Brauer algebras of type C are cellularly stratified

Abstract

In a recent paper Cohen, Liu and Yu introduce the Type C Brauer algebra. We show that this algebra is an iterated inflation of hyperoctahedral groups, and that it is cellularly stratified. This gives an indexing set of the standard modules, results on decomposition numbers, and the conditions under which the algebra is quasi-hereditary.

Brauer algebras of type $$\mathrm{I}_2^n$$ I 2 n ( $$n\ge 5$$ n ≥ 5 )

“…Applying the similar arguments about cellularity in [1,5,7], and [14]; therefore, the theorem below can be obtained.…”

“…By observation, we can transform this equality problem to the elementary problem solved at the beginning of this section in the following way. Consider the paths of a particle starting from (1,1) in the diagram of the left-hand sides of the images under φ of (1.12)-(1.14) with the m − 1 horizontal strands at the top and the m − 1 horizontal strands at the bottom removed and transform the horizontal strands as in Fig. 6.…”

Decomposition numbers for the cyclotomic Brauer algebras in characteristic zero

“…We have that B 2 n (δ) is a subalgebra of the recently defined Brauer algebra of type C n (see [CLY]). This can be seen by 'unfolding' the diagrams (as outlined in [MGP07, Section 4.3]) and using [Bow,Theorem 3.6].…”

Induction and restriction functors for cellular categories

“…It was shown that many important classes of algebras arising in representation theory, invariant theory, knot theory, subfactors and statistical mechanics are cellular (see e.g. [4,17,20,24,40,41,49,50]), and most of their categorical analogues are also cellular, such as Temperley-Lieb categories [48], Brauer diagram categories [30], partition categories [23,33], the categories of invariant tensors for certain quantised enveloping algebras and their highest weight representations [46,47], the categories of Soegrel bimodules [16] and other more general Hecke categories (a strictly object-adapted cellular category due to [15]). As is known, if an algebra admits a cellular structure, one will have a practicable way to describe the representations and homological properties of the algebra [20,8].…”