The Birman–Murakami–Wenzl Algebras of Type D_n

Abstract

The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2 n + 1)n!! − (2 n−1 + 1)n! over a specified commutative ring R, where n!! = 1 • 3 • • • (2n − 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring Z[δ ±1 ]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley-Lieb algebra of type Dn is a subalgebra of the BMW algebra of the same type.

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE Dn

“…Then both ν (8) and ν (7) are reducible and we conclude like in the general case. So, suppose dim(W 8 ) = 14.…”

“…= e j g i + m (e j − e j e i ) when i ∼ j g j e i e j = g −1 i e j = g i e j + m (e j − e i e j ) when i ∼ j e Informations that relate to rank or cellularity can be found in [7]. The CGW algebra of type D n is a generalization of the BMW algebra to type D n .…”

“…Suppose ν (8) is reducible and let W be an irreducible invariant subspace of V 8 . By the discussion of §3.2 of this paper and Appendix B of [20], the degrees of the irreps of H(D 8 ) that are less than 56 are 1,7,8,14,20,21,28,35,42,48 Suppose l ∈ { …”

The Birman-Murakami-Wenzl algebras of type E n

“…This is known for the classical case; see [17]. In [8], it was derived for type D n . In this paper, we prove it for types E 6 , E 7 , E 8 , so that it is established for all spherical simply laced types.…”

“…Although in this paper we provide bases of the BMW algebras of type E n (n = 6,7,8) that are built up from ingredients of the corresponding root systems in the same way as the other types, an interpretation in terms of tangles is still open.…”