Triple crystal action in Fock spaces

Abstract

We make explicit a triple crystal structure on higher level Fock spaces, by investigating at the combinatorial level the actions of two affine quantum groups and of a Heisenberg algebra. To this end, we first determine a new indexation of the basis elements that makes the two quantum group crystals commute. Then, we define a so-called Heisenberg crystal, commuting with the other two. This gives new information about the representation theory of cyclotomic rational Cherednik algebras, relying on some recent results of Shan and Vasserot and of Losev. In particular, we give an explicit labelling of their finite-dimensional simple modules.

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

“…The sl ∞ -crystal is categorified by the action of a Heisenberg algebra on the cyclotomic Cherednik category O [34], [27]; this action is related to parabolic induction with respect to type A parabolics [34]. Previous work by Losev [27] and the first author [13] gave roundabout or indirect constructions of the sl ∞ -crystal, with explicit formulas only in special cases. This left open the problem of formulating a direct combinatorial rule for the arrows in the sl ∞ -crystal.…”

“…Section 5 presents a closed combinatorial formula for each ℓ-partition in a certain connected component of the sl ∞ -crystal component when the charge s ∈ eZ ℓ (this is the lattice c Z of [27, Section 1.2] containing the origin). The connected component in question is the one containing the empty ℓ-partition, and consists of all ℓ-partitions which are simultaneously singular for the action of sl e and of its level-rank dual (see [13]).…”

“…In [13,Section 6.3], the operatorsb σ have been expressed as the composition of more elementary operatorsb + k , where k ∈ Z. Drawing an arrow labeled by k between two abaci A and A ′ whenever A ′ =b + k A gives the description of the sl ∞ -crystal isomorphic to the Young graph.…”

“…The operatorsb ± k are analogues of the Kashiwara crystal operators for the Heisenberg algebra. More precisely, they are the image of the Kashiwara operators of U v (sl ∞ ) under a certain bijection, see [13] and [14].…”

“…We will give a combinatorial formula for the vertices belonging to the connected component of the sl ∞ -crystal with highest weight vertex A • . Such vertices are called doubly highest weight vertices in [13] as they are singular for the action of sl e and sl ℓ (see [36] and [13]) simultaneously.…”