Irreducible Specht modules for Iwahori–Hecke algebras of type $B$

Abstract

Abstract. We consider the problem of classifying irreducible Specht modules for the Iwahori-Hecke algebra of type B with parameters Q, q. We solve this problem completely in the case where q is not a root of unity, and in the case q = −1 we reduce the problem to the corresponding problem in type A.

The reducible Specht modules for the Hecke algebra $\mathcal{H}_{\mathbb{C},{-}1}(\mathfrak{S}_{n})$

“…An interesting special case of this problem is the question of which Specht modules are irreducible. For the symmetric group algebra FS n , the answer to this question is completely known, and for the Hecke algebra H n , the answer is known except in the case where q = −1 [JM1, L1,F1,F2,JLM,L2,F3]. In this paper we prove one half of a conjecture (Conjecture 2.3 below) that describes the irreducible Specht modules when q = −1 and p = 0, and we give a conjecture for the case of positive characteristic.…”

“…We summarise the relevant results here. For details, and to put these results into context, we refer the reader to [FL,Section 5].…”

“…The tableaux T, N are shown in the two diagrams below, and we see that N(λ, µ) = 10. The following lemma follows from [FL,Lemma 5.4 & Lemma 5.5].…”

Kronecker positivity and 2-modular representation theory

“…The classification of simple Specht modules for symmetric groups and their Hecke algebras was a massive undertaking involving over 30 years work [Jam78, JLM06, JM99, Fay05, JM96, Lyl07, JM97, Fay04, FL09, FL13,Fay10]. The pursuit of a description of semisimple and decomposable Specht modules is similarly old [Jam78] and yet has proven a much more difficult nut to crack.…”

On the irreducible Specht modules for Iwahori–Hecke algebras of type A with q=−1

“…This problem has been solved in all cases except when q = −1 and the characteristic of F is not 2. Various partial results are known for this case, together with a conjectured solution [FL,Conjecture 2.2] for the case where F has infinite characteristic (we adopt the convention that the characteristic of a field is the order of its prime subfield). In this note, we concentrate on the case of finite characteristic p. Since the reducibility or not of Specht modules labelled by 2-regular partitions and their conjugates is known, we can concentrate on partitions λ such that neither λ nor λ is 2-regular.…”