An equivalence between truncations of categorified quantum groups and Heisenberg categories

Queffelec, Hoel; Savage, Alistair; Yacobi, Oded

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 197-238.

Nous introduisons une 2-catégorie élémentaire $𝒜$ qui catégorifie l’image de l’espace de Fock comme représentation de l’algèbre de Heisenberg, ainsi que la représentation basique de ${\mathrm{𝔰𝔩}}_{\infty }$. Nous montrons que $𝒜$ est équivalente à une troncation du groupe quantique catégorifié de Khovanov–Lauda $𝒰$ en type $A_{\infty }$, ainsi qu’à une troncation de la 2-catégorie de Heisenberg $\mathscr{H}$ introduite par Khovanov. Cette équivalence se comprend comme une catégorification de la réalisation principale de la représentation basique de ${\mathrm{𝔰𝔩}}_{\infty }$. Il résulte des équivalences catégoriques précédentes que certaines actions de $\mathscr{H}$ induisent des actions de $𝒰$, et vice versa. En particulier, nous obtenons une action explicite de $𝒰$ sur les représentations des groupes symétriques. Nous calculons également explicitement le groupe de Grothendieck de la troncation de $\mathscr{H}$. La 2-catégorie $𝒜$ s’interprète comme un calcul graphique décrivant les foncteurs de $i$-induction et $i$-restriction pour les groupes symétriques, ainsi que les transformations naturelles entre leurs composées. Nous utilisons l’outil de calcul qui en découle pour donner des preuves diagrammatiques simples d’identités (apparemment nouvelles) en théorie des représentations.

We introduce a simple diagrammatic 2-category $𝒜$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of ${\mathrm{𝔰𝔩}}_{\infty }$. We show that $𝒜$ is equivalent to a truncation of the Khovanov–Lauda categorified quantum group $𝒰$ 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 ${\mathrm{𝔰𝔩}}_{\infty }$. As a result of the categorical equivalences described above, certain actions of $\mathscr{H}$ induce actions of $𝒰$, and vice versa. In particular, we obtain an explicit action of $𝒰$ on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of $\mathscr{H}$. The 2-category $𝒜$ 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.

Classification : 17B10, 17B65, 20C30, 16D90
Keywords: Categorification, Heisenberg algebra, Fock space, basic representation, principal realization, symmetric group
Mot clés : Catégorification, algèbre de Heisenberg, espace de Fock, représentation basique, réalisation principale, groupe symétrique

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