Representation theory of symmetric groups and related Hecke algebras

Abstract

Abstract. We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via categorification. We present results on branching rules and crystal graphs, decomposition numbers and canonical bases, graded representation theory, connections with cyclotomic and affine Hecke algebras, Khovanov-Lauda-Rouquier algebras, category O, W -algebras, etc.

On graded decomposition numbers for cyclotomic Hecke algebras in quantum characteristic 2

“…It has been suggested that all graded decomposition numbers of F p S d might have zero coefficients in negative powers of q, which is equivalent to saying that graded adjustment matrices (defined by equation (5)) have integer entries: cf. remarks in [2, Subsection 5.6] and [7,Subsection 10.3]. Using the aforementioned results, we find examples showing that this is not the case when p = 2 (Corollary 6).…”

“…subject to a lengthy list of relations given in [1] (see also, for example, [7]). In particular, the relations ensure that the elements e(i), i ∈ I d , are pairwise orthogonal idempotents summing to 1.…”

“…Section 2 is a brief review of definitions and results about graded cyclotomic Hecke algebras H Λ d and their Specht modules. Generally, we follow the notation of [7], which is a much more detailed survey of the subject. The new results are in Section 3.…”

Relationship between Mullineux involution and the generalized regularization

“…The q-analogue of decomposition numbers d λ,µ (q) is defined by considering the graded multiplicity of D µ inside S λ and we refer to [Kle10] for details regarding its definitions and properties. Lascoux, Leclerc and Thibon conjectured in [LLT96] that when F = C, the d λ,µ (q)'s are the same coefficients in the expansion of the lower global crystal basis {G(µ) | µ ∈ P is b-regular} explained as follows.…”

On signed Young permutation modules and signed p-Kostka numbers

“…By Lemma 4.4, each indecomposable projective summand of W k (γ|δ) is isomorphic to some Q(α|β). By the 'wedge' shape of the decomposition matrix of S n with columns labelled by p-restricted partitions (see for instance [20,Theorem 5.2]) and Brauer reciprocity, the ordinary character of P α contains the irreducible character χ α exactly once. Hence the ordinary character of Q(α|β) contains the character…”