q-Whittaker functions, finite fields, and Jordan forms

• Steven N. Karp ^[1] ; Hugh Thomas ^[2]
  1. [1] University of Notre Dame

     University of Notre Dame

     Township of Portage, Estados Unidos

     
  2. [2] University of Quebec at Montreal

     University of Quebec at Montreal

     Canadá

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 4, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01048-3
• Enlaces
  • Texto completo
• Resumen

  • The q-Whittaker function Wλ(x;q)W_\lambda(\mathbf{x}; q)Wλ(x;q) associated to a partition λ\lambdaλ is a q-analogue of the Schur function sλ(x)s_\lambda(\mathbf{x})sλ(x), and is defined as the t=0t = 0t=0 specialization of the Macdonald polynomial Pλ(x;q,t)P_\lambda(\mathbf{x}; q, t)Pλ(x;q,t). We show combinatorially how to expand Wλ(x;q)W_\lambda(\mathbf{x}; q)Wλ(x;q) in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size 1/q1/q1/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→0q \to 0q→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 GLn\mathrm{GL}_nGLn 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.

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