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1. CJM 2013 (vol 66 pp. 453)
A Remark on BMW algebra, $q$-Schur Algebras and Categorification 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.
Keywords:tangle algebras, BMW algebra, HOMFLY-PT Skein algebra, q-Schur algebra, categorification Categories:57M27, 81R50, 17B37, 16W99 |
2. CJM 2010 (vol 63 pp. 413)
Generating Functions for Hecke Algebra Characters
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 Categories:15A15, 20C08, 81R50 |