Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

Abstract

We associate a monoidal category H λ to each dominant integral weight λ of sl p 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 H λ , where d is the level of λ. The categories H λ 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.Hidden details. For the interested reader, the tex file of the arXiv version of this paper includes hidden details of some straightforward computations and arguments that are omitted in the pdf file. These details can be displayed by switching the details toggle to true in the tex file and recompiling.

Frobenius Heisenberg categorification

“…He showed that the Grothendieck ring of the resulting monoidal category contains an infinitedimensional Heisenberg algebra and conjectured that the two are isomorphic. Since Khovanov's original work, his construction has been generalized to q-deformations [8,14], to categories depending on a graded Frobenius superalgebra [9,16], and to higher level [15]. In addition, a generalization of Khovanov's conjecture [15,Conj.…”

“…In [2], Brundan introduced a new approach to Heisenberg categorification, proving that the higher level Heisenberg categories of [15], which include Khovanov's original category, can be defined using a smaller set of relations, including an "inversion relation". This approach also shows that the affine oriented Brauer category of [4] can be viewed as the level zero Heisenberg category.…”

“…In the current paper, we associate a graded monoidal supercategory Heis F,k to each graded Frobenius superalgebra F and integer k. The Heisenberg supercategories Heis F,k can simultaneously be viewed as graded Frobenius superalgebra deformations of the categories of [15], and as extensions of the graded monoidal supercategories of [16] to higher level. When k = 0, we also obtain new Frobenius deformations of affine oriented Brauer categories.…”

“…(c) Inversion relation: The following matrix of morphisms is an isomorphism in the additive envelope of Heis F,k : (15) is of size (1 + k dim F ) × 1, while the matrix (16) is of size 1 × (1 + k dim F ).) Note that these conditions are independent of the choice of basis B of F .…”

“…Similarly, the split Grothendieck ring K 0 (Kar Heis F,k ) of Kar Heis F,k is naturally a Z q,π -algebra. We equip K 0 (F ) with a nondegenerate symmetric sesquilinear form [9,11,12,15,16]. We expect that the surjectivity statement in Theorem 1.5 also holds without the assumption that ∆ > 0 and without the assumption on R. The assumption on R could likely be dropped if one had a basis theorem for the morphism spaces of Heis F,k .…”