A p-adic interpretation of some integral identities for Hall–Littlewood polynomials

Autor: Vidya Venkateswaran
Rok vydání: 2017
Předmět:
Zdroj: Journal of Combinatorial Theory, Series A. 145:369-399
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2016.03.006
Popis: If one restricts an irreducible representation V λ of GL 2 n to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of λ are even (resp. the conjugate partition λ ' is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered q , t -generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the q = 0 limit (Hall-Littlewood), and provided direct combinatorial arguments for these identities; this approach led to various generalizations and a finite-dimensional analog of a recent summation identity of Warnaar. In this paper, we reformulate some of these results using p-adic representation theory; this parallels the representation-theoretic interpretation in the Schur case. The nonzero values of the identities are interpreted as certain p-adic measure counts. This approach provides a p-adic interpretation of these identities (and a new identity), as well as independent proofs. As an application, we obtain a new Littlewood summation identity that generalizes a classical result due to Littlewood and Macdonald. Finally, our p-adic method also leads to a generalized integral identity in terms of Littlewood-Richardson coefficients and Hall polynomials.
Databáze: OpenAIRE
Externí odkaz: