The Representations of Cyclotomic BMW Algebras, II

Abstract

In this paper, we give a recursive formula to compute the Gram determinant associated to each cell module of the cyclotomic BMW algebras B r,n over an integral domain. As a by-product, we determine explicitly when B r,n is semisimple over a field. This generalizes our previous result on Birman-Murakami-

The Structure of Simple Modules of Birman-Murakami-Wenzl Algebras

“…The following results were proved by Rui and Si for cyclotomic BMW algebras [14] and we only need the special case here.…”

Cyclotomic Birman–Wenzl–Murakami Algebras, II: Admissibility Relations and Freeness

“…We also thank Wilcox and Yu for pointing out that we could remove the hypothesis that the parameter δ 0 is invertible. Recent work of Rui and Xu [29] and Rui and Si [28] also concerns the theory of cyclotomic BMW algebras and has considerable overlap with our work. These authors do not discuss tangle realizations of the algebras, but they obtain additional representation theoretic results.…”

Non-vanishing Gram determinants for cyclotomic Nazarov–Wenzl and Birman–Murakami–Wenzl algebras

“…However, he did assume that ω 0 is invertible. Recently, B r,n have been studying extensively by three groups of mathematicians in [12,17,18,13,36,33,41,[38][39][40]42], etc.…”

“…B r,n ) is cellular over R. We remark that Goodman and Graber [15] give a new proof of the cellularity of B r,n . In this paper, we will make use of JM-basis for B r,n [33] which is a weakly cellular basis. Goodman and Graber [16] constructed JM-basis by using different method, which recovers our results on JM-basis.…”

“…We thank Professor F. Goodman for his comments on the presentation of the paper. been studied extensively by Goodman and Hauschild, Wilcox and Yu, Xu and the authors in [12,17,13,18,33,36,[38][39][40][41][42]. In particular, it has been proved in [38] that the cyclotomic BMW algebras are cellular algebras in the sense of [19].…”