s+t}}\in D\Sigma^{-1}.$ (22)

We may apply $\varphi$ to $\underline{d}^{E_{i}^{-1}}$, using (20) to replace
any negative powers $e^{\prime}_{i,j}$ with positive powers
$\ell-e^{\prime}_{i,j}$ so that
$\varphi(\underline{d}^{E_{i}^{-1}})\in\varphi(D)$, and, using (21) and (22),
it may be verified that

$\varphi(\underline{d}^{E^{-1}_{i}})\varphi(\underline{d}^{E^{-1}_{j}})=q^{m_{i,j}}\varphi(\underline{d}^{E^{-1}_{j}})\varphi(\underline{d}^{E^{-1}_{i}}),$

for all $i,j\in\llbracket 1,N\rrbracket$. Therefore, by the universal property
of algebras with generators and relations, the following map extends to an
algebra homomorphism and hence defines a representation of
$\mathbb{K}_{q^{M}}[T_{1},\ldots,T_{N}]$ on $V$:

$\displaystyle\phi:\mathbb{K}_{q^{M}}[T_{1},\ldots,T_{N}]$
$\displaystyle\longrightarrow\mathrm{End}_{\mathbb{K}}(V)$ $\displaystyle
T_{i}$ $\displaystyle\longmapsto\varphi(\underline{d}^{E^{-1}_{i}}).$

Note that $\operatorname{Im}(\phi)\subseteq\operatorname{Im}(\varphi)$ and,
for all $i\in\llbracket 1,N\rrbracket$,

$\phi(T_{i})^{\ell}=\varphi(\underline{d}^{E^{-1}_{i}})^{\ell}=\varphi(x_{1}^{e^{\prime}_{i,1}}y_{1}^{e^{\prime}_{i,2}}\cdots
x_{s}^{e^{\prime}_{i,2s-1}}y_{s}^{e^{\prime}_{i,2s}}z_{1}^{e^{\prime}_{i,2s+1}}\cdots
z_{t}^{e^{\prime}_{i,2s+t}})^{\ell}=\nu_{i}\operatorname{Id}_{V}$

for some $\nu_{i}\in\mathbb{K}^{*}$, by properties of $\varphi$. Thus
$\phi(T_{i})^{-1}=\nu_{i}^{-1}\phi(T_{i})^{\ell-1}$, which allows us to deal
with negative powers of $T_{i}$ under $\phi$ in a similar manner to how we
dealt with negative powers of $x_{i},\,y_{i},\,z_{j}$ under $\varphi$.

We now wish to show that $(\phi,V)$ is irreducible. In a similar way to above,
we may write each $\varphi(x_{i}),\,\varphi(y_{i}),\,\varphi(z_{j})$ in terms
of the $\phi(T_{i})$. The identity $EME^{T}=S$ allows us to write the entries
$s_{i,j}\in S$ in terms of the entries $m_{i,j}\in M$ and $e_{i,j}\in E$ as
follows:

$s_{i,j}=\sum_{l,k=1}^{N}m_{l,k}e_{i,l}e_{j,k}.$

Letting $E_{i}$ denote the $i^{\text{th}}$ row of $E$, for all $i\in\llbracket
1,N\rrbracket$, and $\underline{T}:=T_{1}\cdots T_{N}$ the ordered monomial of
the generators of $\mathbb{K}_{q^{M}}[T_{1},\ldots,T_{N}]$, we define the
following elements of $\mathbb{K}_{q^{M}}[T_{1}^{\pm 1},\ldots,T_{N}^{\pm
1}]$:

$\displaystyle\underline{T}^{E_{i}}:=T_{1}^{e_{i,1}}\cdots T_{N}^{e_{i,N}}.$
(23)

Using this, one may verify, by explicit calculation on
$\underline{T}^{E_{i}}\underline{T}^{E_{j}}$, that

$\phi(\underline{T}^{E_{i}})\phi(\underline{T}^{E_{j}})=q^{s_{i,j}}\phi(\underline{T}^{E_{j}})\phi(\underline{T}^{E_{i}}),$

for all $i,j\in\llbracket 1,N\rrbracket$. In particular, the elements

$\phi(\underline{T}^{E_{1}}),\;\phi(\underline{T}^{E_{2}}),\;\ldots,\;\phi(\underline{T}^{E_{2s-1}}),\;\phi(\underline{T}^{E_{2s}}),\;\phi(\underline{T}^{E_{2s+1}}),\;\ldots,\,\phi(\underline{T}^{E_{N}})$

share the same relations as the elements
$x_{1},\,y_{1},\,\ldots,\,x_{s},\,y_{s},\,z_{1},\,\ldots,\,z_{t}$. Therefore,
again by the universal property of algebras with generators and relations, the
following map extends to an algebra homomorphism:

$\displaystyle\tau:D$
$\displaystyle{}\longrightarrow\mathrm{End}_{\mathbb{K}}(V)$ $\displaystyle
x_{i}$ $\displaystyle{}\longmapsto\phi(\underline{T}^{E_{2i-1}})$
$\displaystyle y_{i}$ $\displaystyle{}\longmapsto\phi(\underline{T}^{E_{2i}})$
$\displaystyle z_{j}$
$\displaystyle{}\longmapsto\phi(\underline{T}^{E_{2s+j}}),$

with $\operatorname{Im}(\tau)\subseteq\operatorname{Im}(\phi)$. Since
$E^{-1}E=\operatorname{Id}$, where $\operatorname{Id}$ is the identity matrix,
we deduce that $\sum_{k=1}^{N}e^{\prime}_{k,i}e_{j,k}=1$ if $i=j$, and
otherwise the sum is zero. Using this, along with the definition of $\phi$, we
obtain

$\displaystyle\tau(x_{i})$
$\displaystyle{}=\phi(\underline{T}^{E_{2i-1}})=\phi(T_{1}^{e_{2i-1,1}}\cdots
T_{N}^{e_{2i-1,N}})$
$\displaystyle{}=\varphi(\underline{d}^{E^{-1}_{1}})^{e_{2i-1,1}}\ldots\varphi(\underline{d}^{E^{-1}_{N}})^{e_{2i-1,N}}$
$\displaystyle{}=\varphi(x_{1}^{e^{\prime}_{1,1}}y_{1}^{e^{\prime}_{1,2}}\cdots
z_{t}^{e^{\prime}_{1,N}})^{e_{2i-1,1}}\cdots\varphi(x_{1}^{e^{\prime}_{i,1}}y_{1}^{e^{\prime}_{i,2}}\cdots
z_{t}^{e^{\prime}_{N,N}})^{e_{2i-1,N}}$ $\displaystyle{}=\varphi(\nu
x_{1}^{\sum_{k=1}^{N}e^{\prime}_{k,1}e_{2i-1,k}}y_{1}^{\sum_{k=1}^{N}e^{\prime}_{k,2}e_{2i-1,k}}\cdots
z_{t}^{\sum_{k=1}^{N}e^{\prime}_{k,N}e_{2i-1,k}})$
$\displaystyle{}=\varphi(\nu x_{i}),$

where $\nu\in\mathbb{K}^{*}$ is the scalar resulting from reordering the
generators of $D$. Similarly we can show that
$\tau(y_{i})=\varphi(\nu^{\prime}y_{i})$ and
$\tau(z_{j})=\varphi(\nu^{\prime\prime}z_{j})$, for some
$\nu^{\prime},\nu^{\prime\prime}\in\mathbb{K}^{*}$. From this we deduce that
$\operatorname{Im}(\tau)=\operatorname{Im}(\varphi)$ and hence
$\operatorname{Im}(\varphi)\subseteq\operatorname{Im}(\phi)$. It then follows
that $(\phi,V)$ is an irreducible representation of
$\mathbb{K}_{q^{M}}[T_{1},\ldots,T_{N}]$ because $(\varphi,V)$ is irreducible.

∎

Using the irreducible representation of $A$ above, we construct an irreducible
representation of $R_{t}(M_{n})$ of the same dimension.

######  Proposition 4.5.

Take $q$ to be a primitive $\ell^{\mathrm{th}}$ root of unity, with $\ell$
odd. Let $A:=\mathbb{K}\langle\Omega_{n,n}^{t}\rangle\subseteq R_{t}(M_{n})$
be the subalgebra generated by the final quantum minors of size less than or
equal to $t$, denoted by $T_{1},\ldots,T_{2s+t}\in R_{t}(M_{n})$, where
$2s=2nt-t^{2}-t$. Let $(\phi,V)$ be the $\ell^{s}$-dimensional irreducible
representation of $A$ defined in Proposition 4.4.

Then, every element $r\in R_{t}(M_{n})$ may be written as

$r=\sum_{\underline{i}\in\mathbb{Z}^{2s+t}}\alpha_{\underline{i}}T_{1}^{i_{1}}\cdots
T_{2s+t}^{i_{2s+t}}\in\mathbb{K}_{q^{M}}[T_{1}^{\pm 1},\ldots,T_{2s+t}^{\pm
1}]$

where $\alpha_{\underline{i}}\in\mathbb{K}$ and $i_{j}\in\mathbb{Z}$, for all
$j\in\llbracket 1,2s+t\rrbracket$, and there is an algebra homomorphism

$\displaystyle\rho:R_{t}(M_{n})$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{End}_{\mathbb{K}}(V)$
$\displaystyle r$
$\displaystyle{}~{}\longmapsto~{}\sum_{\underline{i}\in\mathbb{Z}^{2s+t}}\alpha_{\underline{i}}\phi(T_{1})^{i_{1}}\cdots\phi(T_{2s+t})^{i_{2s+t}}$

which defines an irreducible representation of $R_{t}(M_{n})$ of dimension
$\ell^{s}$.

###### Proof.

By Lemma 4.2(iii), $A$ is a quantum affine space so, denoting the final
quantum minors in the set $\Omega_{n,n}^{t}$ by $T_{1},\ldots,T_{2s+t}$, this
allows us to write $A=\mathbb{K}_{q^{M^{\prime}}}[T_{1},\ldots,T_{2s+t}]$,
where $M^{\prime}\in M_{2nt-t^{2}}(\mathbb{Z})$ is a skew-symmetric matrix. We
saw in Lemma 4.2(iv) that the dimension of the kernel of $M^{\prime}$ is $t$
and that all its invariant factors, $h_{1},\ldots,h_{s}$, are powers of $2$.
Since $\ell$ is odd then $\gcd(h_{i},\ell)=1$, for all $i\in\llbracket
1,s\rrbracket$, and we may apply Proposition 4.4 to obtain the irreducible
representation $(\phi,V)$ of $A$ of dimension $\ell^{s}$ defined therein.

Recall that $\Sigma\subseteq A$ is the Ore set generated by all $T_{i}$, and
let $\pi:A\rightarrow A\Sigma^{-1}$ be the localisation map which defines
$A\Sigma^{-1}$ as a right ring of fractions of $A$. Recall also, from
Proposition 4.4, that $\phi(T_{i})\subseteq\mathrm{End}_{\mathbb{K}}(V)$ is
invertible for all $T_{i}\in\Sigma$. Then, by the universal property of
localisations (see [GW04, Proposition 10.4]), there exists a unique ring
homomorphism $\hat{\phi}:A\Sigma^{-1}\rightarrow\mathrm{End}_{\mathbb{K}}(V)$
such that $\phi=\hat{\phi}\circ\pi$. Since $\phi$ is a $\mathbb{K}$-algebra
homomorphism then so too are $\hat{\phi}$ and $\pi$. We restrict $\hat{\phi}$
to $R_{t}(M_{n})$ to obtain an algebra homomorphism

$\displaystyle\rho:R_{t}(M_{n})$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{End}_{\mathbb{K}}(V)$
$\displaystyle r$ $\displaystyle{}~{}\longmapsto~{}\hat{\phi}(r\cdot 1^{-1}),$

which defines a representation $(\rho,V)$ of $R_{t}(M_{n})$. In particular,
$\hat{\phi}(T_{i}^{-1})=\hat{\phi}(T_{i})^{-1}=\phi(T_{i})^{-1}$, for all
$T_{i}\in\Sigma$, because of the observation in Proposition 4.4(iii). Thus,
writing $r\in R_{t}(M_{n})$ in terms of $T_{i}^{\pm 1}\in A\Sigma^{-1}$, as is
possible by the inclusions shown in Lemma 4.2(ii), we obtain

$\rho(r)=\hat{\phi}(r\cdot
1^{-1})=\hat{\phi}\left(\sum_{\underline{i}\in\mathbb{Z}^{2s+t}}\alpha_{\underline{i}}T_{1}^{i_{1}}\cdots
T_{2s+t}^{i_{2s+t}}\right)=\sum_{\underline{i}\in\mathbb{Z}^{2s+t}}\alpha_{\underline{i}}\phi(T_{1})^{i_{1}}\cdots\phi(T_{2s+t})^{i_{2s+t}},$

where $i_{j}\in\mathbb{Z}$, for all $j\in\llbracket 1,2s+t\rrbracket$, and
$\alpha_{\underline{i}}\in\mathbb{K}^{*}$.

To show irreducibility of $(\rho,V)$ we use the fact that $A\subseteq
R_{t}(M_{n})$ to deduce
$\operatorname{Im}(\phi)\subseteq\operatorname{Im}(\rho)$. Therefore, if
$(\rho,V)$ were reducible this would force $(\phi,V)$ to be reducible, thus
contradicting Proposition 4.4. Hence $(\rho,V)$ is an irreducible
representation of $R_{t}(M_{n})$ of dimension $\ell^{s}$. ∎

######  Remark 4.6.

General techniques to produce irreducible representations of maximal dimension
for completely prime quotients of Quantum Nilpotent Algebras at roots of unity
were developed in [LLR]. Methods from [LLR] would apply to quantum
determinantal rings viewed as completely prime quotients of quantum matrices
since quantum matrices are an example of Quantum Nilpotent Algebra at roots of
unity.

##  5 Quantum Schubert Varieties

We now return to the quantum Grassmannian and its quantum Schubert varieties
$\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}:=\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))/\langle\Pi_{m,n}^{\gamma}\rangle$
for $\gamma\in\Pi_{m,n}$. It follows from Proposition 2.5 that the PI degree
of $\mathcal{O}_{q}(G_{m,n}(\mathbb{K}))_{\gamma}$ is controlled by the matrix
$M_{\gamma}=\left(\begin{array}[]{cc}M_{\lambda}&{\bf 1}\\\
{\bf-1}&0\end{array}\right)$. This leads us to introduce the following
notation.

######  Definition 5.1.

Given any square matrix $M\in M_{N}(\mathbb{Z})$, we define its _extended
matrix_ as $M^{E}:=\left(\begin{smallmatrix}M&\mathbf{-1}\\\
\mathbf{1}^{T}&0\end{smallmatrix}\right)\in M_{N+1}(\mathbb{Z})$, where
$\mathbf{-1}\in\mathbb{Z}^{N}$ is a column vector consisting of $-1$ in each
entry and $\mathbf{1}^{T}\in\mathbb{Z}^{N}$ is a row vector consisting of $1$
in each entry.

Before we start discussing the properties of the extended matrices
$M_{\gamma}=M_{\lambda}^{E}$, we observe that quantum determinantal rings are
also related to quantum Schubert varieties.

######  Lemma 5.2.

Let $q$ be a primitive $\ell^{\mathrm{th}}$ root of unity and
$\gamma=\\{1,\ldots,t,n+1,\ldots,2n-t\\}$. Then

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(G_{n,2n}(\mathbb{K}))_{\gamma})={\rm PI}\text{-}{\rm
deg}(R_{t}(M_{n})[y;\phi])={\rm PI}\text{-}{\rm
deg}\left(\mathcal{O}_{q^{\mathcal{M}_{n,t}^{E}}}(\mathbb{K}^{2nt-t^{2}+1})\right),$

where $\phi$ is the automorphism on $R_{t}(M_{n})$ defined by
$\phi(X_{i,j})=q^{-1}X_{i,j}$ for all generators $X_{i,j}\in R_{t}(M_{n})$ and
$\mathcal{M}_{n,t}^{E}\in M_{2nt-t^{2}+1}(\mathbb{Z})$ is the skew-symmetric
matrix extended from $\mathcal{M}_{n,t}$.

###### Proof.

We apply the isomorphism in [LR08, Proof of Proposition 4.3], along with
[LR08, Remark 4.2(iv)], to obtain

$R_{t}(M_{n})[y,y^{-1};\phi]\cong\mathcal{O}_{q}(G_{n,2n}(\mathbb{K}))_{\gamma}[[\overline{M}]^{-1}],$
(24)

where $[M]=[\\{n+1,\ldots,3n\\}]\in\mathcal{O}_{q}(G_{n,2n}(\mathbb{K}))$ is a
normal element and $[\overline{M}]$ is its canonical image in the quotient
algebra.

Take $q$ to be a root of unity so that $R_{t}(M_{n})$ is a PI algebra (as a
finite module over a central subalgebra $Z_{0}$ [MR01, Corollary 13.1.13]). It
is clear from the definition of $\phi$ that $y^{\ell}$ is a central element
and hence that the skew-Laurent extension $R_{t}(M_{n})[y,y^{-1};\phi]$ is
finite-dimensional over the central subalgebra $\mathbb{K}\langle
Z_{0},y^{\ell}\rangle$. Thus it is also a PI algebra. Since the PI degree is
invariant under localisation, the isomorphism in (24) allows us to deduce

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(G_{n,2n}(\mathbb{K}))_{\gamma})={\rm PI}\text{-}{\rm
deg}(R_{t}(M_{n})[y;\phi]).$

Observing Corollary 3.12(1) and extending
$\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}})$ by the Ore
extension $[y^{\prime};\phi^{\prime}]$, where
$\phi^{\prime}(T_{i,j})=q^{-1}T_{i,j}$ for all generators
$T_{i,j}\in\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}})$, we
deduce that

$\mathrm{Frac}(R_{t}(M_{n})[y;\phi])\cong\mathrm{Frac}(\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}})[y^{\prime};\phi^{\prime}]).$

Upon noting the commutation rules of $y^{\prime}$ with generators $T_{i,j}$,
as defined by the action of $\phi^{\prime}$, we may rewrite the Ore extension
$\mathcal{O}_{q^{\mathcal{M}_{n,t}}}(\mathbb{K}^{2nt-t^{2}})[y^{\prime};\phi^{\prime}]$
as a quantum affine space in $2nt-t^{2}+1$ indeterminates whose commutation
matrix is $\mathcal{M}_{n,t}^{E}$. The result of the lemma follows
immediately. ∎

By Proposition 2.5 and Lemma 3.1, the key to compute the PI degree of quantum
Schubert varieties is to understand the invariant factors and the dimension of
the kernel of $M_{\lambda}^{E}$ and $\mathcal{M}_{n,t}^{E}$. Since we know the
invariant factors for the matrices $M_{\lambda}$ and $\mathcal{M}_{n,t}$, we
devote the next sections to the study of how these invariants change upon
extending the matrix.

###  5.1 Properties of Arbitrary Extended Matrices

Let $M\in M_{N}(\mathbb{Z})$ be an arbitrary skew-symmetric matrix. Let
$\\{h_{i}(M)\\}_{i=1}^{s}$ denote the set of (nonzero) invariant factors of
$M$ and $\\{h_{i}(M^{E})\\}_{i=1}^{s+1}$ denote the set of not necessarily
nonzero invariant factors of $M^{E}$, where $h_{i}(M^{E})\neq 0$ for all
$i\in\\{1,\ldots,s\\}$ and $h_{s+1}(M^{E})$ is either 0 or it is the last
nonzero invariant factor of $M^{E}$.

######  Proposition 5.3.

For a skew-symmetric matrix $M\in M_{N}(\mathbb{Z})$,

$\dim(\ker(M^{E}))=\dim(\ker(M))+\epsilon~{}~{}\text{ with
}~{}~{}\epsilon=\begin{cases}1&\text{if }\ker(M)\subseteq\mathbf{1}^{\bot}\\\
-1&\text{otherwise}\end{cases},$

where $\mathbf{1}^{\bot}$ denotes the orthogonal complement of $\mathbf{1}$ in
$\mathbb{Z}^{N}$.

###### Proof.

Let $W=\\{\mathbf{w}\in\mathbb{Z}^{N}\mid M\mathbf{w}=0\\}=\ker(M)$,
$V=\\{\mathbf{v}\in\mathbb{Z}^{N}\mid M^{E}\mathbf{v}=0\\}=\ker(M^{E})$, and
$\mathbf{1}^{\bot}:=\\{\mathbf{v}\in\mathbb{Z}^{N}\mid\sum_{i=1}^{N}v_{i}=0\\}$
be the orthogonal complement of $\mathbf{1}$ in $\mathbb{Z}^{N}$.

First we prove the identity for $\epsilon=1$. Suppose
$W\subseteq\mathbf{1}^{\bot}$ so that $\sum_{i=1}^{N}w_{i}=0$ for all
$\mathbf{w}\in W$. Introducing the notation $\mathbf{w}[0]:=(w_{1}w_{2}\ldots
w_{N}0)^{T}$ for any $\mathbf{w}\in W$ we see that $\mathbf{w}[0]\in V$ since

$M^{E}\cdot\mathbf{w}[0]=\left(\begin{matrix}M&\mathbf{-1}\\\
\mathbf{1}^{T}&0\end{matrix}\right)\mathbf{w}[0]=\begin{pmatrix}M\mathbf{w}\\\
\sum_{i=1}^{N}w_{i}\end{pmatrix}=\mathbf{0}.$

This allows us to define an injective map $f:W\rightarrow
V;\;\mathbf{w}\mapsto\mathbf{w}[0]$ which gives the inequality
$\dim(W)\leq\dim(V)-1$, since both $M$ and $M^{E}$ are skew-symmetric so the
parity of the dimensions of their kernels must match the parity of the size of
the matrix. To get equality here we will show that $\dim(V)\leq\dim(W)+1$.

Let $V^{\prime}:=\\{\mathbf{v}\in V\mid v_{N+1}=0\\}\subseteq V$ and let
$U:=V\backslash V^{\prime}$ so that $V=V^{\prime}\sqcup U$. First we note that
$\dim(V^{\prime})=\dim(W)$ since $V^{\prime}$ is precisely the image of the
injective map $f$ defined above. The inequality before shows that
$U\neq\emptyset$ since otherwise $V^{\prime}=V$ and $\dim(V)=\dim(W)$, thus
breaking the inequality $\dim(W)\leq\dim(V)-1$. Therefore there exists some
$\mathbf{u}=(u_{1},\ldots,u_{n+1})\in U$ with $u_{N+1}\neq 0$. If
$U=\mathrm{span}\\{\mathbf{u}\\}$ then we immediately obtain
$\dim(V)=\dim(W)+1$. Suppose instead that there exists some other
$\mathbf{u^{\prime}}=(u^{\prime}_{1},\ldots,u^{\prime}_{N+1})\in U$ with
$u^{\prime}_{N+1}\neq 0$ and $\mathbf{u^{\prime}}\neq\mathbf{u}$. With this we
see that

$u_{N+1}\mathbf{u^{\prime}}-u^{\prime}_{N+1}\mathbf{u}=\begin{pmatrix}u_{N+1}u^{\prime}_{1}-u^{\prime}_{N+1}u_{1}\\\
\vdots\\\ u_{N+1}u^{\prime}_{N}-u^{\prime}_{N+1}u_{N}\\\
u_{N+1}u^{\prime}_{N+1}-u^{\prime}_{N+1}u_{N+1}\end{pmatrix}=\begin{pmatrix}u_{N+1}u^{\prime}_{1}-u^{\prime}_{N+1}u_{1}\\\
\vdots\\\ u_{N+1}u^{\prime}_{N}-u^{\prime}_{N+1}u_{N}\\\ 0\end{pmatrix}\in
V^{\prime}.$

Denoting the basis vectors of $V^{\prime}$ as $\\{\mathbf{b_{i}}\\}_{i}\subset
V^{\prime}$ we write the following:

$u_{N+1}\mathbf{u^{\prime}}-u^{\prime}_{N+1}\mathbf{u}=\sum_{i}\lambda_{i}\mathbf{b_{i}}$

for some $\lambda_{i}\in\mathbb{Z}$. When rearranged, this equation expresses
$u_{N+1}\mathbf{u^{\prime}}$ in terms of these basis vectors as well as
$u^{\prime}_{N+1}\mathbf{u}$. Since $\mathbf{u^{\prime}}$ was arbitrary, we
conclude that the set $\\{\mathbf{b_{i}}\\}_{i}$ does not span $V$ but that
$\\{\mathbf{b_{i}}\\}_{i}\cup\\{\mathbf{u}\\}$ is a linearly independent
spanning set for $V$. Since $\\{\mathbf{b_{i}}\\}_{i}$ is a basis for
$V^{\prime}$ and $\dim(V^{\prime})=\dim(W)$ we conclude that
$\dim(V)\leq\dim(W)+1$ as desired. This proves the statement of the lemma for
$\epsilon=1$.

To prove the statement for $\epsilon=-1$ we suppose
$W\nsubseteq\mathbf{1}^{\bot}$ and let $W^{\prime}\subseteq W$ be the set of
all $\mathbf{w}\in W$ which are orthogonal to $\mathbf{1}$, i.e.
$W^{\prime}=\\{w\in W\mid\sum_{i=1}^{N}w_{i}=0\\}$. Then by the injective map
$f|_{W^{\prime}}:W^{\prime}\rightarrow V$ given by
$\mathbf{w^{\prime}}\mapsto\mathbf{w^{\prime}}[0]$ we obtain the inequality
$\dim(W^{\prime})=\dim(W)-1\leq\dim(V)$. Using parity arguments, this means
that either $\dim(V)=\dim(W)-1$ or $\dim(V)>\dim(W)+1$. Suppose
$\dim(V)>\dim(W)+1$ and and consider $V^{\prime}\subseteq V$ as before so that
$\dim(V^{\prime})=\dim(V)-1$. The injective map $V^{\prime}\rightarrow
W;\;\mathbf{v^{\prime}}=(v_{1}\ldots v_{N}0)\mapsto=(v_{1}\ldots v_{N})$ and
our supposition implies

$\dim(W)\geq\dim(V^{\prime})=\dim(V)-1\geq\dim(W)\implies\dim(V^{\prime})=\dim(W)$

but then the injective map becomes bijective and for each $\mathbf{w}\in W$ we
have some $\mathbf{w}[0]\in V^{\prime}$ meaning that
$\mathbf{w}\in\mathbf{1}^{\bot}$ and $W\subseteq\mathbf{1}^{\bot}$. This
contradicts our initial assumption, therefore we conclude $\dim(V)=\dim(W)-1$.
∎

The proof above used arguments from module theory that relied on $\ker(M)$
being a free $\mathbb{Z}$-module with well-defined rank or dimension. When
working over the finite field $\mathbb{Z}/p\mathbb{Z}$, for prime $p$, the
module arguments translate directly to vector space arguments and a similar
proof can be used to show:

######  Corollary 5.4.

Fix some prime number $p$. Let $\overline{M}:=M\mod{p}\in
M_{N}(\mathbb{Z}/p\mathbb{Z})$ and
$\overline{M^{E}}=\left(\begin{smallmatrix}\overline{M}&-\bar{\mathbf{1}}\\\
\bar{\mathbf{1}}^{T}&0\end{smallmatrix}\right)\in
M_{N+1}(\mathbb{Z}/p\mathbb{Z})$ be its extended matrix. Then

$\dim(\ker(\overline{M^{E}}))=\dim(\ker(\overline{M}))+\epsilon,~{}\text{
where }~{}\epsilon=\begin{cases}1&\text{if
}\ker(\overline{M})\subseteq\bar{\mathbf{1}}^{\bot};\\\
-1&\text{otherwise,}\end{cases}$

where $\bar{\mathbf{1}}^{\bot}$ denotes the orthogonal complement of
$\bar{\mathbf{1}}$ in $(\mathbb{Z}/p\mathbb{Z})^{N}$.

######  Proposition 5.5.

Fix a prime number $p$ and let $\overline{M}:=M\mod{p}\in
M_{N}(\mathbb{Z}/p\mathbb{Z})$. Then

$\ker(\overline{M})\subseteq\bar{\mathbf{1}}^{\bot}\iff p\mid
h_{s^{\prime}+1}(M^{E}),$

where $s^{\prime}$ is the number of nonzero invariant factors of
$\overline{M}$, with $s^{\prime}\leq s$, and $\overline{\mathbf{1}}^{\bot}$
denotes the orthogonal complement of $\mathbf{1}$ in
$(\mathbb{Z}/p\mathbb{Z})^{N}$.

###### Proof.

From Corollary 5.4 we see that
$\ker(\overline{M})\subseteq\overline{\mathbf{1}}^{\bot}$ if and only if
$\dim(\ker(\overline{M^{E}}))=\dim(\ker(\overline{M}))+1$. Applying [Tho79,
Theorem 3] to $\overline{M}$ and $\overline{M^{E}}$ to get

$h_{i}(\overline{M^{E}})|h_{i}(\overline{M})|h_{i+1}(\overline{M^{E}})\quad\forall\;i\in[1,s^{\prime}]$
(25)

we see that $\overline{M^{E}}$ has at least as many nonzero invariant factors
as $\overline{M}$. Hence the dimension of the kernel increases in the extended
matrix if and only if $h_{s^{\prime}+1}(\overline{M^{E}})=\overline{0}$. That
is, if and only if $p\mid h_{s^{\prime}+1}(M^{E})$. ∎

###  5.2 Properties of Extended Matrices Coming From (Cauchon-Le) Diagrams

Our results in this section use notation and results proved in [BCL12]. In
particular, for a permutation $\sigma\in S_{k}$, we denote by $P_{\sigma}$ the
corresponding $k\times k$ permutation matrix defined by
$P_{\sigma}[i,j]:=\delta_{j,\sigma(i)}$, where $\delta$ denotes the Kronecker
symbol.

We now take $M=M(D)\in M_{N}(\mathbb{Z})$ to be the matrix corresponding to an
$m\times n$ diagram $D$ with $N$ white squares labelled $1,\ldots,N$. Let
$\tau=\sigma\omega^{-1}$ be the toric permutation associated to $D$, where
$\sigma$ is the restricted permutation associated to $D$ and $\omega$ is the
restricted permutation associated to the $m\times n$ diagram consisting only
of black squares. As noted in the proof of [BCL12, Lemma 4.3], the odd cycles
$\\{\eta_{1},\ldots,\eta_{r}\\}$ in $\tau$ give a basis
$B=\\{\mathbf{v}^{\eta_{i}}\mid i\in\llbracket 1,r\rrbracket\\}$ for
$\ker(P_{\omega}+P_{\sigma})$. Recall that
$\ker(P_{\omega}+P_{\sigma})\cong\ker(M)$ via injective functions
$\phi:\ker(P_{\omega}+P_{\sigma})\rightarrow\ker(M(D))$ and
$\phi:\ker(M(D))\rightarrow\ker(P_{\omega}+P_{\sigma})$ as defined in [BCL12,
Theorem 4.6]. This allows us to transfer the basis $B$ to a basis $\phi(B)$ of
$\ker(M)$.

######  Lemma 5.6.

For any basis element $\mathbf{w}:=\phi(\mathbf{v})\in\ker(M)$ corresponding
to basis element $\mathbf{v}:=\mathbf{v}^{\eta}\in\ker(P_{\omega}+P_{\sigma})$
and odd cycle $\eta$ in $\tau$, we have that

$\sum_{i=1}^{N}w_{i}=\sum_{\begin{subarray}{c}\\{j\in\llbracket
m+1,m+n\rrbracket|\\\ \eta(j)\in\llbracket
1,m\rrbracket\\}\end{subarray}}v_{\eta(j)}~{}~{}-\sum_{\begin{subarray}{c}\\{k\in\llbracket
1,m\rrbracket|\\\ \eta(k)\in\llbracket
m+1,m+n\rrbracket\\}\end{subarray}}v_{\eta(k)}.$ (26)

Writing $\eta=(R_{1},\,C_{1},\,R_{2},\,C_{2},\,\ldots\,R_{k},\,C_{k})$ as
contiguous sequences which alternate between increasing subsequences of row-
labels (the $R_{i}$, which form a partition of $\llbracket 1,m\rrbracket$) and
decreasing subsequences of column-labels (the $C_{i}$, which form a partition
of $\llbracket m+1,m+n\rrbracket$), and letting $v_{1^{\prime}}$ be the first
nonzero entry in $\mathbf{v}$, we can rewrite the above sum as

$\sum_{i=1}^{N}w_{i}=\left(-(-1)^{|R_{1}|}+(-1)^{|R_{1}|+|C_{1}|}-\cdots+(-1)^{|R_{1}|+|C_{1}|+\cdots+|R_{k}|+|C_{k}|}\right)v_{1^{\prime}}.$

###### Proof.

From [BCL12, Lemmas 4.2 & 4.3], $\mathbf{v}\in\ker(P_{\omega}+P_{\sigma})$ is
a vector with entries in $\\{-1,0,1\\}$ satisfying
$v_{i}=(-1)^{k}v_{\eta^{k}(i)}$ for all $k\in\mathbb{N}$ and $i\in\llbracket
1,m+n\rrbracket$. The element $\mathbf{w}\in\ker(M)$ is defined in [BCL12,
Theorem 4.6] to be the vector with entries
$w_{i}=v_{\text{left}(i)}-v_{\text{up}(i)}$, for all $i\in\llbracket
1,N\rrbracket$, where $\text{left}(i)$ and $\text{up}(i)$ is the row- or
column-label reached by following the pipe exiting white square $i$ to the
left and at the top, respectively. Figure 4 illustrates this construction in
the case where $D$ is the $3\times 5$ diagram from Figure 1.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$1$$2$$3$$1$$2$$3$$4$$5$$6$$7$$8$$4$$5$$6$$7$$8$
Figure 4: In this example, $\text{left}(7)=7$ and $\text{up}(7)=6$, so that
$w_{7}=v_{7}-v_{6}$.

From this we obtain

$\sum_{i=1}^{N}w_{i}=\sum_{i=1}^{N}v_{\text{left}(i)}-v_{\text{up}(i)}.$ (27)

Notice that we can split this sum up into the contributions coming from paths
$j\rightarrow\tau(j)$ in $D$, for all $j\in\llbracket 1,m+n\rrbracket$, and if
$j\notin\eta$ then $v_{j}=v_{\tau(j)}=0$. Therefore we only need to consider
contributions from paths $j\rightarrow\eta(j)$ for $j\in\eta$. Label the white
squares that the path $j\rightarrow\eta(j)$ passes through as
$i_{j_{1}},\ldots,i_{j_{s}}$. For each such path we sum the contributions
coming from the white squares $i_{j_{1}},\ldots,i_{j_{s}}$. If $j$ and
$\eta(j)$ are both column-labels then the pipe must exit square $i_{j_{1}}$ to
the left and traverse an even number of white squares before exiting square
$i_{j_{s}}$ at the top. Thus the contribution of this path to the sum (27) is:

$v_{\text{left}(i_{j_{1}})}-v_{\text{up}(i_{j_{2}})}+\ldots+v_{\text{left}(i_{j_{s-1}})}-v_{\text{up}(i_{j_{s}})}=v_{\eta(j)}-v_{\eta(j)}+\ldots+v_{\eta(j)}-v_{\eta(j)}=0.$

Similarly, if $j$ and $\eta(j)$ are both row-labels then the contribution of
the path to the sum (27) is also $0$. If, however, $j$ is a row-label and
$\eta(j)$ is a column-label then $s$ is odd and the contribution becomes:

$-v_{\text{up}(i_{j_{1}})}+v_{\text{left}(i_{j_{2}})}-\ldots+v_{\text{left}(i_{j_{s-1}})}-v_{\text{up}(i_{j_{s}})}=-v_{\eta(j)}+v_{\eta(j)}-\ldots+v_{\eta(j)}-v_{\eta(j)}=-v_{\eta(j)}.$

With similar calculations we see that if $j$ is a column-label and $\eta(j)$
is a row-label then the contribution of the path is $v_{\eta(j)}$. Hence the
only contributions to the sum in (27) come from the paths
$j\rightarrow\eta(j)$ going from a row-label to a column-label, or vice-versa,
and these contributions are $-v_{\eta(j)}$ when $j$ is a row-label and
$v_{\eta(j)}$ when $j$ is a column-label. This proves identity (26).

We now group the consecutive entries of $\eta$ into contiguous subsequences as
discussed in [BCL12, Section 5] and write the permutation as
$\eta=(R_{1},\,C_{1},\,R_{2},\,C_{2},\,\ldots\,R_{k},\,C_{k})$. If we let
$v_{1^{\prime}}$ be the first nonzero entry of $\mathbf{v}$ then these
subsequences become:

$R_{1}=(v_{1^{\prime}},\,v_{\eta(1^{\prime})},\,\ldots,\,v_{\eta^{|R_{1}|-1}(1^{\prime})}),\quad
C_{1}=(v_{\eta^{|R_{1}|}(1^{\prime})},\,\ldots,\,v_{\eta^{|R_{1}|+|C_{1}|-1}(1^{\prime})})$

and, in general, for any $i\in\llbracket 1,k-1\rrbracket$:

$\displaystyle
R_{i+1}=(v_{\eta^{|R_{1}|+|C_{1}|+\cdots+|R_{i}|+|C_{i}|}(1^{\prime})},\,\ldots,\,v_{\eta^{|R_{1}|+|C_{1}|+\cdots+|R_{i}|+|C_{i}|+|R_{i+1}|-1}(1^{\prime})}),$
$\displaystyle
C_{i+1}=(v_{\eta^{|R_{1}|+|C_{1}|+\cdots+|C_{i}|+|R_{i+1}|}(1^{\prime})},\,\ldots,\,v_{\eta^{|R_{1}|+|C_{1}|+\cdots+|R_{i+1}|+|C_{i+1}|-1}(1^{\prime})}).$

The sums on the right-hand side of equation (26) then correspond to summing
the first entries in $R_{1},\ldots,R_{k}$ and the first entries in
$C_{1},\ldots,C_{k}$ respectively. That is, we may write these sums as

$\displaystyle\sum_{\begin{subarray}{c}\\{j\in\llbracket m+1,m+n\rrbracket|\\\
\eta(j)\in\llbracket 1,m\rrbracket\\}\end{subarray}}v_{\eta(j)}$
$\displaystyle=v_{1^{\prime}}+\sum_{i=1}^{k-1}v_{\eta^{|R_{1}|+|C_{1}|+\cdots+|R_{i}|+|C_{i}|}(1^{\prime})}$
$\displaystyle\sum_{\begin{subarray}{c}\\{k\in\llbracket 1,m\rrbracket|\\\
\eta(k)\in\llbracket m+1,m+n\rrbracket\\}\end{subarray}}v_{\eta(k)}$
$\displaystyle=v_{\eta^{|R_{1}|}(1^{\prime})}+\sum_{i=1}^{k-1}v_{\eta^{|R_{1}|+|C_{1}|+\cdots+|C_{i}|+|R_{i+1}|}(1^{\prime})}.$

Upon applying the identity $(-1)^{i}v_{1^{\prime}}=v_{\eta^{i}(1^{\prime})}$,
for all $i\in\mathbb{N}$, we can rewrite the sums above in terms of
$v_{1^{\prime}}$ only, thus completing the proof. ∎

######  Proposition 5.7.

Take $D$ to be an $m\times n$ diagram. $M:=M(D)$, and let
$\\{h_{i}(M)\\}_{i=1}^{s}$ be the nonzero invariant factors of the associated
matrix. Then $h_{i}(M^{E})$ is a power of 2, for all $i\in\llbracket
1,s\rrbracket$, and $h_{s+1}(M^{E})$ is either 0 or its odd prime factors are
bounded above by $\min(m,n)$.

###### Proof.

Let $M=M(D)$ be the matrix corresponding to an $m\times n$ diagram. By [Tho79,
Theorem 3] we obtain the following interlacing inequalities:

$h_{1}(M^{E})|h_{1}(M),~{}h_{2}(M^{E})|h_{2}(M),~{}\ldots,h_{s}(M^{E})|h_{s}(M),$
(28)

where the $|$ means ‘divides’. From this we see that $h_{i}(M^{E})$ are all
powers of 2 for $i\in\llbracket 1,s\rrbracket$, but not necessarily for
$i=s+1$.

Suppose that $h_{s+1}(M^{E})\neq 0$ and that $p|h_{s+1}(M^{E})$ for some odd
prime $p>2$. In this case, by Proposition 5.3 we deduce that
$\ker(M)\nsubseteq\mathbf{1}^{\bot}$ and $\ker(M\mod
p)\subseteq\mathbf{1}^{\bot}\mod p$. To see this we note that when
$h_{s+1}(M^{E})\neq 0$ then $\dim(\ker(M^{E}))=\dim(\ker(M))-1$ (hence
$\ker(M)\nsubseteq\mathbf{1}^{\bot}$, by Proposition 5.3) and when
$p|h_{s+1}(M^{E})$ then $\dim(\ker(M^{E}))<\dim(\ker(M^{E}\mod p))$ and
$\dim(\ker(M))=\dim(\ker(M\mod p))$, since all $h_{i}(M)$ are powers of 2.
Applying Proposition 5.3 to $M\mod p$ and $M^{E}\mod p$, we obtain

$\displaystyle\dim(\ker(M^{E}\mod p))$ $\displaystyle{}>\dim(\ker(M^{E}))$
$\displaystyle{}=\dim(\ker(M))-1$ $\displaystyle{}=\dim(\ker(M\mod p))-1$
$\displaystyle{}=\dim(\ker(M^{E}\mod p))-\epsilon-1,$

where $\epsilon$ is defined as in Corollary 5.4. This implies $\epsilon=1$ and
thus $\ker(M\mod p)\subseteq\mathbf{1}^{\bot}\mod p$.

We now find an upper bound for $p$ given that
$\ker(M)\nsubseteq\mathbf{1}^{\bot}$ and $\ker(M\mod
p)\subseteq\mathbf{1}^{\bot}\mod p$. Without loss of generality we will take
$m<n$ so that $\min(m,n)=m$. Suppose that the disjoint cycle decomposition of
$\tau$ consists of exactly one odd cycle
$\tau=(1^{\prime}\;\tau(1^{\prime})\;\ldots\;\tau^{m+n-1}(1^{\prime}))$, where
$1^{\prime}$ is the first nonzero entry of $\tau$ and we let each successive
label $\tau^{i}(1^{\prime})$ alternate between row- and column-labels for as
long as this is possible. That is, we suppose that
$1^{\prime},\tau^{2}(1^{\prime}),\ldots,\tau^{2m-2}(1^{\prime})$ are row-
labels and
$\tau(1^{\prime}),\tau^{3}(1^{\prime}),\ldots,\tau^{2m-1}(1^{\prime}),\tau^{2m}(1^{\prime}),\allowbreak\ldots,\tau^{m+n-1}(1^{\prime})$
are column-labels. Thus the pair
$\\{\tau^{j}(1^{\prime}),\,\tau^{j+1}(1^{\prime})\\}$ denotes a switch from a
row-label to a column-label $m$ times (when $j=0,2,\ldots,2m-2$) and from a
column-label to a row-label $m$ times (when $j=1,3,\ldots,2m-3,m+n-1$). In
this case, the kernel of $M$ has 1 basis element $\mathbf{w}=\phi(\mathbf{v})$
associated to the odd cycle $\tau$ and vector
$\mathbf{v}=\mathbf{v^{\tau}}\in\ker(P_{\omega}+P_{\sigma})$. Applying
Proposition 5.7 to $\mathbf{w}$ we obtain the alternating sum:

$\displaystyle\sum_{i=1}^{N}w_{k}=v_{1^{\prime}}-v_{\tau(1^{\prime})}+v_{\tau^{2}(1^{\prime})}-\ldots+v_{\tau^{2m-2}(1^{\prime})}-v_{\tau^{m+n-1}(1^{\prime})}=2mv_{1^{\prime}}=\pm
2m.$ (29)

Thus, since $\ker(M\mod p)\subseteq\mathbf{1}^{\bot}\mod p$, the sum above
must be congruent to $0\mod p$. That is, $p$ must be less than $m$, since $p$
is odd. A similar argument can be used if $n\leq m$ to conclude that $p<n$
hence the general result concludes that $p<\min(m,n)$.

It is obvious that if the disjoint cycle decomposition of $\tau$ consists of
shorter length odd cycles $\eta_{1},\ldots,\eta_{r}$ then the absolute value
of the sum (29) corresponding to basis element $\mathbf{w}$ associated to any
$\eta_{i}$ will be less than $2\min(m,n)$. Hence $2\min(m,n)$ is the upper
bound for any toric permutation, $\tau$, and hence any diagram, $D$. ∎

We note that the invariant factors of a matrix $M(D)^{E}$ do not need to be
powers of $2$ as the following example shows.

######  Example 5.8.

An example of a diagram with a matrix whose extended matrix has an invariant
factor which is not a power of 2 is given in Figure 5. The $3\times 3$ diagram
produces a matrix $M(D)$, which has nonzero invariant factors $(1,1)$, whereas
its extended matrix has nonzero invariant factors $(1,1,3)$. This still obeys
the theorem above as the invariant factor $h_{3}(M^{E})$ has prime factor
$3\leq\min(3,3)$.

$D:$ | $\mapsto$$\mathrm{Sk}(M(D)^{E})=\left(\begin{matrix}0&1&0&0&0&0\\\ -1&0&0&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&-1&0&0&0\\\ 0&0&0&0&0&3\\\ 0&0&0&0&-3&0\end{matrix}\right)$
---|---
Figure 5: A diagram $D$ with the skew-normal form of its extended adjacency
matrix $M(D)^{E}$.

###  5.3 PI degree of Quantum Schubert Varieties associated to quantum
determinantal ideals

Recall the matrix $\mathcal{M}_{n,t}$ defining the quantum affine space
associated to $R_{t}(M_{n})$, for $1\leq t<n$. Let $2s_{t}:=2nt-t^{2}-t$ be
its rank so that it has $s_{t}$ (pairs of nonzero) invariant factors, which we
will denote by $\\{h_{i}^{(t)}\\}_{i}$.

The aim of this section is to compute the PI degree of the quantum Schubert
variety $\mathcal{O}_{q}(G_{n,2n}(\mathbb{K}))_{\gamma}$ when
$\gamma=\\{1,\ldots,t,n+1,\ldots,2n-t\\}$.

######  Proposition 5.9.

  1. 1.

$h_{s_{t}+1}(\mathcal{M}^{E}_{n,t})=0$ if and only if $(n=2m\text{ and }t|m)$.

  2. 2.

All invariant factors of $\mathcal{M}_{n,t}^{E}$ are powers of $2$.

  3. 3.

Let $\gamma=\\{1,\ldots,t,n+1,\ldots,2n-t\\}\in\Pi_{n,2n}$. Let $q$ be a
primitive $\ell^{\mathrm{th}}$ root of unity with $\ell$ odd. Then,

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(G_{n,2n}(\mathbb{K}))_{\gamma})=\begin{cases}\ell^{\frac{2nt-t^{2}-t}{2}}&\text{if
}n=2m\text{ and }t|m,\\\
\ell^{\frac{2nt-t^{2}-t}{2}+1}&\text{otherwise.}\end{cases}$

###### Proof.

  1. 1.

For each of the $t$ odd cycles
$\eta_{i}=(i\;\tau(i)\;\ldots\;\tau^{l}(i))=(R_{1},C_{1},R_{2},C_{2},\ldots,R_{k},C_{k})$