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

Algebra & Discrete Mathematics

Speaker: Dr. Vidya Venkateswaran, UC Davis
Location: 2112 MSB
Start time: Mon, May 19 2014, 12:10PM

If one restricts an irreducible representation of $GL_{n}$ to the orthogonal subgroup (respectively, the symplectic subgroup), classical branching rules tell us when the trivial representation is contained in the restricted representation. In both cases, the partition $\lambda$ that indexes the original representation must satisfy a particular condition: in the orthogonal (respectively, symplectic) case, $\lambda$ (resp. $\lambda'$) must have all even parts. Using character theory, these results may be rephrased in terms of integrals involving the Schur functions. Since Hall-Littlewood polynomials are $t$-generalizations of Schur functions, one may consider $t$-analogs of these results. We will discuss these identities, focusing on an interpretation using $p$-adic representation theory that parallels the Schur case.