Generating Functions for Hecke Algebra Characters

Konvalinka, Matjaž ; Skandera, Mark

Canadian journal of mathematics, Tome 63 (2011) no. 2, pp. 413-435

Résumé

Certain polynomials in ${{n}^{2}}$ variables that serve as generating functions for symmetric group characters are sometimes called $\left( {{S}_{n}} \right)$ 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 ${{H}_{n}}\left( q \right)$ , 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 ${{H}_{n}}\left( q \right)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis–Lê–Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.

Détail

DOI : 10.4153/CJM-2010-082-7

Mots-clés : 15A15, 20C08, 81R50, determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring