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- Trihedral Soergel bimodulesPublication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Tubbenhauer, DanielThe quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum sl(2) representation category. It also establishes a precise relation between the simple transitive 2-representations of both monoidal cate-gories, which are indexed by bicolored ADE Dynldn diagrams. Using the quantum Satake correspondence between affine A(2) Soergel bimodules and the semisimple quotient of the quantum sl(3)representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive 2-representations corresponding to tricolored generalized ADE Dynkin diagrams.
- Analogues of centralizer subalgebras for fiat 2-categories and their 2-representationsPublication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Zhang, XiaotingThe main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex J of a fiat 2-category C and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of C associated with a diagonal H-cell in J. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types B-3 and B-4.
- Simple transitive $2$-representations of Soergel bimodules for finite Coxeter typesPublication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Tubbenhauer, Daniel; Zhang, XiaotingIn this paper we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive $2$-representations and we complete their classification in all types but $H_{3}$ and $H_{4}$.
- Categorifications of the extended affine Hecke algebra and the affine q-Schur algebra S(n, r) for 3 <= r < nPublication . Mackaay, Marco; Thiel, Anne-LaureWe categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n, r) for 3 <= r < n, using results on diagrammatic categorification in affine type A by Elias-Williamson, that extend the work of Elias-Khovanov for finite type A, and Khovanov-Lauda respectively. We also define 2-representations of these categorifications on an extension of the 2-category of affine (singular) Soergel bimodules. These results are the affine analogue of the results in [28].
- Erratum - A remark on Rasmussen's Invariant of knots (vol 16, pg 333, 2007)Publication . Mackaay, Marco; Turner, P.; Vaz, P.
- A remark on Rasmussen's invariant of knotsPublication . Mackaay, Marco; Turner, Paul; Vaz, PedroWe show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.
- Categorified skew Howe duality and comparison of knot homologiesPublication . Mackaay, Marco; Webster, BenIn this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for sl(n). Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category O, affine Grassmannians, and diagrammatic categorifications of tensor products. While the definitions of these theories are quite different, there is a key commonality between them which makes it possible to prove that they are all isomorphic: they arise from a skew Howe dual action of gl(l) for some l. In this paper, we show that the construction of knot homology based on categorifying tensor products (from earlier work of the second author) fits into this framework, and thus agrees with other such homologies, such as Khovanov-Rozansky homology. We accomplish this by categorifying the action of gl(l) x gl(n) on Lambda(P)(C-l circle times C-n) using diagrammatic bimodules. In this action, the functors corresponding to gl(l) and gl(n) are quite different in nature, but they will switch roles under Koszul duality.
- Simple transitive 2-representations of small quotients of Soergel bimodulesPublication . Kildetoft, Tobias; Mackaay, Marco; Mazorchuk, Volodymyr; Zimmermann, JakobIn all finite Coxeter types but I-2(12), I-2(18), and I-2(30), we classify simple transitive 2-representations for the quotient of the 2-category of Soergel bimodules over the coinvariant algebra which is associated with the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive 2-representations are exhausted by cell 2-representations. However, in Coxeter types I-2(2k), where k >= 3, there exist simple transitive 2-representations which are not equivalent to cell 2-representations.
- 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.
- Analogues of centralizer subalgebras for fiat 2-categories and their 2-representationsPublication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Zhang, XiaotingThe main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex J of a fiat 2-category C and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of C associated with a diagonal H-cell in J. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types B-3 and B-4.