We establish a fundamental connection between the geometric Robinson–Schensted–Knuth (RSK) correspondence and -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 -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 -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 -dimensional directed polymer model with log-gamma weights recently introduced by one of the authors.
Ivan Corwin. Neil O’Connell. Timo Seppäläinen. Nikolaos Zygouras. "Tropical combinatorics and Whittaker functions." Duke Math. J. 163 (3) 513 - 563, 15 February 2014. https://doi.org/10.1215/00127094-2410289