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- Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorificationPublication . Mackaay, Marco; Savage, AlistairWe associate a monoidal category H-lambda to each dominant integral weight lambda of sl(p) or sl(infinity). 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 sl infinity case, the level d Heisenberg algebra embeds into the Grothendieck ring of H-lambda, where d is the level of lambda. The categories 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.
- TheslN-web algebras and dual canonical basesPublication . Mackaay, MarcoIn this paper, which is a follow-up to [38], I define and study SIN-web algebras, for any N >= 2. For N = 2 these algebras are isomorphic to Khovanov's [22] arc algebras and for N = 3 they are Morita equivalent to the sl(3)-web algebras which I defined and studied together with Pan and Tubbenhauer [34]. The main result of this paper is that the SIN-web algebras are Morita equivalent to blocks of certain level-N cyclotomic KLR algebras, for which I use the categorified quantum skew Howe duality defined in [38]. Using this Morita equivalence and Brundan and Kleshchev's [4] work on cyclotomic KLR-algebras, I show that there exists an isomorphism between a certain space of SIN-webs and the split Grothendieck group of the corresponding SIN-web algebra, which maps the dual canonical basis elements to the Grothendieck classes of the indecomposable projective modules (with a certain normalization of their grading).
- A diagrammatic categorification of the affine q-Schur algebra S(n,n) for n > 2Publication . Mackaay, Marco; Thiel, Anne-LaureThis is a follow-up to the paper in which we categorified the affine quantum Schur algebra S(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's categorification of the affine quantum sl_n. In this paper we categorify S(n,n) for n > 2, using an extension of the aforementioned quotient.
- Two-color Soergel Calculus and Simple transitive 2-representationsPublication . Mackaaij, Marco; Tubbenhauer, DanielIn this paper we complete the ADE-like classification of simple transitive 2-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to give an explicit construction of a graded (non-strict) version of all these 2-representations. Moreover, we give simple combinatorial criteria for when two such 2-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations. Finally, our construction also gives a large class of simple transitive 2-representations in infinite dihedral type for general bipartite graphs.