Tropical combinatorics and Whittaker functions

Abstract

We establish a fundamental connection between the geometric Robinson–Schensted–Knuth (RSK) correspondence and $\mathrm{GL}(N,\mathbb{R})$-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 $\mathrm{GL}(N,\mathbb{R})$-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 generalization of an integral identity for $\mathrm{GL}(N,\mathbb{R})$-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 $1$-dimensional directed polymer model with log-gamma weights recently introduced by one of the authors.