Vanishing integrals for Hall-Littlewood polynomials - Mathematics > Combinatorics

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Abstract: It is well known that if one integrates a Schur function indexed by apartition $\lambda$ over the symplectic resp. orthogonal group, the integralvanishes unless all parts of $\lambda$ have even multiplicity resp. all partsof $\lambda$ are even. In a recent paper of Rains and Vazirani, Macdonaldpolynomial generalizations of these identities and several others weredeveloped and proved using Hecke algebra techniques. However at $q=0$ theHall-Littlewood level, these approaches do not work, although one can obtainthe results by taking the appropriate limit. In this paper, we develop a directapproach for dealing with this special case. This technique allows us to provesome identities that were not amenable to the Hecke algebra approach, as wellas to explicitly control the nonzero values. Moreover, we are able togeneralize some of the identities by introducing extra parameters. This leadsus to a finite-dimensional analog of a recent result of Warnaar, which uses theRogers-Szeg\-o polynomials to unify some existing summation type formulas forHall-Littlewood functions.

Autor: Vidya Venkateswaran

Fuente: https://arxiv.org/