Canad. J. Math. 63(2011), 413-435
Printed: Apr 2011
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
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
We also obtain a generalization of Zhang's quantization of Muir's
determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring
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]