On the structure of cyclotomic Nazarov–Wenzl algebras

Abstract

Ariki, Mathas and Rui [S. Ariki, A. Mathas, H. Rui, Cyclotomic Nazarov–Wenzl algebras, Nagoya Math. J. 182 (2006) 47–134 (special volume in honor of Professor G. Lusztig)] introduced a class of finite dimensional algebras W r , n , called the cyclotomic Nazarov–Wenzl algebras which are associative algebras over a commutative ring R generated by { S i , E i , X j ∣ 1 ≤ i < n and 1 ≤ j ≤ n } satisfying the defining relations given in this paper. In particular, E i 2 = ω 0 E i for any positive integers i ≤ n − 1 . Note that W r , n ′ s are quotients of affine Wenzl algebras in [M. Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra 182 (1996) 664–693]. It has been proved in the first cited reference above that W r , n is cellular in the sense of [J.J. Graham, G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996) 1–34]. Using the representation theory of cellular algebras, Ariki, Mathas and Rui have classified the irreducible W r , n -modules under the assumption ω 0 ≠ 0 in their above-cited work. In this paper, we are going to classify the irreducible W r , n -modules under the assumption ω 0 = 0 . We will compute the Gram determinant associated to each cell module for W r , n no matter whether ω 0 is zero or not. At the end of this paper, we use our formulae for Gram determinants to determine the semisimplicity of W r , n for arbitrary parameters over an arbitrary field F with char F ≠ 2 .