Quiver Schur algebras and q-Fock space

By C. Stroppel, Ben Webster. 2011

78 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9acff22a9f629953e1d547c942fb38e1c6821134

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In Algebra and Number Theory

1. I. Gordon
2. Ben Webster
3. I. Losev
4. V. Ginzburg
5. Aaron D. Lauda
6. E. Gorsky
7. T. Arakawa
8. T. Nevins

In Discrete Mathematics and Combinatorics

1. I. Losev
2. Ben Webster
3. I. Gordon
4. Stephen Griffeth
5. Pavel Etingof
6. François Bergeron
7. Victor Reiner
8. Anne V. Shepler

In Geometry and Topology

1. I. Losev
2. I. Gordon
3. Ben Webster
4. V. Ginzburg
5. Pavel Etingof
6. Stephen Griffeth
7. E. Gorsky
8. Mikhail Khovanov

Citations in the corpus

• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products (2010, methodology): “geofprojective modulesoverSλ m isthesubcategory additivelygenerated by ψ tξtξT λ= y iξ,κξ T λas we range over all multipartitions with m boxes fitting inside 2Thisisanunfortunateterminologicalclashwith[SW],whereanon-equivalent,butgradedMorita equivalentalgebraisgiventhesamename;afterforgettingthegrad”
• ROUQUIER’S CONJECTURE AND DIAGRAMMATIC ALGEBRA (2013, methodology): “When n = 0, the category Sν agrees with the representations of a quiver Schur algebra as in [SW]; thus, statement (1) and thus (2) hold by [SW, 7.”
• On category O for cyclotomic rational Cherednik algebras (2011, background): “ee for instance [5, Theorem 3.11.4]. Recently Stroppel and Webster have determined the decomposition numbers of the cyclotomic q-Schur algebras at roots of unity in terms of Fock space combinatorics, [55]. 58 6.9.3. Support of irreducible modules. Itisshown in[51, Theorem 6.3]thatthesetIrrep(κ,(s⋆,m”