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# Generating Functions for Hecke Algebra Characters

Published:2010-12-29
Printed: Apr 2011
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
• Mark Skandera,
Department of Mathematics, Lehigh University, Bethlehem, PA 18015, U.S.A.
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## Abstract

Certain polynomials in $n^2$ variables that serve as generating functions for symmetric group characters are sometimes called ($S_n$) character immanants. We point out a close connection between the identities of Littlewood--Merris--Watkins and Goulden--Jackson, which relate $S_n$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra $H_n(q)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood--Merris--Watkins identities and selected Goulden--Jackson identities that relate $H_n(q)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis--L\^e--Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.
 Keywords: determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring
 MSC Classifications: 15A15 - Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 20C08 - Hecke algebras and their representations 81R50 - Quantum groups and related algebraic methods [See also 16T20, 17B37]

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