Showing 1–42 of 42 results for author: Miemietz, V

Categorification in Representation Theory

Authors: Vanessa Miemietz

Abstract: In this survey article, we give an introduction to the relatively young subject of $2$-representation theory, which studies categorifications of important players in classical representation theory. In particular, we provide a streamlined exposition of the main results leading to the classification of simple $2$-representations for Soergel bimodules associated to finite Weyl groups in characterist… ▽ More In this survey article, we give an introduction to the relatively young subject of $2$-representation theory, which studies categorifications of important players in classical representation theory. In particular, we provide a streamlined exposition of the main results leading to the classification of simple $2$-representations for Soergel bimodules associated to finite Weyl groups in characteristic $0$, obtained in [M. Mackaay, V. Mazorchuk, V. Miemietz, D. Tubbenhauer and X. Zhang, Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types. Proc. London Math. Soc. 126 (2023)]. △ Less

Submitted 20 August, 2026; originally announced August 2026.

Comments: survey article for Proceedings from ICRA 2024

The subregular and submaximal $p$-cells

Authors: Vanessa Miemietz, Marie Roth, Daniel Tubbenhauer

Abstract: Cells for the canonical and $p$-canonical bases organise the representation theory and geometry of Hecke categories. We determine the relevant $p$-canonical basis elements and the resulting $p$-cell structure for the subregular and submaximal cells in all classical types. Cells for the canonical and $p$-canonical bases organise the representation theory and geometry of Hecke categories. We determine the relevant $p$-canonical basis elements and the resulting $p$-cell structure for the subregular and submaximal cells in all classical types. △ Less

Submitted 20 August, 2026; originally announced August 2026.

Comments: 33 pages

MSC Class: Primary 20C08; 20C20; Secondary 14M15; 18N10

Almost finitary birepresentation theory and applications to affine Soergel bimodules

Authors: Marco Mackaay, Vanessa Miemietz, Pedro Vaz

Abstract: In this article, we develop a generalization of finitary birepresentation theory applicable to Soergel bimodules for infinite Coxeter groups. We establish a reduction process for the classification of simple birepresentations of almost finitary bicategories, and consider in detail the case of Soergel bimodules in extended affine type A. In this article, we develop a generalization of finitary birepresentation theory applicable to Soergel bimodules for infinite Coxeter groups. We establish a reduction process for the classification of simple birepresentations of almost finitary bicategories, and consider in detail the case of Soergel bimodules in extended affine type A. △ Less

Submitted 22 April, 2026; originally announced April 2026.

Comments: 55 pages

Induction for extended affine type A Soergel bimodules: first steps

Authors: Marco Mackaay, Vanessa Miemietz, Pedro Vaz

Abstract: In this paper we take the first steps towards the categorification of the Zelevinsky tensor product of finite dimensional representations of extended affine type A Hecke algebras. In this paper we take the first steps towards the categorification of the Zelevinsky tensor product of finite dimensional representations of extended affine type A Hecke algebras. △ Less

Submitted 5 June, 2026; v1 submitted 3 July, 2025; originally announced July 2025.

Comments: 91 pages, lots of pictures, comments welcome. V2. accepted version to appear in J. Inst. Math. Jussieu

MSC Class: 18N25; 20C08

Hochschild cohomology for finitary 2-representations

Authors: James Macpherson, Vanessa Miemietz, Mateusz Stroiński

Abstract: In this article, we define and investigate Hochschild cohomology for finitary 2-representations of quasi-fiat 2-categories. In this article, we define and investigate Hochschild cohomology for finitary 2-representations of quasi-fiat 2-categories. △ Less

Submitted 12 March, 2025; originally announced March 2025.

Applying projective functors to arbitrary holonomic simple modules

Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: We prove that applying a projective functor to a holonomic simple module over a semi-simple finite dimensional complex Lie algebra produces a module that has an essential semi-simple 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… ▽ More We prove that applying a projective functor to a holonomic simple module over a semi-simple finite dimensional complex Lie algebra produces a module that has an essential semi-simple 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. △ Less

Submitted 25 January, 2024; v1 submitted 7 November, 2023; originally announced November 2023.

Comments: v2: significant changes compared to v1, the main result is restricted to the case of holonomic modules

Differential graded cell 2-representations

Authors: Robert Laugwitz, Vanessa Miemietz

Abstract: This article develops a theory of cell combinatorics and cell 2-representations for differential graded 2-categories. We introduce two types of partial preorders, called the strong and weak preorder. We then analyse and compare them. The weak preorder is more easily tractable, while the strong preorder is more closely related to the combinatorics of the associated homotopy 2-representations. To ea… ▽ More This article develops a theory of cell combinatorics and cell 2-representations for differential graded 2-categories. We introduce two types of partial preorders, called the strong and weak preorder. We then analyse and compare them. The weak preorder is more easily tractable, while the strong preorder is more closely related to the combinatorics of the associated homotopy 2-representations. To each left cell, we associate a maximal ideal spectrum, and each maximal ideal gives rise to a differential graded cell 2-representation. We prove that any strong cell is contained in a weak cell and that there is a bijection between the corresponding maximal ideal spectra. Finally, we classify weak and strong cell 2-representations for dg 2-categories of projective bimodules over finite-dimensional differential graded algebras. △ Less

Submitted 17 January, 2025; v1 submitted 15 December, 2022; originally announced December 2022.

MSC Class: 18N10; 18N25; 16E45

Basic Hopf algebras and symmetric bimodules

Authors: Katerina Hristova, Vanessa Miemietz

Abstract: Motivated by the so-called H-cell reduction theorems, we investigate certain classes of bicategories which have only one H-cell apart from possibly the identity. We show that H_0-simple quasi fiab bicategories with unique H-cell H_0 are fusion categories. We further study two classes of non-semisimple quasi-fiab bicategories with a single H-cell apart from the identity. The first is $\cH_A$, index… ▽ More Motivated by the so-called H-cell reduction theorems, we investigate certain classes of bicategories which have only one H-cell apart from possibly the identity. We show that H_0-simple quasi fiab bicategories with unique H-cell H_0 are fusion categories. We further study two classes of non-semisimple quasi-fiab bicategories with a single H-cell apart from the identity. The first is $\cH_A$, indexed by a finite-dimensional radically graded basic Hopf algebra A, and the second is $\cG_A$, consisting of symmetric projective A-A-bimodules. We show that $\cH_A$ can be viewed as a 1-full subbicategory of $\cG_A$ and classify simple transitive birepresentations for $\cG_A$. We point out that the number of equivalence classes of the latter is finite, while that for $\cH_A$ is generally not. △ Less

Submitted 26 July, 2022; originally announced July 2022.

Comments: 28 pages

Evaluation birepresentations of affine type A Soergel bimodules

Authors: M. Mackaay, V. Miemietz, P. Vaz

Abstract: 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… ▽ More 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. △ Less

Submitted 7 November, 2023; v1 submitted 6 July, 2022; originally announced July 2022.

Comments: 61 pages, lots of colored pictures. v2, accepted version

Pretriangulated 2-representations via dg algebra 1-morphisms

Authors: Robert Laugwitz, Vanessa Miemietz

Abstract: This paper develops a theory of pretriangulated 2-representations of dg 2-categories. We characterize cyclic pretriangulated 2-representations, under certain compactness assumptions, in terms of dg modules over dg algebra 1-morphisms internal to associated dg 2-categories of compact objects. Further, we investigate the Morita theory and quasi-equivalences for such dg 2-representations. We relate t… ▽ More This paper develops a theory of pretriangulated 2-representations of dg 2-categories. We characterize cyclic pretriangulated 2-representations, under certain compactness assumptions, in terms of dg modules over dg algebra 1-morphisms internal to associated dg 2-categories of compact objects. Further, we investigate the Morita theory and quasi-equivalences for such dg 2-representations. We relate this theory to various classes of examples of dg categorifications from the literature. △ Less

Submitted 23 April, 2025; v1 submitted 20 May, 2022; originally announced May 2022.

Comments: Final version

MSC Class: 18N10; 18N25; 16E45

Kostant's problem for fully commutative permutations

Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: 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 generalising the Barbasch--Vogan theorem for Lie algebras. 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 generalising the Barbasch--Vogan theorem for Lie algebras. △ Less

Submitted 13 April, 2023; v1 submitted 18 May, 2022; originally announced May 2022.

Comments: Minor revision following comments by referees. To appear in Rev. Mat. Iberoam

The $N$-Stable Category

Authors: Jeremy R. B. Brightbill, Vanessa Miemietz

Abstract: A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to $N$-complexes, one must find an appropriate candidate for the $N$-analogue of the stable category. We identify this "$N$-stab… ▽ More A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to $N$-complexes, one must find an appropriate candidate for the $N$-analogue of the stable category. We identify this "$N$-stable category" via the monomorphism category and prove Buchweitz's theorem for $N$-complexes over a Frobenius exact abelian category. We also compute the Serre functor on the $N$-stable category over a self-injective algebra and study the resultant fractional Calabi-Yau properties. △ Less

Submitted 15 November, 2021; v1 submitted 16 September, 2021; originally announced September 2021.

Comments: 51 pages. References added to introduction, results in Section 3 strengthened

MSC Class: 16E35

Uniqueness of exact Borel subalgebras and bocses

Authors: Julian Külshammer, Vanessa Miemietz

Abstract: Together with Koenig and Ovsienko, the first author showed that every quasi-hereditary algebra can be obtained as the (left or right) dual of a directed bocs. In this monograph, we prove that if one additionally assumes that the bocs is basic, a notion we define, then this bocs is unique up to isomorphism. This should be seen as a generalisation of the statement that the basic algebra of an arbitr… ▽ More Together with Koenig and Ovsienko, the first author showed that every quasi-hereditary algebra can be obtained as the (left or right) dual of a directed bocs. In this monograph, we prove that if one additionally assumes that the bocs is basic, a notion we define, then this bocs is unique up to isomorphism. This should be seen as a generalisation of the statement that the basic algebra of an arbitrary associative algebra is unique up to isomorphism. The proof associates to a given presentation of the bocs an $A_\infty$-structure on the $\operatorname{Ext}$-algebra of the standard modules of the corresponding quasi-hereditary algebra. Uniqueness then follows from an application of Kadeishvili's theorem. △ Less

Submitted 18 September, 2023; v1 submitted 8 September, 2021; originally announced September 2021.

Comments: 71 pages with an appendix of 119 pages

MSC Class: 16G10 (Primary); 16E45; 18C20; 18G55 (Secondary)

Finitary birepresentations of finitary bicategories

Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer, Xiaoting Zhang

Abstract: In this paper, we discuss the generalization of finitary $2$-representation theory of finitary $2$-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive $2$-representations of a given $2$-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections betwe… ▽ More In this paper, we discuss the generalization of finitary $2$-representation theory of finitary $2$-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive $2$-representations of a given $2$-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of $2$-representations. In this paper, we generalize them to biequivalences between certain $2$-categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory. △ Less

Submitted 24 September, 2021; v1 submitted 4 August, 2020; originally announced August 2020.

Comments: Significant revision of the original version

Journal ref: Forum Math. 33 (2021), no. 5, 1261-1320

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

Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer, Xiaoting Zhang

Abstract: 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 $H_{4}$. 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 $H_{4}$. △ Less

Submitted 12 February, 2023; v1 submitted 27 June, 2019; originally announced June 2019.

Comments: Final version, to appear in the Proceedings of the LMS

Journal ref: Proc. Lond. Math. Soc. (3) 126 (2023), no. 5, 1585-1655

2-categories of symmetric bimodules and their 2-representations

Authors: Volodymyr Mazorchuk, Vanessa Miemietz, Xiaoting Zhang

Abstract: In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting $2$-category is fiat (in the sense of \cite{MM1}) and classify simple transitive $2$-representations of this $2$-category (under some mild technical assumption). We also… ▽ More In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting $2$-category is fiat (in the sense of \cite{MM1}) and classify simple transitive $2$-representations of this $2$-category (under some mild technical assumption). We also study several classes of examples in detail. △ Less

Submitted 11 April, 2019; originally announced April 2019.

Coidempotent subcoalgebras and short exact sequences of finitary 2-representations

Authors: Aaron Chan, Vanessa Miemietz

Abstract: In this article, we study short exact sequences of finitary 2-representations of a weakly fiat 2-category. We provide a correspondence between such short exact sequences with fixed middle term and coidempotent subcoalgebras of a coalgebra 1-morphism defining this middle term. We additionally relate these to recollements of the underlying abelian 2-representations. In this article, we study short exact sequences of finitary 2-representations of a weakly fiat 2-category. We provide a correspondence between such short exact sequences with fixed middle term and coidempotent subcoalgebras of a coalgebra 1-morphism defining this middle term. We additionally relate these to recollements of the underlying abelian 2-representations. △ Less

Submitted 15 January, 2019; originally announced January 2019.

Trihedral Soergel bimodules

Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer

Abstract: The 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… ▽ More The 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. △ Less

Submitted 19 June, 2019; v1 submitted 24 April, 2018; originally announced April 2018.

Comments: 61 pages, many colored figures, revised version, comments welcome, to appear in Fund. Math

Journal ref: Fund. Math. 248 (2020), no. 3, 219-300

Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations

Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Xiaoting Zhang

Abstract: The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $\mathcal{J}$ of a fiat $2$-category $\cC$ and the set of equivalence classes of faithful simple transitive $2$-representations of the fiat $2$-subquotient of $\cC$ associated with a diagonal $\mathcal{H}$-cell in $\mathcal{J}$. As an application,… ▽ More The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $\mathcal{J}$ of a fiat $2$-category $\cC$ and the set of equivalence classes of faithful simple transitive $2$-representations of the fiat $2$-subquotient of $\cC$ associated with a diagonal $\mathcal{H}$-cell in $\mathcal{J}$. 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 $B_4$. △ Less

Submitted 6 February, 2018; originally announced February 2018.

Cell 2-Representations and Categorification at Prime Roots of Unity

Authors: Robert Laugwitz, Vanessa Miemietz

Abstract: Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in d… ▽ More Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type $\mathfrak{sl}_2$ at prime roots of unity by Elias-Qi [Advances in Mathematics 288 (2016)]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones. △ Less

Submitted 11 December, 2019; v1 submitted 23 June, 2017; originally announced June 2017.

Comments: V4: Accepted manuscript with minor edits. To appear in Adv. Math

Journal ref: Adv. Math. 361 (2020), 106937, 66 pp

Pyramids and 2-representations

Authors: Volodymyr Mazorchuk, Vanessa Miemietz, Xiaoting Zhang

Abstract: We describe a diagrammatic procedure which lifts strict monoidal actions from additive categories to categories of complexes avoiding any use of direct sums. As an application, we prove that every simple transitive $2$-representation of the $2$-category of projective bimodules over a finite dimensional algebra is equivalent to a cell $2$-representation. We describe a diagrammatic procedure which lifts strict monoidal actions from additive categories to categories of complexes avoiding any use of direct sums. As an application, we prove that every simple transitive $2$-representation of the $2$-category of projective bimodules over a finite dimensional algebra is equivalent to a cell $2$-representation. △ Less

Submitted 9 May, 2017; originally announced May 2017.

Characterisation and applications of $\Bbbk$-split bimodules

Authors: Volodymyr Mazorchuk, Vanessa Miemietz, Xiaoting Zhang

Abstract: We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk$-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk$-vector spaces. As one application, we show that any simple $2$-c… ▽ More We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk$-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk$-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\Bbbk$-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $\Bbbk[x,y]/(x^2,y^2,xy)$. △ Less

Submitted 11 January, 2017; originally announced January 2017.

Journal ref: MATH. SCAND. 124 (2019), 161--178

Simple transitive 2-representations via (co)algebra 1-morphisms

Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer

Abstract: For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using coalgebra 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 explicitly. For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using coalgebra 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 explicitly. △ Less

Submitted 6 February, 2019; v1 submitted 19 December, 2016; originally announced December 2016.

Comments: Revised version to appear in Indiana Univ. Math. J. the numbering of theorems etc. is changed to adjust to the published version

Journal ref: Indiana Univ. Math. J. 68 (2019), no. 1, 1-33

Affine quiver Schur algebras and $p$-adic $GL_n$

Authors: Vanessa Miemietz, Catharina Stroppel

Abstract: In this paper we consider the (affine) Schur algebra introduced by Vignéras as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a $p$-adic field, a large part of the unipotent block over fields of characteristic different from $p$. We show that this Schur algebra is, after a suitable completion,… ▽ More In this paper we consider the (affine) Schur algebra introduced by Vignéras as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a $p$-adic field, a large part of the unipotent block over fields of characteristic different from $p$. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a byproduct we obtain a detailed description of the algebra with a basis adapted to the geometric basis of quiver Schur algebras. We illustrate the grading in the explicit example of $\mathrm{GL}_2(\mathbb{Q}_5)$ in characteristic $3$ △ Less

Submitted 20 February, 2019; v1 submitted 27 January, 2016; originally announced January 2016.

Comments: to appear in Selecta Mathematica

Isotypic faithful $2$-representations of $\mathcal{J}$-simple fiat $2$-categories

Authors: Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: We introduce the class of isotypic $2$-representations for finitary $2$-categories and the notion of inflation of $2$-representations. Under some natural assumptions we show that isotypic $2$-representations are equivalent to inflations of cell $2$-representations. We introduce the class of isotypic $2$-representations for finitary $2$-categories and the notion of inflation of $2$-representations. Under some natural assumptions we show that isotypic $2$-representations are equivalent to inflations of cell $2$-representations. △ Less

Submitted 28 September, 2015; v1 submitted 26 August, 2014; originally announced August 2014.

Comments: to appear in Math Z

Journal ref: Math. Z. 282 (2016), no.1-2, 411-434

Transitive 2-representations of finitary 2-categories

Authors: Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: In this article, we define and study the class of simple transitive $2$-representations of finitary $2$-categories. We prove a weak version of the classical Jordan-Hölder Theorem where the weak composition subquotients are given by simple transitive $2$-representations. For a large class of finitary $2$-categories we prove that simple transitive $2$-representations are exhausted by cell $2$-repres… ▽ More In this article, we define and study the class of simple transitive $2$-representations of finitary $2$-categories. We prove a weak version of the classical Jordan-Hölder Theorem where the weak composition subquotients are given by simple transitive $2$-representations. For a large class of finitary $2$-categories we prove that simple transitive $2$-representations are exhausted by cell $2$-representations. Finally, we show that this large class contains finitary quotients of $2$-Kac-Moody algebras. △ Less

Submitted 16 December, 2015; v1 submitted 30 April, 2014; originally announced April 2014.

Comments: revised version to appear in Transactions of the AMS, proof of Proposition 22 corrected

Journal ref: Transactions of the American Mathematical Society 368 (2016), no. 11, 7623-7644

Dimensions of Ext-groups of Weyl modules for GL_2

Authors: Stephan Baier, Sergey Lamzin, Vanessa Miemietz

Abstract: Let F be an algebraically closed field of positive characteristic p. The third author and Will Turner gave an explicit description of the extension algebra of Weyl modules for GL_2(F). This, in particular, produced an explicit basis. We examine this basis and use it to give upper and lower bounds for the growth behaviour of the dimensions of Ext-groups. Moreover, we implement an algorithm to deter… ▽ More Let F be an algebraically closed field of positive characteristic p. The third author and Will Turner gave an explicit description of the extension algebra of Weyl modules for GL_2(F). This, in particular, produced an explicit basis. We examine this basis and use it to give upper and lower bounds for the growth behaviour of the dimensions of Ext-groups. Moreover, we implement an algorithm to determine the dimension of any given Ext-group. △ Less

Submitted 31 May, 2013; v1 submitted 25 May, 2013; originally announced May 2013.

Comments: 43 pages, 1 figure, 8 pages of source code of C programs

MSC Class: 20G05

Morita theory for finitary 2-categories

Authors: Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: We develop Morita theory for finitary additive 2-representations of finitary 2-categories. As an application we describe Morita equivalence classes for 2-categories of projective functors associated to finite dimensional algebras and for 2-categories of Soergel bimodules. We develop Morita theory for finitary additive 2-representations of finitary 2-categories. As an application we describe Morita equivalence classes for 2-categories of projective functors associated to finite dimensional algebras and for 2-categories of Soergel bimodules. △ Less

Submitted 17 December, 2013; v1 submitted 17 April, 2013; originally announced April 2013.

Comments: To appear in Quantum Topology

Journal ref: Quantum Topol. 7 (2016), no. 1, 1-28

Affine Cellularity of Khovanov-Lauda-Rouquier algebras in type A

Authors: Alexander S. Kleshchev, Joseph W. Loubert, Vanessa Miemietz

Abstract: We prove that the Khovanov-Lauda-Rouquier algebras $R_\al$ of type $A_\infty$ are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\al$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\al$ is finite, and yields a theory of standard and reduced stand… ▽ More We prove that the Khovanov-Lauda-Rouquier algebras $R_\al$ of type $A_\infty$ are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\al$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\al$ is finite, and yields a theory of standard and reduced standard modules for $R_\al$. △ Less

Submitted 24 October, 2012; originally announced October 2012.

Endomorphisms of cell 2-representations

Authors: Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: We determine the endomorphism categories of cell 2-representations of fiat 2-categories associated with strongly regular two-sided cells under some natural assumptions. Along the way, we completely describe J-simple fiat 2-categories which have only one two-sided cell J apart from the identities, under the same conditions as above. For positively graded 2-categories, we show that the additional re… ▽ More We determine the endomorphism categories of cell 2-representations of fiat 2-categories associated with strongly regular two-sided cells under some natural assumptions. Along the way, we completely describe J-simple fiat 2-categories which have only one two-sided cell J apart from the identities, under the same conditions as above. For positively graded 2-categories, we show that the additional restrictions are redundant. △ Less

Submitted 7 November, 2025; v1 submitted 26 July, 2012; originally announced July 2012.

Comments: Corrections to the published version: The main result and its consequences are only valid under some additional assumption. Both these additional assumptions and the corrected proof of the main result are collected in the new last section

Additive versus abelian 2-representations of fiat 2-categories

Authors: Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to $\mathfrak{sl}_2$-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur alg… ▽ More We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to $\mathfrak{sl}_2$-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras. △ Less

Submitted 21 December, 2011; originally announced December 2011.

Comments: 20 pages

Journal ref: Mosc. Math. J. 14 (2014), no. 3, 595-615

Hochschild cohomology of polynomial representations of $GL_2(\bar{\mathbb{F}}_p)$

Authors: Vanessa Miemietz, Will Turner

Abstract: We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of $\GL_2$ over an algebraically closed field of characteristic $p>2$, that is, of any block whose number of simple modules is a power of $p$. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications. We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of $\GL_2$ over an algebraically closed field of characteristic $p>2$, that is, of any block whose number of simple modules is a power of $p$. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications. △ Less

Submitted 5 March, 2018; v1 submitted 12 July, 2011; originally announced July 2011.

Comments: revised version, to appear in Documenta Mathematica

The Weyl extension algebra of $GL_2(\bar{\mathbb{F}}_p)$

Authors: Vanessa Miemietz, Will Turner

Abstract: We compute the Yoneda extension algebra of the collection of Weyl modules for $GL_2$ over an algebraically closed field of positive characteristic p by developing a theory of generalised Koszul duality for certain 2-functors, one of which controls the rational representation theory of $GL_2$ over such a field. We compute the Yoneda extension algebra of the collection of Weyl modules for $GL_2$ over an algebraically closed field of positive characteristic p by developing a theory of generalised Koszul duality for certain 2-functors, one of which controls the rational representation theory of $GL_2$ over such a field. △ Less

Submitted 5 March, 2018; v1 submitted 28 June, 2011; originally announced June 2011.

Comments: 55 pages, Adv. Math. 246 (2013), 144-197; this new version corrects a mistake in Lemma 3 by adding a new Appendix to replace Section 5

Koszul dual 2-functors and extension algebras of simple modules for $GL_2$

Authors: Vanessa Miemietz, Will Turner

Abstract: Let p be a prime number. We compute the Yoneda extension algebra of $GL_2$ over an algebraically closed field of characteristic p by developing a theory of Koszul duality for a certain class of 2-functors, one of which controls the category of rational representations of $GL_2$ over such a field. Let p be a prime number. We compute the Yoneda extension algebra of $GL_2$ over an algebraically closed field of characteristic p by developing a theory of Koszul duality for a certain class of 2-functors, one of which controls the category of rational representations of $GL_2$ over such a field. △ Less

Submitted 9 July, 2014; v1 submitted 27 June, 2011; originally announced June 2011.

Comments: 39 pages, title changed in second version, to appear in Selecta Math. (N.S.)

Cell 2-representations of finitary 2-categories

Authors: Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by Kazhdan-Lusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not… ▽ More We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by Kazhdan-Lusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorification of Kazhdan-Lusztig cell modules for Hecke algebras of type $A$ from \cite{MS2}. △ Less

Submitted 15 November, 2010; originally announced November 2010.

Comments: 25 pages

Journal ref: Compositio Math. 147 (2011) 1519-1545

Affine cellularity of affine Hecke algebras of rank two

Authors: Jeremie Guilhot, Vanessa Miemietz

Abstract: We show that affine Hecke algebras of rank two with generic parameters are affine cellular in the sense of Koenig-Xi. We show that affine Hecke algebras of rank two with generic parameters are affine cellular in the sense of Koenig-Xi. △ Less

Submitted 22 March, 2011; v1 submitted 9 November, 2010; originally announced November 2010.

Comments: 24 pages, 4 figures and 14 tables. New version: added references, corrected typos. Final version

Serre functors for Lie algebras and superalgebras

Authors: Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $\mathcal{O}$ and its parabolic generalizations for classical Lie superalgebras a… ▽ More We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $\mathcal{O}$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category $\mathcal{O}$ for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra $\mathfrak{q}(2)$. △ Less

Submitted 10 November, 2010; v1 submitted 6 August, 2010; originally announced August 2010.

Comments: 19 pages, to appear in Annales de l'Institut Fourier in 2011

MSC Class: 17B10

Journal ref: Annales de l'institut Fourier, 62 (2012), No. 1, 47--75

Symmetric quasi-hereditary envelopes

Authors: Volodymyr Mazorchuk, Vanessa Miemietz

Abstract: We show how any finite-dimensional algebra can be realized as an idempotent subquotient of some symmetric quasi-hereditary algebra. In the special case of rigid symmetric algebras we show that they can be realized as centralizer subalgebras of symmetric quasi-hereditary algebras. We also show that the infinite-dimensional symmetric quasi-hereditary algebras we construct admit quasi-hereditary st… ▽ More We show how any finite-dimensional algebra can be realized as an idempotent subquotient of some symmetric quasi-hereditary algebra. In the special case of rigid symmetric algebras we show that they can be realized as centralizer subalgebras of symmetric quasi-hereditary algebras. We also show that the infinite-dimensional symmetric quasi-hereditary algebras we construct admit quasi-hereditary structure with respect to two opposite orders, that they have strong exact Borel and $Δ$-subalgebras and the corresponding triangular decompositions. △ Less

Submitted 17 December, 2008; originally announced December 2008.

Comments: 19 pages

MSC Class: 16S99

Homotopy, homology, and $GL_2$

Authors: Vanessa Miemietz, Will Turner

Abstract: We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\mathbb{O}_p$ of these operators control all homological aspects of the rational representation theory of the algebraic group $GL_2$, over a field of positive characteristic. We prove that when $x$ is a Rickard… ▽ More We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\mathbb{O}_p$ of these operators control all homological aspects of the rational representation theory of the algebraic group $GL_2$, over a field of positive characteristic. We prove that when $x$ is a Rickard tilting complex, the operators $\mathbb{O}_{(c,x)}$ honour derived equivalences, in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight $\mathbb{Z}_+$-gradings on Schur algebras $S(2,r)$, and the existence of braid group actions on the derived categories of blocks of these Schur algebras. △ Less

Submitted 5 September, 2008; originally announced September 2008.

Comments: 28 pages

MSC Class: 20G05; 18D05

Rational representations of $GL_2$

Authors: Vanessa Miemietz, Will Turner

Abstract: Let $F$ be an algebraically closed field of characteristic $p$. We fashion an infinite dimensional basic algebra $\underleftarrow{\mathcal{C}}_p(F)$, with a transparent combinatorial structure, which we expect to control the rational representation theory of $GL_2(F)$. Let $F$ be an algebraically closed field of characteristic $p$. We fashion an infinite dimensional basic algebra $\underleftarrow{\mathcal{C}}_p(F)$, with a transparent combinatorial structure, which we expect to control the rational representation theory of $GL_2(F)$. △ Less

Submitted 5 September, 2008; originally announced September 2008.

Comments: 27 pages

MSC Class: 20G05; 16GXX

On representation theory of affine Hecke algebras of type B

Authors: Vanessa Miemietz

Abstract: Ariki's and Grojnowski's approach to the representation theory of affine Hecke algebras of type $A$ is applied to type $B$ with unequal parameters to obtain -- under certain restrictions on the eigenvalues of the lattice operators -- analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type $A$. Ariki's and Grojnowski's approach to the representation theory of affine Hecke algebras of type $A$ is applied to type $B$ with unequal parameters to obtain -- under certain restrictions on the eigenvalues of the lattice operators -- analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type $A$. △ Less

Submitted 26 September, 2007; originally announced September 2007.

Comments: to appear in Algebras and Representation theory

MSC Class: 20C08; 16S80

Crystals and affine Hecke algebras of type D

Authors: Masaki Kashiwara, Vanessa Miemietz

Abstract: The Lascoux-Leclerc-Thibon-Ariki theory asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra $g$ where $g$ is $gl_\infty$ or the affine Lie algebra $A^{(1)}_\ell$, and the irreducible representations correspond to the upper global bases. Recentl… ▽ More The Lascoux-Leclerc-Thibon-Ariki theory asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra $g$ where $g$ is $gl_\infty$ or the affine Lie algebra $A^{(1)}_\ell$, and the irreducible representations correspond to the upper global bases. Recently, N. Enomoto and the first author presented the notion of symmetric crystals and formulated analogous conjectures for the affine Hecke algebras of type B. In this note, we present similar conjectures for certain classes of irreducible representations of affine Hecke algebras of type D. The crystal for type D is a double cover of the one for type B. △ Less

Submitted 10 March, 2007; originally announced March 2007.

Comments: 8 pages

MSC Class: 17B37; 20C08

Journal ref: Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 7, 135--139