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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 $H_{4}$.
- 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.
- 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 $\mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $\mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $\mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $\mathfrak{sl}_3$ 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 $\mathsf{ADE}$ Dynkin diagrams.
- Evaluation birepresentations of affine type a soergel bimodulesPublication . Mackaay, Marco; Miemietz, Vanessa; Vaz, PedroIn this paper, we use Soergel calculus to define a monoidal functor, called the evaluation functor, from extended affine type A Soergel bimodules to the homotopy category of bounded complexes in finite type A Soergel bimodules. This functor categorifies the well-known evaluation homomorphism from the extended affine type A Hecke algebra to the finite type A Hecke algebra. Through it, one can pull back the triangulated birepresentation induced by any finitary birepresentation of finite type A Soergel bimodules to obtain a triangulated birepresentation of extended affine type A Soergel bimodules. We show that if the initial finitary birepresentation in finite type A is a cell birepresentation, the evaluation birepresentation in extended affine type A has a finitary cover, which we illustrate by working out the case of cell birepresentations with subregular apex in detail. (c) 2023 Elsevier Inc. All rights reserved.
- Kostant’s problem for fully commutative permutationsPublication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, VanessaWe give a complete combinatorial answer to Kostant's problem for simple highest weight modules indexed by fully commutative permutations. We also propose a reformulation of Kostant's problem in the context of fiab bicategories and classify annihilators of simple objects in the principal birepresentations of such bicategories generalizing the Barbasch-Vogan theorem for Lie algebras.
- Simple transitive 2-representations via (Co-)Algebra 1-MorphismsPublication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Tubbenhauer, DanielFor any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using co-algebra 1-morphisms in the injective abelianization of C. Dually, we show that these can also be constructed using algebra 1-morphisms in the projective abelianization of C. We also extend Morita-Takeuchi theory to our setup and work out several examples, including that of Soergel bimodules for dihedral groups, explicitly.