Tom Halverson

q-ROOK MONOID ALGEBRAS, HECKE ALGEBRAS, AND SCHUR-WEYL DUALITY

When we were at the beginnings of our careers, Sergeis support helped us to believe in our work. He generously encouraged us to publish our results on Brauer and Birman–Murakami–Wenzl algebras, results which had in part, or possibly in total, been obtained earlier by Sergei himself. He remains a great inspiration for us, both mathematically and in our memory of his kindness, modesty, generosity, and encouragement to the younger generation. Bibliography: 19 titles.

Representations of the Rook-Brauer Algebra

We study the representation theory of the rook-Brauer algebra , which has a of Brauer diagrams that allow for the possibility of missing edges. The Brauer, Temperley–Lieb, Motzkin, rook monoid, and symmetric group algebras are subalgebras of . We prove that is the centralizer of the orthogonal group on tensor space and that and are in Schur–Weyl duality. When x ∈ ℂ is chosen so that is semisimple, we use its Bratteli diagram to explicitly construct a complete set of irreducible representations for .

Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of classical type

In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of the Iwahori-Hecke algebras of types An−1, Bn, and Dn. Our method is a generalization of a derivation of the Murnaghan-Nakayama formula for the irreducible characters of the symmetric group given by Curtis Greene. Greene’s approach is to sum up the diagonal entries of the matrices of certain cycle permutations in Young’s seminormal representations. The analogues of the Young seminormal representations for the Iwahori-Hecke algebras of types An−1, Bn, and Dn were given by Hoefsmit.

Character Formulas for q -Rook Monoid Algebras

AbstractThe q-rook monoid Rn(q) is a semisimple ℂ(q)-algebra that specializes when q → 1 to ℂ[Rn], where Rn is the monoid of n × n matrices with entries from {0, 1} and at most one nonzero entry in each row and column. We use a Schur-Weyl duality between Rn(q) and the quantum general linear group

Commuting families in Hecke and Temperley-Lieb algebras

U_q {\mathfrak{g}}{\mathfrak{l}}(r)

Teaching computational science in a liberal arts environment

to compute a Frobenius formula, in the ring of symmetric functions, for the irreducible characters of Rn(q). We then derive a recursive Murnaghan-Nakayama rule for these characters, and we use Robinson-Schensted-Knuth insertion to derive a Roichman rule for these characters. We also define a class of standard elements on which it is sufficient to compute characters. The results for Rn(q) specialize when q = 1 to analogous results for Rn.

Iwahori-Hecke algebras of type A, bitraces and symmetric functions

The partition algebra