Abstract
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 $\mathit{GL}(n,{\mathbb{R}})$ -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.
Highlights
In the paper [19] it was shown that there is a fundamental connection between the geometric RSK (gRSK) correspondence and GL(n, R)-Whittaker functions, analogous to the well-known connection between the RSK correspondence and Schur functions
In the paper [19], an explicit integral formula is obtained for the Laplace transform of the law of the partition function associated with a random directed polymer model on the two-dimensional lattice with log-gamma weights introduced in [42]
We first provide further insight into the results of [19] by showing: (a) the gRSK mapping is volume preserving with respect to the product measure ij dxij /xij on (R>0)n×m, and (b) the integrand in Givental’s integral formula for GL(n, R)-Whittaker functions [26, 32] arises naturally through the application of the gRSK map
Summary
These operators were introduced in the papers [24, 25]. We remark that, in those papers, they are referred to as Baxter Q-type operators by analogy with similar operators that were originally introduced by Baxter (see, for example, [6, 7]) as a tool for solving the eight-vertex model. Define Ψλn(x) for λ ∈ Cn, x ∈ (R>0)n recursively as follows. These functions were first introduced by Jacquet [31] They play an important role in the theory of automorphic forms [14,15,16, 27, 30, 43, 44] and the quantum Toda lattice [24,25,26, 32, 34, 36, 41]. For λ ∈ Ch and s > 0 (this condition is only required if h > n) This formula is just a re-writing of the above definition (2.7) of Ψλn;s. It is not obvious from the above definition, the function Ψλn is invariant under permutations of the indices λ1, . Γ (λi − λj )−1, i=j is the Sklyanin measure
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