Certain polynomials in $n^2$ variables that serve as generatingfunctions for symmetric group characters are sometimes called($S_n$) character immanants.We point out a close connection between the identities ofLittlewood--Merris--Watkinsand Goulden--Jackson, which relate $S_n$ character immanantsto the determinant, the permanent and MacMahon's Master Theorem.From these results we obtain a generalizationof Muir's identity.Working with the quantum polynomial ring and the Hecke algebra$H_n(q)$, we define quantum immanants that are generatingfunctions for Hecke algebra characters.We then prove quantum analogs of the Littlewood--Merris--Watkins identitiesand selected Goulden--Jackson identitiesthat relate $H_n(q)$ character immanants tothe quantum determinant, quantum permanent, and quantum Master Theoremof Garoufalidis--L^e--Zeilberger.We also obtain a generalization of Zhang's quantization of Muir'sidentity.
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