$$G(\ell ,k,d)$$ G ( ℓ , k , d ) -modules via groupoids

Abstract

In this note we describe a seemingly new approach to the complex representation theory of the wreath product G ≀ S d where G is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of G ≀ S d. This directly implies a classification of simple modules. As an application, we get a Gelfand model for G ≀ S d from the classical involutive Gelfand model for the symmetric group. We describe the Schur-Weyl duality which motivates our approach and relate it to various Schur-Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type G(ℓ, k, d).

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

“…In the latter case it follows from the classical Schur-Weyl duality for wreath products, see e.g. [63,Theorem 9] that the two actions centralize each other, that is C(im α C ) = im β C and C(im β C ) = im α C where C denotes the centralizer. To obtain the lemma it suffices then to show…”

“…On the uncategorified level this duality can also be found for instance in [11], [82] and implicitly in [9]. It is the coideal version of the well-known duality for wreath products, [63]. We present a categorified version:…”

Semisimplicity of Hecke and (Walled) Brauer Algebras

“…Proof. The statements (a) and (b), in the classical case, are proven in [40,Theorem 9] (see also [40,Remark 12] for the isomorphism criterion). The arguments given there go through for arbitrary K and q ∈ K * as well.…”

“…The arguments given there go through for arbitrary K and q ∈ K * as well. Statement (c) can be deduced from [40,Lemma 11], which again works in the semisimple, quantized case as well. See also [23,Theorem 4.3].…”

“…All of them contribute at most once. We get (for λ = 4ε 1 + 2ε 2 + ε 3 ) λ + ρ − pε 1 = 1 40 H. H. Andersen, C. Stroppel and D. Tubbenhauer [40] Fix a Z-algebra A Z . Recall that there is a trace form · , · : A Z ⊗ A Z → Z on A Z given as follows.…”