Blocks of Birman–Murakami–Wenzl Algebras

Birman, Wenzl[3], and independently Murakami[19] introduced a class of finite dimensional algebras $\mathfrak B_n$, which are known as the Birman–Murakami–Wenzl algebras or BMWalgebras. When the ground field κ contains invertible r and q such that o(q²), the multiplicative order of q², is strictly greater than n, and char(κ), the characteristic of κ is not 2, we determine the structure of the cell module Δ(1, λ) of $\mathfrak B_n$, where λ is any partition of n −2. In particular, we compute the dimension of the simple head of Δ(1, λ). Cox, De Visscher, and Martin have classified the blocks of Brauer algebras Bn in characteristic zero[5]. We solve the similar problem for $\mathfrak B_n$ over the aforementioned field κ. As a by-product, we give a criterion for each module of $\mathfrak B_n$ being equal to its simple head over an arbitrary field.