# Quiver Schur algebras and q-Fock space

Catharina Stroppel, Ben Webster

Number | 33 |
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Author | Catharina Stroppel |

Year | 2011 |

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.