# Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

## TL;DR

This paper constructs monoidal categories using planar diagrams that act on modules of degenerate cyclotomic Hecke algebras, embedding Heisenberg algebras and providing new insights into their structure and centralizers.

## Contribution

It introduces a graphical calculus for categories related to degenerate cyclotomic Hecke algebras and demonstrates their connection to higher level Heisenberg categorification.

## Key findings

- Embedding of level d Heisenberg algebra into Grothendieck ring for sl_ case
- Graphical calculus describes induction and restriction functors
- New results on centralizers of degenerate cyclotomic Hecke algebras

## 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.