The representations of cyclotomic BMW algebras

Abstract

a b s t r a c tIn this paper, we classify the irreducible B r,n -modules over a field.

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

“…It is shown in [7,10,37], and [29] that cyclotomic BMW algebras over an admissible ground ring are cellular in the sense of Graham and Lehrer [13]. The proof in [7] depends on the isomorphism of cyclotomic BMW algebras and cyclotomic Kaufmann tangle algebras established here.…”

“…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.…”

“…An appropriate condition, called admissibility, was introduced by Wilcox and Yu in [34], and another, called u-admissibility, by Rui and Xu [29]. It turns out that, for integral ground rings with q − q −1 = 0, admissibility and u-admissibility are equivalent [8].…”

“…Appropriate conditions can be found by considering the representation theory of the 2-strand algebra, in particular representations in which the generator e 1 acts non-trivially. This idea was used by Ariki, Mathas, and Rui in a related context [3]; the analogue of their condition for cyclotomic BMW algebras was introduced by Rui and Xu [29]. Another notion of admissibility for cyclotomic BMW algebras was discovered by Wilcox and Yu [34], again by analyzing the representation of the 2-strand algebra.…”

“…Another notion of admissibility for cyclotomic BMW algebras was discovered by Wilcox and Yu [34], again by analyzing the representation of the 2-strand algebra. It has been shown that the admissibility conditions of [29] and [34] coincide for integral ground rings with q − q −1 = 0 [8].…”