A general approach to Heisenberg categorification via wreath product algebras

Abstract

We associate a monoidal category H B , defined in terms of planar diagrams, to any graded Frobenius superalgebra B. This category acts naturally on modules over the wreath product algebras associated to B. To B we also associate a (quantum) lattice Heisenberg algebra h B . We show that, provided B is not concentrated in degree zero, the Grothendieck group of H B is isomorphic, as an algebra, to h B . For specific choices of Frobenius algebra B, we recover existing results, including those of Khovanov and Cautis-Licata. We also prove that certain morphism spaces in the category H B contain generalizations of the degenerate affine Hecke algebra. Specializing B, this proves an open conjecture of Cautis-Licata.

Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

“…We note that higher level Heisenberg algebras are also categorified by the categories H B of [RS17] for suitable choices of the Frobenius algebra B on which these categories depend.…”

Affine Wreath Product Algebras

“…the Z-grading on F is nontrivial) is of particular interest in Heisenberg categorification. In particular, it is exactly this property that allows one to conclude in [CL12,RS17] that the Grothendieck groups of the categories defined there are isomorphic to Heisenberg algebras. For the original Heisenberg category of [Kho14], where the grading is trivial, Khovanov proves that the Heisenberg algebra embeds into the Grothendieck group, and it is still an open conjecture that one has equality.…”

“…Before investigating the properties of the algebra A n (F ), we first discuss how various choices of the Frobenius algebra F recover well-studied algebras. In its full generality, the algebra A n (F ) first appeared in [RS17], where it occurred naturally in the endomorphism space of the object Q n of the diagrammatic category H F . More precisely, A n (F ) is isomorphic to the opposite algebra of the algebra D n defined in [RS17, Def.…”

“…We expect that the cyclotomic wreath product algebras introduced in Section 6 provide a framework for unifying these two generalizations. In particular, one should be able to define a graphical category based on these cyclotomic quotients that specializes to the categories of [MS] when F = k and to the categories of [RS17] when the quotient is of level one. 1 More ambitiously, q-deformations as in §7.1 might allow one to also incorporate the q-deformed Heisenberg category of [LS13].…”

A graphical calculus for the Jack inner product on symmetric functions

“…Let B be a nonnegatively graded Frobenius superalgebra over an algebraically closed field F of characteristic 0 (for example, the cohomology over F of a compact connected manifold). Inspired by constructions of Khovanov in [Kho14] and Cautis and the first author in [CL12], the second and third author, in [RS17], associated to B a graded pivotal monoidal category H * B . The objects of H * B are formal direct sums of compact oriented 0-manifolds, and the morphisms are linear combinations of immersed oriented planar 1-manifolds, decorated by elements of the Frobenius algebra B, and subject to certain local relations.…”