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Hoel Queffelec; Alistair Savage; Oded Yacobi
An equivalence between truncations of categorified quantum groups and Heisenberg categories
(Une équivalence entre des troncations de groupes quantiques catégorifiés et des catégories de Heisenberg)
Journal de l'École polytechnique — Mathématiques, 5 (2018), p. 197-238, doi: 10.5802/jep.68
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Class. Math.: 17B10, 17B65, 20C30, 16D90
Keywords: Categorification, Heisenberg algebra, Fock space, basic representation, principal realization, symmetric group

Résumé - Abstract

We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty $. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov–Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty $, and also to a truncation of Khovanov’s Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty $. As a result of the categorical equivalences described above, certain actions of $\mathscr{H}$ induce actions of $\mathscr{U}$, and vice versa. In particular, we obtain an explicit action of $\mathscr{U}$ on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of $\mathscr{H}$. The 2-category $\mathscr{A}$ can be viewed as a graphical calculus describing the functors of $i$-induction and $i$-restriction for symmetric groups, together with the natural transformations between their compositions. The resulting computational tool is used to give simple diagrammatic proofs of (apparently new) representation theoretic identities.

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