Center for Mathematical Analysis, Geometry and Dynamical Systems

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Publications

Kostant’s problem for fully commutative permutations

Publication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa

We 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.

2023Journal article

Open Access

Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types

Publication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Tubbenhauer, Daniel; Zhang, Xiaoting

In 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 $H_{4}$.

2019-06-27Journal article

Open Access

Evaluation birepresentations of affine type a soergel bimodules

Publication . Mackaay, Marco; Miemietz, Vanessa; Vaz, Pedro

In 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.

2024Journal article

Restricted

Applying projective functors to arbitrary holonomic simple modules

Publication . Mazorchuk, Volodymyr; Miemietz, Vanessa; Mackaay, Marco

We prove that applying a projective functor to a holonomic simple module over a semisimple finite‐dimensional complex Lie algebra produces a module that has an essential semisimple submodule of finite length. This implies that holonomic simple supermodules over certain Lie superalgebras are quotients of modules that are induced from simple modules over the even part. We also provide some further insight into the structure of Lie algebra modules that are obtained by applying projective functors to simple modules.

2024-07-18Journal article

Open Access