Search: MSC category 15A15
( Determinants, permanents, other special matrix functions [See also 19B10, 19B14] )
1. 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
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
Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring
Categories:15A15, 20C08, 81R50
2. CJM 2001 (vol 53 pp. 758)
||Inequivalent Transitive Factorizations into Transpositions |
The question of counting minimal factorizations of permutations into
transpositions that act transitively on a set has been studied extensively
in the geometrical setting of ramified coverings of the sphere and in the
algebraic setting of symmetric functions.
It is natural, however, from a combinatorial point of view to ask how such
results are affected by counting up to equivalence of factorizations, where
two factorizations are equivalent if they differ only by the interchange of
adjacent factors that commute. We obtain an explicit and elegant result for
the number of such factorizations of permutations with precisely two
factors. The approach used is a combinatorial one that rests on two
We believe that this approach, and the combinatorial primitives that have
been developed for the ``cut and join'' analysis, will also assist with the
Keywords:transitive, transposition, factorization, commutation, cut-and-join
Categories:05C38, 15A15, 05A15, 15A18