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

详细信息    查看全文

• 作者：Vidya Venkateswaran ^{vidyav@math.mit.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Hall&ndash ; Littlewood polynomials ; Integral identities ; p-adic representation theory
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：145
• 期：Complete
• 页码：369-399
• 全文大小：489 K

文摘

If one restricts an irreducible representation V_λ$V_{\lambda }$ of GL_2n${\mathrm{GL}}_{2n}$ 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^′${\lambda }^{\prime }$ is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered q,t$q,t$-generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the q=0$q=0$ 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.