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 socalled 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 finitedimensional simple modules.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1601.00581
 Bibcode:
 2016arXiv160100581G
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics;
 17B37;
 05E10;
 20C08
 EPrint:
 Version 2: Proof of Lemma 5.5 improved, Section 6 simplified and additional minor changes. Version 3: Section 4 rewritten (including simpler proof of Lemma 4.7 and Theorem 4.8), Theorem 6.26 replaced by Corollary 6.25, notation for "levelrank" dual objects modified, Section 7.1 slightly rewritten