Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

q-Whittaker functions, finite fields, and Jordan forms

Steven N. Karp, Hugh Thomas

TL;DR

The paper studies $q$-Whittaker functions $W_\lambda(\mathbf{x};q)$ as $q$-analogues of Schur functions, constructing a geometric proof of the $q$-Cauchy identity via a probabilistic bijection called the $q$-Burge correspondence, defined through pairs of partial flags and nilpotent endomorphisms over $\mathbb{F}_{1/q}$. It provides two expansions of $W_\lambda$—one by counting flags strictly compatible with a fixed nilpotent and one dual expansion by counting nilpotent endomorphisms with fixed flags—thereby linking symmetric-function data to geometry of nilpotent orbits and double cosets. The $q$-Burge correspondence yields a weight-preserving bijection between contingency tables and pairs of semistandard tableaux, proving the $q$-Cauchy identity and, in the limit $q\to0$, recovering Rosso’s classical Burge correspondence. The work further connects these combinatorics to representation theory by enumerating isomorphism classes of modules over type $A$ preprojective algebras refined by socle filtrations, thereby bridging symmetric-function combinatorics, flag-variety geometry, and quiver representations. It also discusses growth diagrams and polynomiality questions, highlighting how the $q$-Burge framework differs from insertion-based bijections and proposing avenues for future exploration in related algebraic geometry and representation theory contexts.

Abstract

The $q$-Whittaker function $W_λ(\mathbf{x};q)$ associated to a partition $λ$ is a $q$-analogue of the Schur function $s_λ(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial $P_λ(\mathbf{x};q,t)$. We show combinatorially how to expand $W_λ(\mathbf{x};q)$ in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size $1/q$. This yields an expression analogous to a well-known formula for the Hall-Littlewood functions. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, proving the Cauchy identity for $q$-Whittaker functions. We call our probabilistic bijection the $q$-Burge correspondence, and prove that in the limit as $q\to 0$, we recover a description of the classical Burge correspondence (also known as column RSK) due to Rosso (2012). A key step in the proof is the enumeration of an arbitrary double coset of $\text{GL}_n$ modulo two parabolic subgroups, which we find to be of independent interest. As an application, we use the $q$-Burge correspondence to count isomorphism classes of certain modules over the preprojective algebra of a type $A$ quiver (i.e. a path), refined according to their socle filtrations. This develops a connection between the combinatorics of symmetric functions and the representation theory of preprojective algebras.

q-Whittaker functions, finite fields, and Jordan forms

TL;DR

The paper studies -Whittaker functions as -analogues of Schur functions, constructing a geometric proof of the -Cauchy identity via a probabilistic bijection called the -Burge correspondence, defined through pairs of partial flags and nilpotent endomorphisms over . It provides two expansions of —one by counting flags strictly compatible with a fixed nilpotent and one dual expansion by counting nilpotent endomorphisms with fixed flags—thereby linking symmetric-function data to geometry of nilpotent orbits and double cosets. The -Burge correspondence yields a weight-preserving bijection between contingency tables and pairs of semistandard tableaux, proving the -Cauchy identity and, in the limit , recovering Rosso’s classical Burge correspondence. The work further connects these combinatorics to representation theory by enumerating isomorphism classes of modules over type preprojective algebras refined by socle filtrations, thereby bridging symmetric-function combinatorics, flag-variety geometry, and quiver representations. It also discusses growth diagrams and polynomiality questions, highlighting how the -Burge framework differs from insertion-based bijections and proposing avenues for future exploration in related algebraic geometry and representation theory contexts.

Abstract

The -Whittaker function associated to a partition is a -analogue of the Schur function , and is defined as the specialization of the Macdonald polynomial . We show combinatorially how to expand in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size . This yields an expression analogous to a well-known formula for the Hall-Littlewood functions. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, proving the Cauchy identity for -Whittaker functions. We call our probabilistic bijection the -Burge correspondence, and prove that in the limit as , we recover a description of the classical Burge correspondence (also known as column RSK) due to Rosso (2012). A key step in the proof is the enumeration of an arbitrary double coset of modulo two parabolic subgroups, which we find to be of independent interest. As an application, we use the -Burge correspondence to count isomorphism classes of certain modules over the preprojective algebra of a type quiver (i.e. a path), refined according to their socle filtrations. This develops a connection between the combinatorics of symmetric functions and the representation theory of preprojective algebras.
Paper Structure (35 sections, 61 theorems, 268 equations, 1 table)

This paper contains 35 sections, 61 theorems, 268 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Vector spaces
  4. Weak compositions and partitions
  5. Tableaux
  6. q-analogues
  7. q-Whittaker functions
  8. Nilpotent endomorphisms and Jordan forms
  9. Pairs of partial flags and relative position
  10. Partial flag varieties
  11. Schubert decomposition
  12. Parabolic subgroup decomposition
  13. Relative position
  14. Parabolic double cosets of GL(n)
  15. q-Whittaker functions via partial flags
  16. ...and 20 more sections

Key Result

Theorem 1.1

We have the Cauchy identity for $q$-Whittaker functions where the sum on the right-hand side is over all partitions $\lambda$. Equivalently, let $\alpha = (\alpha_1, \dots, \alpha_k)$ and $\beta = (\beta_1, \dots, \beta_l)$ be weak compositions of $n$. Taking the coefficient of $x_1^{\alpha_1}\cdots x_k^{\alpha_k}y_1^{\beta_1}\cdots y_l^{\beta_l}$ on eac where the sum on the left-hand side is ove

Theorems & Definitions (230)

  • Theorem 1.1: Macdonald macdonald95
  • Theorem 1.2
  • Example 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 220 more