[Paper Review] Proof of Stasinski and Voll's Hyperoctahedral Group Conjecture

Aaron Landesman|arXiv (Cornell University)|Aug 29, 2014

Advanced Combinatorial Mathematics3 references3 citations

TL;DR

This paper proves a conjecture by Stasinski and Voll on the signed generating function of the L-statistic over quotients of the hyperoctahedral group $B_n$, establishing a product formula for all quotients. Using recursive structure and sign-reversing involutions to cancel terms, the authors show that the generating function matches a rational product formula, with applications to Poincaré polynomials of symmetric matrices and representation zeta functions.

ABSTRACT

In a recent paper, Stasinski and Voll introduced a length-like statistic on hyperoctahedral groups and conjectured a product formula for this statistic's signed distribution over arbitrary quotients. Stasinski and Voll proved this conjecture for a few special types of quotients. We prove this conjecture in full, showing it holds for all quotients. In the case of signed permutations with at most one descent, this formula gives the Poincare polynomials for the varieties of symmetric matrices of a fixed rank.

Motivation & Objective

• To prove the full conjecture of Stasinski and Voll on the signed distribution of the L-statistic over all quotients of the hyperoctahedral group $B_n$.
• To establish a connection between this generating function and Poincaré polynomials of symmetric matrices over $\mathbb{F}_q$ of fixed rank.
• To demonstrate that the result implies the validity of an identity in representation zeta functions as described in Brenti and Carnevale.
• To generalize the generating function to a two-variable version and conjecture a divisibility condition involving $tX + 1$ when $0 \in I$.

Proposed method

• Define the L-statistic on $B_n$ as half the number of pairs $(i,j)$ with $i < j$, $w(i) > w(j)$, and $i \not\equiv j \pmod{2}$, measuring signed distance from identity over opposite-parity indices.
• Use induction on $n$ to prove the generating function $\sum_{w \in B_n^{I^c}} (-1)^{l(w)} X^{L(w)}$ satisfies the same recurrence as the product formula $f_{n,I}(X)$.
• Introduce sign-reversing involutions called "swaps" to cancel terms in the sum, reducing the support to only "initially uncancelled" and "finally uncancelled" elements.
• Characterize initially uncancelled and finally uncancelled elements as those fixed under certain swap operations, and construct a bijection between these sets to verify the recurrence.
• Leverage the Coxeter group structure of $B_n$, including descent sets $D(w)$ and standard generators $s_i$, to analyze the recursive behavior of $l(w)$ and $L(w)$.
• Use the notation $f_{n,I}(X) = \frac{(\underline{n})!}{(\underline{i_1})! \cdot \prod_{k=1}^l (\underline{\delta_k})!!}$ with $\delta_k = i_{k+1} - i_k$, and $\underline{n} = 1 - X^n$ for $n > 0$, $\underline{0} = 1$.

Experimental results

Research questions

• RQ1 Does the signed generating function $\sum_{w \in B_n^{I^c}} (-1)^{l(w)} X^{L(w)}$ equal the product formula $f_{n,I}(X)$ for all subsets $I \subset [n-1]_0$?
• RQ2 Can the L-statistic's distribution over $B_n^{I^c}$ be fully characterized via recursive structure and sign-reversing involutions?
• RQ3 What is the connection between this generating function and the Poincaré polynomials of symmetric matrices over $\mathbb{F}_q$ of fixed rank?
• RQ4 Does the two-variable generating function $\sum_{w \in B_n^{I^c}} t^{l(w)} X^{L(w)}$ admit a factorization involving $tX + 1$ if and only if $0 \in I$?
• RQ5 How do the descent sets $D(w)$ and the quotient structure $B_n^{I^c}$ influence the recursive decomposition of the generating function?

Key findings

• The conjecture of Stasinski and Voll is fully proven: $\sum_{w \in B_n^{I^c}} (-1)^{l(w)} X^{L(w)} = f_{n,I}(X)$ holds for all $n \in \mathbb{N}$ and all $I \subset [n-1]_0$.
• The generating function for the L-statistic over $B_n^{I^c}$ matches the rational product formula $f_{n,I}(X)$, which is defined via $\underline{n} = 1 - X^n$ and double factorials of even indices.
• For signed permutations with at most one descent, the formula yields the Poincaré polynomials of the variety of $n \times n$ symmetric matrices over $\mathbb{F}_q$ of fixed rank.
• The result implies the validity of an identity in [5, Proposition 5.5], confirming a previously unproven relation in representation zeta functions.
• When $0 \in I$, the two-variable generating function $\sum_{w \in B_n^{I^c}} t^{l(w)} X^{L(w)}$ is divisible by $tX + 1$, as shown via an involution on elements with $w_{1,1} = \pm 1$.
• The conjecture that $tX + 1$ divides the two-variable generating function if and only if $0 \in I$ is supported by computational verification for $n \leq 6$, though a general proof remains open.