Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

We associate a monoidal category Hλ to each dominant integral weight λ of slˆp or sl∞. These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to λ. We show that, in the sl∞ case, the level d Heisenberg algebra embeds into the Grothendieck ring of Hλ, where d is the level of λ. The categories Hλ 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.