Quiver Schur algebras and 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.

68 pages. Section 4 changed. Several arguments revised, more details. Typos fixed

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• Canonical bases and higher representation theory
• Rouquier's conjecture and diagrammatic algebra
• On category O for cyclotomic rational Cherednik algebras
• Categorified skew Howe duality and comparison of knot homologies
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• On graded presentations of Hecke algebras and their generalizations
• Canonical basis, KLR-algebras and parity sheaves
• Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra
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• Quiver Schur algebras and Koszul duality
• Seminormal forms and cyclotomic quiver Hecke algebras of type $A$
• Highest weight sl_2-categorifications II: structure theory
• p-DG cyclotomic nilHecke algebras
• Towards multiplicities for categories O of cyclotomic rational Cherednik algebras
• Polynomial representations and categorifications of Fock Space
• On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution
• Quiver Schur algebras and cohomological Hall algebras
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• A Survey on Springer Theory
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• KLR and Schur algebras for curves and semi-cuspidal representations
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• Fayers' conjecture and the socles of cyclotomic Weyl modules
• Hall algebras of cyclic quivers and $q$-deformed Fock spaces
• DG-Enhanced Hecke and KLR Algebras
• Tensor product algebras in type A are Koszul
• On a higher level extension of Leclerc-Thibon product theorem in q-deformed Fock spaces
• Modified affine Hecke algebras and quiver Hecke algebras of type $A$
• Bi-orbital sheaves and affine Hecke algebras at roots of unity
• Parallelotope tilings and $q$-decomposition numbers