Kleshchev multipartitions and extended Young diagrams

Abstract

We give a new simple characterization of the set of Kleshchev multipartitions, and more generally of the set of Uglov multipartitions. These combinatorial objects play an important role in various areas of representation theory of quantum groups, Hecke algebras or finite reductive groups. As a consequence, we obtain a proof of a generalization of a conjecture by Dipper, James and Murphy and a generalization of the LLT algorithm for arbitrary level.2010 Mathematics Subject Classification: 20C08,05E10,17B37• in the context of the representation theory of rational Cherednik algebras: one can identify the so called KZ-component of the "Cherednik crystal of irreducible representations" with these multipartitions (see for example [15,7])• in the context of the representation theory for finite reductive groups (see [11]). For l = 2, Uglov 2-partitions are known to label some unipotent modules of the unitary group, and the associated graph has a remarkable interpretation in Harish-Chandra theory [14].All these connections give strong motivations for studying theses types of multipartitions. If l = 1, the lpartitions may be identified with the partitions and the Uglov l-partitions have then a very simple definition: they are given by the e-regular partitions (that is the partition when no nonzero part are repeated e or more times.) In contrast, when l > 1, the definition is far more complicated. If l = 2, S. Ariki, V. Kreiman, and S. Tsuchioka [3] have given an alternative non recursive definition of the set of Kleshchev 2-partitions using abaci display. In the general case, a simple characterization was still missing.In this paper, we give a new simple characterization of the set of Uglov l-partitions for all l ∈ Z >0 and all s ∈ Z l : see Theorem 5.1.1. It thus also concerns the set of Kleshchev multipartitions, as a special case (see §3.2.1.) This characterization is still recursive on n but easier than the original definition and it does not use the crystal graph. The proof will then be largely combinatorial. It is based on extensions of classical combinatorial definitions around the combinatorics of Young diagrams and on the study of certain crystal isomorphisms already introduced in [22]. In the context of Cherednik algebras, such isomorphisms can be interpreted as wall crossing functors in the sense of Losev [27] as it is shown in [24].We also develop two consequences of our main result. The first one is a proof of a generalization of a conjecture by Dipper, James and Murphy [9] stated by Graham and Lehrer in [16,§5] concerning the set of Kleshchev multipartitions. A proof has been previously given in the case l = 2 by S. Ariki and the author [2], for l ∈ N and e = ∞ by J. Hu [17], and for e = 2 by J. Hu, K. Zhou and K. Wang [18]. We here treat the most general case l ∈ N >0 and e ∈ N >1 (without using these previous works.) The second consequence of our main result is a direct generalization of the LLT algorithm computing the canonical bases of irreducible highest weight U( sl e )-modules (see §10.2). Previously, ...

Keys and Demazure crystals for Kac–Moody algebras

“…The vertices of the associated crystal are called the Kleshchev multipartitions. They have, in principle, a non trivial inductive definition but an elementary characterization of them has been recently given in [15].…”

Uglov Bipartitions and Extended Young Diagrams

“…Point 1 just follows from the definition of FLOTW bipartitions. Point 2 has already been stated (it also follows from [13,Lemma 4.2.5] and [13,Lemma 4.2.6]). Let us consider the last point.…”

“…The specialization at v = 1 of the above expression corresponds to the elements of the Theorem. As a consequence, as in [13], we obtain a LLT algorithm-like for the computation of the canonical basis elements.…”

“…The aim of this note is to continue the work of [13] and to obtain a general proof of the Dipper-James-Murphy conjecture for the whole class of Uglov bipartitions (see Theorem 3.5.1 or its reformulation in Corollary 4.4.1). To do this, we use the fact that this conjecture may be easily proved for the class of FLOTW bipartitions and we use the bijections between classes of Uglov bipartitions that we have already studied and defined in a number of papers.…”

On modular Harish-Chandra series of finite unitary groups

“…By "largest" we mean again, largest with respect to the order that b > b if ct(b) > ct(b ) or ct(b) = ct(b ) and b ∈ λ 1 , b ∈ λ 2 , extended to an order on addable vertical strips in the obvious way. Technically, we should use the extended Young diagram of [39] to make this precise, which translates into a total order on the beads of the abacus: if (β, j), (β , j ) ∈ A|λ, s are distinct beads in rows j, j and columns β, β , respectively, then (β, j) > (β , j ) if β > β or β = β and j = 1, j = 2.…”