Tropical combinatorics and Whittaker functions. (English) Zbl 1288.82022

Corwin, Ivan; O’Connell, Neil; Seppäläinen, Timo; Zygouras, Nikolaos

Duke Math. J. 163, No. 3, 513-563 (2014).

Summary: 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 D. Bump [Automorphic forms on \(\mathrm{GL}(3,\mathbb R)\). Berlin etc.: Springer (1984; Zbl 0543.22005)] and E. Stade [Isr. J. Math. 127, 201–219 (2002; Zbl 1032.11020)]. 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.

MSC:

82B23	Exactly solvable models; Bethe ansatz
60B20	Random matrices (probabilistic aspects)
05E05	Symmetric functions and generalizations
05E10	Combinatorial aspects of representation theory
82D60	Statistical mechanics of polymers
33C15	Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)

Keywords:

Robinson-Schensted-Knuth (RSK) correspondence; Whittaker functions; polymer model

Citations:

Zbl 0543.22005; Zbl 1032.11020

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