Hebing Rui

Cyclotomic Nazarov-Wenzl algebras

Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra , in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank r n ( 2n −1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

A criterion on the semisimple Brauer algebras II

In [H. Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser. A 111 (2005) 78-88], the first author gave an algorithm for determining the pairs (n, δ) such that the Brauer algebra Bn(δ) over a field F is semisimple. Such an algorithm involves a subset Z(n) ⊂ Z. In this note, we give an explicit description about Z(n). Using [H. Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser. A 111 (2005) 78-88, 1.3] we verify Enyangs conjecture given in [J. Enyang, Specht modules and semisimplicity criteria for Brauer and Birman-Murakami-Wenzl algebras, preprint, 2005, 12.2].

Based algebras and standard bases for quasi-hereditary algebras

Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly fullbased. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.

Gram determinants and semisimplicity criteria for Birman-Wenzl algebras

Abstract In this paper, we compute all Gram determinants associated to all cell modules of Birman-Wenzl algebras. As a by-product, we give a necessary and sufficient condition for Birman-Wenzl algebras being semisimple over an arbitrary field.

SPECHT MODULES FOR ARIKI-KOIKE ALGEBRAS

Specht modules for an Ariki-Koike algebra H of type G(m, 1, r) have been investigated recently in the context of cellular algebras (see [GL] and [DJM]). Thus, these modules are defined as quotient modules of certain “permutation” modules, that is, defined as “cell modules” via cellular bases. So cellular bases play a decisive roˆle in these work. However, the classical theory [C] or the work [DJ1], [DJ2] in the case when m = 1,2 (i.e., the case of type A and B Hecke algebras) suggest that a construction as submodules without using cellular bases should exist. Following our previous work [DR], we shall introduce in this paper Specht modules for an Ariki-Koike algebra as submodules of those “permutation” modules, and prove that they are isomorphic to those (as quotient modules) defined in [DJM]. This results in some new versions of several known results such as standard basis and branching theorems. Under the hypothesis of the Morita equivalence theorem given in [DR], we shall also prove that Brundans result which generalizes Kleshchevs branching rules [K] for symmetric groups to type A Hecke algebras can be lifted to Ariki-Koike algebras in this case. We point out that, using quantum group approach, I. Grojnowski [[G], 9.14] has established branching rules for Ariki-Koike algebras in general. However, it is not yet clear even in the type A case if one can use these rules to obtain Kleshchevs branching rules or vise versa.

Markov traces on cyclotomic Temperley-Lieb algebras

In this note, we use generalized Tchebychev polynomials to define a trace function which satisfies certain conditions. Such a trace will be called the Markov trace. In particular, we obtain formulae for the weights of the Markov trace. As a corollary, we get a combinatorial identity. This generalizes Joness 1983 result on Temperley-Lieb algebras.