An equivalence between truncations of categorified quantum groups and Heisenberg categories

Abstract

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 sl∞. We show that A is equivalent to a truncation of the Khovanov-Lauda categorified quantum group U of type A∞, and also to a truncation of Khovanov's Heisenberg 2-category H . This equivalence is a categorification of the principal realization of the basic representation of sl∞.As a result of the categorical equivalences described above, certain actions of 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 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. Contents 1. Introduction 2.1. Bosonic Fock space and the category A 2.2. The basic representation 2.3. A Kac-Moody presentation of A 2.4. Notation and conventions for 2-categories 3. Modules for symmetric groups 3.1. Module categories 3.2. Decategorification 3.3. Biadjunction and the fundamental bimodule decomposition 3.4. The Jucys-Murphy elements and their eigenspaces 3.5. Combinatorial formulas 4. The 2-category A 4.1. Definition 4.2. Truncated categorified quantum groups 4.3. 1-morphism spaces 4.4. 2-morphism spaces 4.5. Decategorification 5. The 2-category H tr 5.1. Definition 5.2. A decomposition H tr = H ǫ ⊕ H δ 5.3. Region shifting

Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

“…Replacing Khovanov's Heisenberg category by the more general categories H λ of the current paper should yield a generalization of the results of [QSY18]. The 2-category version H λ of H λ has objects labelled by integers.…”

“…This corresponds to the fact that the restriction of V (λ) to the principal Heisenberg subalgebra of g is a direct sum of Fock spaces. If λ is of level one, then V (λ) remains irreducible as a module over the Heisenberg algebra, and we are in the setting of [QSY18]. See Section 8.1 for further comments in this direction.…”

“…Truncations and categorified quantum groups. In [QSY18], it is shown that a certain truncation of a 2-category version of Khovanov's Heisenberg category is equivalent to a truncation of the Khovanov-Lauda categorified quantum group of type A ∞ , introduced in [KL10]. (See also the related work [Rou].)…”

Heisenberg and Kac–Moody categorification

“…So now Theorem A gives a new proof of the existence of a categorical action of g, without any need to appeal to combinatorial facts from the representation theory of symmetric groups. (See also [32] for a different point of view. )…”