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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.) (English. French summary) Zbl 1433.17011
Summary: We introduce a simple diagrammatic 2-category \(\mathcal{A}\) that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of \(\mathfrak{sl}_\infty\). We show that \(\mathcal{A}\) is equivalent to a truncation of the Khovanov-Lauda categorified quantum group \(\mathcal{U}\) of type \(A_\infty\), and also to a truncation of Khovanov’s Heisenberg 2-category \(\mathcal{H}\). 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 \(\mathcal{H}\) induce actions of \(\mathcal{U}\), and vice versa. In particular, we obtain an explicit action of \(\mathcal{U}\) on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of \(\mathcal{H}\). The 2-category \(\mathcal{A}\) 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.

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B65 Infinite-dimensional Lie (super)algebras
16D90 Module categories in associative algebras
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