Quiver Schur algebras and q-Fock space

TL;DR

This paper introduces a graded version of cyclotomic q-Schur algebras, connecting their structure to higher level q-Fock spaces and canonical bases, with applications in representation theory and algebraic geometry.

Contribution

It develops a new graded, diagrammatic, and geometric framework for cyclotomic q-Schur algebras, linking them to higher q-Fock spaces and canonical bases.

Findings

Identified coefficients of canonical basis with q-analogues of decomposition numbers.

Constructed a graded cellular basis for the quiver Schur algebra.

Established a connection between projective modules and canonical bases in higher q-Fock space.

Abstract

We develop a graded version of the theory of cyclotomic q-Schur algebras, in the spirit of the work of Brundan-Kleshchev on Hecke algebras and of Ariki on q-Schur algebras. As an application, we identify the coefficients of the canonical basis on a higher level Fock space with q-analogues of the decomposition numbers of cyclotomic q-Schur algebras. We present cyclotomic q-Schur 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 q-Schur algebra with a non-obvious 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 q-Fock space (defined as a tensor product of level 1 q-deformed 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.