Semisimplicity of Hecke and (Walled) Brauer Algebras

Abstract

We show how to use Jantzen's sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of U q -tilting modules (for any field K and any parameter q ∈ K − {0, −1}). As an application, we recover the semisimplicity criteria for the Hecke algebras of types A and B, the walled Brauer algebras and the Brauer algebras from our more general approach.2010 Mathematics subject classification: primary 17B37; secondary 17B10, 20C08.

Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality

“…[38], in particular Br d (N ) is semisimple. In fact, Br d (δ) is generically semisimple, [84]; for δ ≥ 0 it is semisimple in precisely the following cases, see [16] or [71], [4]: δ = 0, and δ ≥ d − 1 or δ = 0, and d = 1, 3, 5.…”

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

“…Remark 3.3 Generically, the algebra Br d (δ) is semi-simple, but not semi-simple for specific integral values for δ (dependent on d), see [24,29] and also [1]. For a detailed study of the non semi-simple algebras over the complex numbers we refer to [12, 2.2].…”

“…Then define (k,0) the surjectivity of the theorem follows if we show that f ∈ im(Φ). Thanks to Claim (1) we know that f is contained in the span of the oriented generalized Brauer diagrams (r, c, p) with c ∈ B[d] (l,q) where l ≤ k − 1 or l = k and q = 0. But then by Claim (1) we have l ≤ k − 1 and some q.…”

On Grothendieck ring of the sequence of walled Brauer algebras

“…As a generalization of the classical Schur-Weyl duality, the centralizer of the natural action of GL n (C) on a mixed tensor space V ⊗r ⊗ W ⊗s , with V = C n and W = V * , was characterized as the walled Brauer algebra B r,s (n), for n r + s. Walled Brauer algebras have been extensively studied, including the cellularity, semi-simplicity, decomposition numbers, Jucys-Murphy elements, block theory, Kazhdan-Lusztig theory and so on. We refer the reader to [1,3,4,5,9,12] for details.…”