Quiver Schur algebras and qFock space
Abstract
We develop a graded version of the theory of cyclotomic qSchur algebras, in the spirit of the work of BrundanKleshchev on Hecke algebras and of Ariki on qSchur algebras. As an application, we identify the coefficients of the canonical basis on a higher level Fock space with qanalogues of the decomposition numbers of cyclotomic qSchur algebras. We present cyclotomic qSchur algebras as a quotient of a convolution algebra arising in the geometry of quivers; hence we call these quiver Schur algebras. These algebras are also presented diagrammatically, similar in flavor to a recent construction of Khovanov and Lauda. They are also manifestly graded and so equip the cyclotomic qSchur algebra with a nonobvious grading. On the way we construct a graded cellular basis of this algebra, resembling the constructions for cyclotomic Hecke algebras by Mathas, Hu, Brundan and the first author. The quiver Schur algebra is also interesting from the perspective of higher representation theory. The sum of Grothendieck groups of certain cyclotomic quotients is known to agree with a higher level Fock space. We show that our graded version defines a higher qFock space (defined as a tensor product of level 1 qdeformed Fock spaces). Under this identification, the indecomposable projective modules are identified with the canonical basis and the Weyl modules with the standard basis. This allows us to prove the already described relation between decomposition numbers and canonical bases.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 arXiv:
 arXiv:1110.1115
 Bibcode:
 2011arXiv1110.1115S
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Representation Theory
 EPrint:
 68 pages. Section 4 changed. Several arguments revised, more details. Typos fixed