[Paper Review] Generating functions and statistics on spaces of maximal tori in classical Lie groups

Jason Fulman, Rita Jiménez Rolland+1 more|arXiv (Cornell University)|Oct 21, 2016

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper uses generating function techniques to derive explicit formulas for stable values of polynomial statistics on $F$-stable maximal tori in classical Lie groups of type A, B, and C. It establishes twisted homological stability and computes generating functions for stable twisted Betti numbers, showing they are quasipolynomials satisfying linear recurrence relations.

ABSTRACT

In this paper we use generating function methods to obtain new asymptotic results about spaces of $F$-stable maximal tori in $GL_n(\overline{F_q})$, $Sp_{2n}(\overline{F_q})$, and $SO_{2n+1}(\overline{F_q})$. We recover stability results of Church--Ellenberg--Farb and Jiménez Rolland--Wilson for "polynomial" statistics on these spaces, and we compute explicit formulas for their stable values. We derive a double generating function for the characters of the cohomology of flag varieties in type B/C, which we use to obtain analogs in type B/C of results of Chen: we recover "twisted homological stability" for the spaces of maximal tori in $Sp_{2n}(\mathbb{C})$ and $SO_{2n+1}(\mathbb{C})$, and we compute a generating function for their "stable twisted Betti numbers". We also give a new proof of a result of Lehrer using symmetric function theory.

Motivation & Objective

• To provide new generating function-based proofs of asymptotic stability results for polynomial statistics on $F$-stable maximal tori in classical Lie groups.
• To compute explicit stable values for these statistics using generating functions, extending prior results by Church–Ellenberg–Farb and Jiménez Rolland–Wilson.
• To derive a double generating function for the cohomology characters of flag varieties in type B/C, enabling analogs of Chen’s results in type A.
• To establish that stable twisted Betti numbers in type B/C are quasipolynomials with bounded quasiperiod and finite recurrence order.
• To give a new symmetric function-theoretic proof of Lehrer’s theorem on character polynomials and point counts in type A.

Proposed method

• Utilizes generating functions to analyze the left-hand side of Lehrer’s character formula for $F$-stable maximal tori in $GL_n$, $Sp_{2n}$, and $SO_{2n+1}$ over finite fields.
• Applies symmetric function theory to reprove Lehrer’s result in type A, focusing on the coinvariant algebra and inner products of class functions.
• Derives a double generating function for the characters of the cohomology of flag varieties in type B/C, using the structure of the hyperoctahedral group $B_n$.
• Computes generating functions for stable twisted Betti numbers as rational functions with denominators involving $1 - z^i$ and $1 + z^i$, revealing quasipolynomial structure.
• Analyzes the generating function’s poles to deduce that stable Betti numbers are quasipolynomials of degree at most $\deg(P) - 1$ with quasiperiod dividing $\text{lcm}(2, 4, \dots, 2\deg(P))$.
• Uses recurrence relations derived from the generating function to characterize the stable Betti numbers for representations such as $\mathbb{C}^n$, $\bigwedge^2\mathbb{C}^n$, and $\bigwedge^3\mathbb{C}^n$.

Experimental results

Research questions

• RQ1 What are the explicit stable values of polynomial statistics on $F$-stable maximal tori in classical Lie groups of type A, B, and C?
• RQ2 How can generating functions be used to derive twisted homological stability results in type B/C analogous to Chen’s results in type A?
• RQ3 What is the structure of the generating function for stable twisted Betti numbers in type B/C, and what does it imply about their functional form?
• RQ4 Do stable twisted Betti numbers in type B/C satisfy linear recurrence relations, and if so, what determines their order and coefficients?
• RQ5 Can symmetric function theory be used to give a new proof of Lehrer’s character formula for type A?

Key findings

• The stable values of polynomial statistics on $F$-stable maximal tori in $GL_n$, $Sp_{2n}$, and $SO_{2n+1}$ are explicitly computed via generating functions.
• The generating function for stable twisted Betti numbers in type B/C is a rational function whose denominator is a product of terms $1 - z^i$ and $1 + z^i$, with degrees bounded by the degree of the character polynomial.
• Stable twisted Betti numbers are quasipolynomials of degree at most $\deg(P) - 1$, with quasiperiod dividing $\text{lcm}(2, 4, \dots, 2\deg(P))$.
• For the canonical representation $\mathbb{C}^n$, the stable Betti numbers are $\beta_d = 1$ if $d$ is odd and $0$ if $d$ is even, satisfying $\beta_d = \beta_{d-2}$ for $d \geq 3$.
• For $\bigwedge^2\mathbb{C}^n$, the stable Betti numbers satisfy $\beta_d = \beta_{d-2} + \beta_{d-4} - \beta_{d-6}$ for $d \geq 6$, and are given by a piecewise quasipolynomial with period 4.
• For $\bigwedge^3\mathbb{C}^n$, the stable Betti numbers are quasipolynomials of degree 2 with period 12, satisfying a recurrence of order 12, and exhibit a quadratic dependence on $d$ modulo 12.