A general approach to Heisenberg categorification via wreath product algebras

Abstract

We associate a monoidal category ${\mathcal{H}}_B$, 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 ${\mathfrak{h}}_B$. We show that, provided $B$ is not concentrated in degree zero, the Grothendieck group of ${\mathcal{H}}_B$ is isomorphic, as an algebra, to ${\mathfrak{h}}_B$. 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 ${\mathcal{H}}_B$ contain generalizations of the degenerate affine Hecke algebra. Specializing $B$, this proves an open conjecture of Cautis--Licata.

46 pages. v2: Several sign errors and other minor typos corrected. v3: Minor corrections, published version

References in corpus (4)

• On a categorical Boson-Fermion correspondence
• Twisted Frobenius extensions of graded superrings
• Towers of graded superalgebras categorify the twisted Heisenberg double
• Twisted Heisenberg doubles

Cited by in corpus (18)

• Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification
• Frobenius Heisenberg categorification
• Affine wreath product algebras
• Foundations of Frobenius Heisenberg categories
• A basis theorem for the degenerate affine oriented Brauer-Clifford supercategory
• Quantum Frobenius Heisenberg categorification
• A graphical calculus for the Jack inner product on symmetric functions
• Quantum affine wreath algebras
• String diagrams and categorification
• Trace of the twisted Heisenberg Category
• Group partition categories
• Affine oriented Frobenius Brauer categories
• Skew-Zigzag Algebras
• Embedding Deligne's category $\underline{\mathrm{Re}}\mathrm{p}(S_t)$ in the Heisenberg category
• An equivalence between truncations of categorified quantum groups and Heisenberg categories
• Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category
• Planar algebras for the Young graph and the Khovanov Heisenberg category
• Frobenius W-algebras and traces of Frobenius Heisenberg categories