The $\mathfrak{sl}_\infty$-crystal combinatorics of higher level Fock spaces

Abstract

For integers e, ℓ ≥ 2, the level ℓ Fock space has an sl ∞ -crystal structure arising from the action of a Heisenberg algebra, intertwining the sl e -crystal. The vertices of these crystals are charged ℓ-partitions. We give the combinatorial rule for computing the arrows anywhere in the sl ∞crystal. This allows us to pinpoint the location of any charged ℓ-partition. As an application, we compute the support of the spherical representation of a cyclotomic rational Cherednik algebra, and in particular, the set of parameters such that it is finite-dimensional. We also give an easy abacus characterization of all finite-dimensional representations of type B Cherednik algebras.(T.G.) Lehrstuhl

On modular Harish-Chandra series of finite unitary groups

“…The device of e-periods is very close to the device of yokes that appears in [19]; however the two are distinct, as the β-numbers appearing in a yoke do not have to be consecutive; see [19,Figure 3]. In [31] we followed [40] where e-periods and totally e-periodic abaci were introduced and [27] where they were interpreted in terms of vertical strips. The cylindric multipartitions studied using yokes in [19] and the totally e-periodic abaci as above turn out to be related to each other by level-rank duality [25].…”

“…We focus on the situation that A|λ, s is a totally e-periodic abacus of some charged bipartition |λ, s . The description of the edges in the sl ∞ -crystal on F e,s is a bit more complicated in full generality and we refer to [31] for the general case. The idea, however, is similar to the totally e-periodic case.…”

“…The modification of the edge rule in Definition 1.5 consists in specifying that the right shift of the kth fore period should be the kth aft period of the resulting abacus. We refer to [31,Theorem 4.15], and simply illustrate the rule here with an example. We note the following facts about the sl ∞ -crystal on F e,s :…”

“…This crystal was first defined indirectly using the categorical action of an infinite-dimensional Heisenberg Lie algebra on cyclotomic Cherednik category O [49]. A combinatorial description crystallized through the work of several authors over the following years [43], [26], [27], [31]. As a directed graph, each of its connected components is isomorphic to Young's lattice, the branching graph of the symmetric group in characteristic 0 [49], [43].…”

“…Each arrow in the crystal adds a vertical strip of e boxes to a bipartition [43], [27]. The precise rule is subtle but similar in spirit to the rule for adding a good box in the sl e -crystal [31] 2 .…”