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Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification. (English) Zbl 1437.20004
In this paper, the authors give a generalization of Khovanov’s Heisenberg category from level one to higher levels case. More precisely, they associate a monoidal category \(\mathcal{H}_\lambda\) to each dominant integral weight \(\lambda\) of \(sl_p\) or \(sl_\infty\) and use the degenerate affine Hecke algebras and their corresponding cyclotomic quotients to give a categorification of the higher level Heisenberg algebra. These categories \(\mathcal{H}_\lambda\), defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to \(\lambda\). As an application, they prove a new result concerning centralizers for degenerate cyclotomic Hecke algebras which generalize a result of G. I. Ol’shanskij [Sov. Math., Dokl. 36, No. 3, 569–573 (1988; Zbl 0662.22016); translation from Dokl. Akad. Nauk SSSR 297, 1050–1054 (1987)] on centralizers for group algebras of symmetric groups.
Reviewer: Hu Jun (Beijing)

MSC:
20C08 Hecke algebras and their representations
18N25 Categorification
19A22 Frobenius induction, Burnside and representation rings
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