## Duke Mathematical Journal

### 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.

#### Article information

Source
Duke Math. J., Volume 163, Number 3 (2014), 513-563.

Dates
First available in Project Euclid: 11 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1392128877

Digital Object Identifier
doi:10.1215/00127094-2410289

Mathematical Reviews number (MathSciNet)
MR3165422

Zentralblatt MATH identifier
1288.82022

#### Citation

Corwin, Ivan; O’Connell, Neil; Seppäläinen, Timo; Zygouras, Nikolaos. Tropical combinatorics and Whittaker functions. Duke Math. J. 163 (2014), no. 3, 513--563. doi:10.1215/00127094-2410289. https://projecteuclid.org/euclid.dmj/1392128877

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