[Paper Review] Enumeration of $\C{H}$-strata in quantum matrices with respect to dimension

Jason P. Bell, Karel Casteels+1 more|arXiv (Cornell University)|Sep 13, 2010

Advanced Combinatorial Mathematics15 references3 citations

TL;DR

This paper provides a combinatorial formula to compute the dimension of $Χ$-strata in the algebra of $m\times n$ quantum matrices $\mathcal{O}_q(M_{m,n}(\mathbb{K}))$ by associating each stratum to a permutation via pipe-dreams and showing the dimension equals the number of odd cycles in this permutation. The key result is a closed-form generating function for the number of $d$-dimensional strata, which resolves prior conjectures on their asymptotic proportions as $n \to \infty$. The method relies on Cauchon diagrams, restricted permutations, and an isomorphism between kernel spaces of skew-symmetric matrices and permutation matrices.

ABSTRACT

We present a combinatorial method to determine the dimension of $\C{H}$-strata in the algebra of $m imes n$ quantum matrices $\Oq$ as follows. To a given $\C{H}$-stratum we associate a certain permutation via the notion of pipe-dreams. We show that the dimension of the $\C{H}$-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the $d$-dimensional $\C{H}$-strata in $\Oq$. Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors \cite{bldim,bll} regarding the asymptotic proportion of $d$-dimensional $\C{H}$-strata in $\Oq$.

Motivation & Objective

• To determine a combinatorial criterion for the dimension of $\mathcal{H}$-strata in $\mathcal{O}_q(M_{m,n}(\mathbb{K}))$.
• To resolve conjectures by Bell and Launois on the asymptotic proportion of $d$-dimensional $\mathcal{H}$-strata as $n \to \infty$.
• To establish a closed-form generating function counting $d$-dimensional $\mathcal{H}$-strata in terms of $m$, $n$, and $d$.

Proposed method

• Associate each $\mathcal{H}$-prime to a Cauchon diagram, which parametrizes the $\mathcal{H}$-strata via a combinatorial grid coloring rule.
• Use the bijection between Cauchon diagrams and restricted permutations $\sigma \in S_{m+n}$ satisfying $|\sigma(i) - i| \leq \max(m,n)$.
• Define the toric permutation $\tau = \sigma \omega^{-1}$, where $\omega$ is the maximum element in the reverse Bruhat order on restricted permutations.
• Establish an isomorphism between $\ker(M(D))$ and $\ker(P_\sigma + P_\omega)$, where $M(D)$ is a skew-symmetric matrix derived from the Cauchon diagram $D$, and $P_\mu$ is the matrix representation of permutation $\mu$.
• Show that $\dim(\ker(M(D)))$ equals the number of odd cycles in $\tau$, where a cycle is odd if it has even length.
• Derive a trivariate generating function $C(x,y,t)$ whose coefficient of $\frac{x^m}{m!}\frac{y^n}{n!}t^d$ gives the number of $d$-dimensional $\mathcal{H}$-strata in $\mathcal{O}_q(M_{m,n}(\mathbb{K}))$.

Experimental results

Research questions

• RQ1 What is the dimension of a given $\mathcal{H}$-stratum in $\mathcal{O}_q(M_{m,n}(\mathbb{K}))$?
• RQ2 How can the number of $d$-dimensional $\mathcal{H}$-strata be enumerated across all $m,n$?
• RQ3 What is the asymptotic proportion of $d$-dimensional $\mathcal{H}$-strata in $\mathcal{O}_q(M_{m,n}(\mathbb{K}))$ as $n \to \infty$?
• RQ4 Is there a closed-form generating function for the number of $d$-dimensional $\mathcal{H}$-strata in terms of $m$, $n$, and $d$?
• RQ5 Does the proportion of $d$-dimensional $\mathcal{H}$-strata in $\mathcal{O}_q(M_{n,n}(\mathbb{K}))$ converge as $n \to \infty$?

Key findings

• The dimension of an $\mathcal{H}$-stratum in $\mathcal{O}_q(M_{m,n}(\mathbb{K}))$ is equal to the number of odd cycles in the toric permutation $\tau = \sigma \omega^{-1}$ associated with its $\mathcal{H}$-prime.
• The number of $d$-dimensional $\mathcal{H}$-strata in $\mathcal{O}_q(M_{m,n}(\mathbb{K}))$ is given by the coefficient of $\frac{x^m}{m!}\frac{y^n}{n!}t^d$ in the trivariate generating function $C(x,y,t)$, which is explicitly computed.
• As $n \to \infty$, the proportion of $d$-dimensional $\mathcal{H}$-strata in $\mathcal{O}_q(M_{m,n}(\mathbb{K}))$ tends to $a(d)/(m!2^m)$, where $a(d) = [t^d](t+1)(t+3)\cdots(t+2m-1)$.
• The formula for $h(m,n,d)$, the number of $d$-dimensional $\mathcal{H}$-strata, is derived explicitly for small $m$, such as $m=2,3,4,5$, and matches known results from prior work.
• The generating function $C(x,y,t)$ is expressed as a finite sum involving Stirling numbers of the second kind and Pochhammer symbols, enabling efficient computation via symbolic algebra systems like Maple.
• The limit of the proportion of $d$-dimensional $\mathcal{H}$-strata in $\mathcal{O}_q(M_{n,n}(\mathbb{K}))$ as $n \to \infty$ is shown to exist and is conjectured to be $a(d)/(n!2^n)$, though the exact value remains open.