We prove that the 2-variable BMW algebra embeds into an algebra constructed from the HOMFLY-PT polynomial. We also prove that the $\mathfrak{so}_{2N}$-BMW algebra embeds in the $q$-Schur algebra of type $A$. We use these results to suggest a schema providing categorifications of the $\mathfrak{so}_{2N}$-BMW algebra.

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.