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Hecke algebras, finite general linear groups, and Heisenberg categorification. (English) Zbl 1279.20006
Quantum Topol. 4, No. 2, 125-185 (2013); erratum ibid. 6, No. 1, 183 (2015).
The authors define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. This category, which is a \(q\)-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type \(A\) and finite general linear groups. In this way, they obtain a categorification of the bosonic Fock space. They also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting.
Reviewer: Hu Jun (Sydney)

MSC:
20C08 Hecke algebras and their representations
17B65 Infinite-dimensional Lie (super)algebras
16D90 Module categories in associative algebras
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
16S80 Deformations of associative rings
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