Graded cellular bases for the cyclotomic Khovanov–Lauda–Rouquier algebras of type A

Abstract

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.

Graded decomposition numbers for the blob algebra

“…Proof: Claims (a), (b) and (c) are [8,Lemma 2.13]. Part (d) is a direct consequence of the definitions.…”

“…Then, the node α is below β if and only if t λ (α) > t λ (β). Using the dominance order there is a similar interpretation of the concept of to be below introduced in [8,Section 4]. Since t λ does not coincide with the unique maximal bitableau for the dominance order, the two concepts does not coincide in general.…”

Projective modules for the symmetric group and Young's seminormal form

“…This is the key Lemma 4.1 of [HuMa1]. Hence via Lemma 5, we get that e λ is the symmetrized idempotent projector on a generalized weight space for the JucysMurphy operators.…”

“…Our solution to the problem uses once again Young's seminormal form and goes back to ideas of Hu and Mathas. In fact, the arguments that lead to equation (40) below are closely related to the arguments leading to the crucial Theorem 4.14 of [HuMa1].…”

Nilpotency in type A cyclotomic quotients

“…We also remark that prior to Brundan and Kleshchev's work, Brundan and Stroppel [5] [6] have defined a graded cellular basis for the algebras R Λ ν in type A. We hope that understanding these nilpotency degrees will be a step towards constructing explicit homogeneous monomial bases for these quotients.…”