Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

Abstract

We associate a monoidal category ${\mathcal{H}}^{\lambda }$ to each dominant integral weight $\lambda$ of ${\widehat{\mathfrak{sl}}}_p$ or ${\mathfrak{sl}}_{\infty }$. These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to $\lambda$. We show that, in the ${\mathfrak{sl}}_{\infty }$ case, the level $d$ Heisenberg algebra embeds into the Grothendieck ring of ${\mathcal{H}}^{\lambda }$, where $d$ is the level of $\lambda$. The categories ${\mathcal{H}}^{\lambda }$ can be viewed as a graphical calculus describing induction and restriction functors between categories of modules for degenerate cyclotomic Hecke algebras, together with their natural transformations. As an application of this tool, we prove a new result concerning centralizers for degenerate cyclotomic Hecke algebras.

35 pages; v2: published version

References in corpus (9)

• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Graded decomposition numbers for cyclotomic Hecke algebras
• Frobenius Heisenberg categorification
• Affine wreath product algebras
• The Elliptic Hall algebra and the deformed Khovanov Heisenberg category
• Centers of degenerate cyclotomic Hecke algebras and parabolic category O
• A graphical calculus for the Jack inner product on symmetric functions
• An equivalence between truncations of categorified quantum groups and Heisenberg categories

Cited by in corpus (14)

• On the definition of quantum Heisenberg category
• Frobenius Heisenberg categorification
• Affine wreath product algebras
• The degenerate Heisenberg category and its Grothendieck ring
• Heisenberg and Kac-Moody categorification
• Foundations of Frobenius Heisenberg categories
• Quantum Frobenius Heisenberg categorification
• String diagrams and categorification
• Affine oriented Frobenius Brauer categories
• An equivalence between truncations of categorified quantum groups and Heisenberg categories
• Embedding Deligne's category $\underline{\mathrm{Re}}\mathrm{p}(S_t)$ in the Heisenberg category
• Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category
• Frobenius W-algebras and traces of Frobenius Heisenberg categories
• Planar algebras for the Young graph and the Khovanov Heisenberg category