Representation theory of towers of recollement: Theory, notes, and examples

Abstract

We give an axiomatic framework for studying the representation theory of towers of algebras. We introduce a new class of algebras, contour algebras, generalising (and interpolating between) blob algebras and cyclotomic Temperley-Lieb algebras. We demonstrate the utility of our formalism by applying it to this class.

The blocks of the Brauer algebra in characteristic zero

“…After reviewing the definition of the Brauer algebra, we will show that families of such algebras form towers of recollement in the sense of [CMPX06] (which we will see follows from various results of Doran et al [DWH99]). This will be the framework in which we base our analysis of these algebras.…”

“…To prove these results we use the theory of towers of recollement developed in [CMPX06]. This approach is already closely modelled, for , in work of Doran, Wales, and Hanlon [DWH99] (since both papers use the methods developed in [Mar96]).…”

“…The paper begins with a section defining the various objects of interest, and a review of their basic properties in the spirit of [CMPX06]. This is followed by a brief section describing some basic results about Littlewood-Richardson coefficients which will be needed in what follows.…”

Decomposition numbers for Brauer algebras of type in characteristic zero

“…In this section we study B m,p,n via Clifford theory. By verifying the first two conditions of a tower of recollement, we deduce conditions under which the algebra is quasi-hereditary; we leave it as an exercise for the reader to check that the algebra is a tower of recollement (in the sense of [CMPX06]) by checking that it obeys the remaining conditions. 3.1.…”

“…The idempotent e n−2 (for n = 6) in the cases that δ = 0, δ = 0 respectively. A tower of recollement was defined in [CMPX06] to be a family of algebras (with idempotents) satisfying six conditions (A1-6). It is easy to see that (3.1.1) e n−2 B m,p,n e n−2 ∼ = B m,p,n−2 and that (3.1.2) B m,p,n /B m,p,n e n−2 B m,p,n ∼ = kG(m, p, n).…”