Cellular bases for the Brauer and Birman–Murakami–Wenzl algebras

Abstract

An explicit combinatorial construction is given for cellular bases (in the sense of Graham and Lehrer) for the Birman-Murakami-Wenzl and Brauer algebra. We provide cell modules for the Birman-Murakami-Wenzl and Brauer algebras with bases index by certain bitableaux, generalising the Murphy basis for the Specht modules of the Iwahori-Hecke algebra of the symmetric group. The bases for the cell modules given here are constructed non-diagrammatically and hence are relatively amenable to computation.

Specht modules and semisimplicity criteria for Brauer and Birman–Murakami–Wenzl algebras

“…The proof of Theorem 3.1 given in [4] rests upon the following facts, respectively Proposition 3.2 of [12] and Proposition 3.3 of [4], stated below for later reference.…”

“…, and applying Proposition 3.7 or Corollary 3.1 of [4], we may assume that v ∈ D f,n , whenever a u,v = 0 in the above expression.…”

“…(2) In Section 3, we define a generic version of the B-M-W algebras and restate in a more transparent notation the main results of [4] on cellular bases of the same algebras. (3) In Section 4, we state for reference some consequences following from the statements in Section 3 and the theory of cellular algebras given in [5].…”

Cellular Bases for Algebras with a Jones Basic Construction

“…The version of the presentation given here follows [31]. Cellularity of the BMW algebras was established in [8,9,43]. Morton and Wassermann [31] give a realization of the BMW algebra as an algebra of (n, n)-tangle diagrams modulo regular isotopy and the following Kauffman skein relations:…”

BMW algebra, quantized coordinate algebra and type 𝐶 Schur–Weyl duality

“…Note that in the notation of [22,Proposition 3.7], it is easy to check (in all the three cases listed in [22,Proposition 3.7]…”