Tropical Combinatorics and Whittaker functions

TL;DR

This paper reveals a deep connection between geometric RSK correspondence and GL(N,R)-Whittaker functions, leading to new measures and integral formulas relevant for directed polymer models with log-gamma weights.

Contribution

It establishes a fundamental link between geometric RSK and Whittaker functions, generalizing classical identities and providing explicit formulas for polymer partition functions.

Findings

Derived a family of measures related to Whittaker functions

Generalized the Cauchy-Littlewood identity for this setting

Obtained an explicit integral formula for the Laplace transform of the polymer partition function

Abstract

We establish a fundamental connection between the geometric RSK correspondence and GL(N,R)-Whittaker functions, analogous to the well known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with GL(N,R)-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy-Littlewood identity can be seen as a generalisation of an integral identity for GL(N,R)-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integral formula for the Laplace transform of the law of the partition function associated with a one-dimensional directed polymer model with log-gamma weights recently introduced by one of the authors (TS).