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

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• 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 rc">rmulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300024&_mathId=si1.gif&_user=111111111&_pii=S0097316516300024&_rdoc=1&_issn=00973165&md5=53ceb952f36d1526ffbcb22944671e13" title="Click to view the MathML source">V_λr hidden">roll">row>Vrow>row>λrow> of rc">rmulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300024&_mathId=si2.gif&_user=111111111&_pii=S0097316516300024&_rdoc=1&_issn=00973165&md5=f2c82166183af2d106b5ebdfe1e8987a" title="Click to view the MathML source">GL_2nr hidden">roll">row>riant="normal">GLrow>row>2nrow> 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 rc">rmulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300024&_mathId=si3.gif&_user=111111111&_pii=S0097316516300024&_rdoc=1&_issn=00973165&md5=775aac74887aa76245c138f0218b933d" title="Click to view the MathML source">λ^′r hidden">roll">row>λrow>row>′row> is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered rc">rmulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300024&_mathId=si4.gif&_user=111111111&_pii=S0097316516300024&_rdoc=1&_issn=00973165&md5=ba3cfb11e2d1ff464b3514d2cfb1bf4a" title="Click to view the MathML source">q,tr hidden">roll">q,t-generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the rc">rmulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300024&_mathId=si25.gif&_user=111111111&_pii=S0097316516300024&_rdoc=1&_issn=00973165&md5=9986c95a941df88c3fee596745dcd08f" title="Click to view the MathML source">q=0r hidden">roll">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.