TY - JOUR

T1 - Geometric RSK correspondence, Whittaker functions and symmetrized random polymers

AU - O'Connell, Neil

AU - Seppäläinen, Timo

AU - Zygouras, Nikos

PY - 2014

Y1 - 2014

N2 - We show that the geometric lifting of the RSK correspondence introduced by A.N. Kirillov (Physics and Combinatorics. Proc. Nagoya 2000 2nd Internat Workshop, pp. 82-150, 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. Apart from providing further evidence that Whittaker functions are the natural analogue of Schur polynomials in this setting, our results also provide a new 'combinatorial' framework for the study of random polymers. When the input matrix consists of random inverse gamma distributed weights, the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of Whittaker functions. Next we restrict the geometric RSK mapping to symmetric matrices and show that the volume preserving property continues to hold. We determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal, with an additional factor on the main diagonal. The third combinatorial mapping studied is a variant of the geometric RSK mapping for triangular arrays, which is again showed to be volume preserving. This leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal. We also show that the analogues of the Cauchy-Littlewood identity in the setting of this paper are equivalent to a collection of Whittaker integral identities conjectured by Bump (Number Theory, Trace Formulas, and Discrete Groups, pp. 49-109, 1989) and Bump and Friedberg (Festschrift in Honor of Piatetski-Shapiro, Part II, pp. 47-65, 1990) and proved by Stade (Am. J. Math. 123:121-161, 2001; Israel J. Math. 127:201-219, 2002). Our approach leads to new 'combinatorial' proofs and generalizations of these identities, with some restrictions on the parameters.

AB - We show that the geometric lifting of the RSK correspondence introduced by A.N. Kirillov (Physics and Combinatorics. Proc. Nagoya 2000 2nd Internat Workshop, pp. 82-150, 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. Apart from providing further evidence that Whittaker functions are the natural analogue of Schur polynomials in this setting, our results also provide a new 'combinatorial' framework for the study of random polymers. When the input matrix consists of random inverse gamma distributed weights, the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of Whittaker functions. Next we restrict the geometric RSK mapping to symmetric matrices and show that the volume preserving property continues to hold. We determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal, with an additional factor on the main diagonal. The third combinatorial mapping studied is a variant of the geometric RSK mapping for triangular arrays, which is again showed to be volume preserving. This leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal. We also show that the analogues of the Cauchy-Littlewood identity in the setting of this paper are equivalent to a collection of Whittaker integral identities conjectured by Bump (Number Theory, Trace Formulas, and Discrete Groups, pp. 49-109, 1989) and Bump and Friedberg (Festschrift in Honor of Piatetski-Shapiro, Part II, pp. 47-65, 1990) and proved by Stade (Am. J. Math. 123:121-161, 2001; Israel J. Math. 127:201-219, 2002). Our approach leads to new 'combinatorial' proofs and generalizations of these identities, with some restrictions on the parameters.

UR - http://www.scopus.com/inward/record.url?scp=84904512803&partnerID=8YFLogxK

U2 - 10.1007/s00222-013-0485-9

DO - 10.1007/s00222-013-0485-9

M3 - Article (Academic Journal)

AN - SCOPUS:84904512803

VL - 197

SP - 361

EP - 416

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 2

ER -