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Séminaire Lotharingien de Combinatoire, 85B.57 (2021), 12 pp.

# Steven N. Karp and Hugh Thomas

*q*-Whittaker Functions, Finite Fields, and Jordan Forms

**Abstract.**
The *q*-Whittaker function *W*_{λ}(**x**;*q*) associated to a partition λ is a *q*-analogue of the Schur function
*s*_{λ}(**x**), and the *t*=0 specialization of the Macdonald polynomial
*P*_{λ}(**x**;*q*,*t*). We give a new formula for *W*_{λ}(**x**;*q*) in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size 1/*q*, 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, which we call the *q*-Burge correspondence. In the *q* → 0 limit, we recover a description of the classical Burge correspondence (also known as column RSK) due independently to Gansner (1981), Spaltenstein (1982), and Steinberg (1988) for permutation matrices, and to Rosso (2012) in general. Finally, we apply the *q*-Burge correspondence to prove enumerative formulas for certain modules over the preprojective algebra of a path quiver.

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

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