Friday, October 19, 2012

1210.5126 (Neil O'Connell et al.)

Geometric RSK correspondence, Whittaker functions and symmetrized random

Neil O'Connell, Timo Seppäläinen, Nikos Zygouras
We show that the geometric lifing of the RSK correspondence introduced by A.N. Kirillov (2001) is volume preserving with respect to a natural product measure on its domain, and that the integrand in Givental's integral formula for GL(n,R)-Whittaker functions arises naturally in this context. This yields a new proof (and generalisation) of Stade's Whittaker integral identity, which can be seen as the analogue of the Cauchy-Littlewood identity in this setting. We also consider the restriction of the geometric RSK mapping to symmetric matrices and show that the volume preserving property continues to hold. The corresponding Whittaker integral identity involves only a single Whittaker function. As an application, we determine the law of the partition function for a random directed polymer model with log-gamma weights which are constrained to be symmetric about the main diagonal, with an additional `pinning' factor on the main diagonal.
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