This paper constructs an explicit homogeneous cellular basis for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A.Specht modules of Brundan, Kleshchev and Wang to give a graded basis of H Λ n and then in section 6 we construct the dual graded basis and use this to show that the blocks of H Λ n are graded symmetric algebras. As an application we construct an isomorphism between the graded Specht modules and the dual of the dual graded Specht modules, which are defined using our second graded cellular basis of H Λ n . In an appendix, which was actually the starting point for this work, we use a different approach to explicitly describe the homogeneous elements which span the one dimensional two-sided ideals of H Λ n .
Graded cellular algebrasThis section defines graded cellular algebras and develops their representation theory, extending Graham and Lehrer's [20] theory of cellular algebras. Most of the arguments of Graham and Lehrer apply with minimal change in the graded setting. In particular, we obtain graded cell modules, graded simple and projective modules and a graded analogue of Brauer-Humphreys reciprocity. §2.1. Graded algebras Let R be a commutative integral domain with 1. In this paper a graded R-module is an R-module M which has a direct sum decompositionthen m is homogeneous of degree d and we set deg m = d. If M is a graded R-module let M be the ungraded R-module obtained by forgetting the grading on M . If M is a graded R-module and s ∈ Z let M s be the graded R-module obtained by shifting the grading on M up by s; that is, M s d = M d−s , for d ∈ Z.A graded R-algebra is a unital associative R-algebra A = d∈Z A d which is a graded R-module such that A d A e ⊆ A d+e , for all d, e ∈ Z. It follows that 1 ∈ A 0 and that A 0 is a graded subalgebra of A. A graded (right) A-module is a graded R-module M such that M is an A-module and M d A e ⊆ M d+e , for all d, e ∈ Z. Graded submodules, graded left A-modules and so on are all defined in the obvious way. Let A-Mod be the category of all finitely generated graded A-modules together with degree preserving homomorphisms; that is,for all M, N ∈ A-Mod. The elements of Hom A (M, N ) are homogeneous maps of degree 0. More generally, if f ∈ Hom A (M d , N ) ∼ = Hom A (M, N −d ) then f is a homogeneous map from M to N of degree d and we write deg f = d. Set Hom Z A (M, N ) = d∈Z Hom A (M d , N ) ∼ = d∈Z Hom A (M, N −d ) for M, N ∈ A-Mod. §2.2. Graded cellular algebras Following Graham and Lehrer [20] we now define graded cellular algebras.
Abstract. Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain "cyclotomic quotients" of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank r n (2n − 1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.
In this paper we use the Hecke algebra of type B to define a new algebra S which is an analogue of the q‐Schur algebra. We show that S has ‘generic’ basis which is independent of the choice of ring and the parameters q and Q. We then construct Weyl modules for S and obtain, as factor modules, a family of irreducible S‐modules defined over any field. 1991 Mathematics Subject Classification: 16G99, 20C20, 20G05.
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