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Catharina Stroppel

Publications and source records attributed to Catharina Stroppel.

At least 37 records · Page 2Linked to original sources
arXiv2020

Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.

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arXiv2019

Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley--Lieb algebra with infinitely many generators and defining parameter $0$ on the category $\mathcal{F}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley--Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of $\mathcal{F}_n$. We also prove that indecomposable projective modules in this category are multiplicity-free.

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arXiv2019

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

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 $3$

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arXiv2018

Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

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arXiv2018

Yang-Baxter algebras as convolution algebras: The Grassmannian case

We present a simple but explicit example of a recent development which connects quantum integrable models with Schubert calculus: there is a purely geometric construction of solutions to the Yang-Baxter equation and their associated Yang-Baxter algebras which play a central role in quantum integrable systems and exactly solvable lattice models in statistical physics. We consider the degenerate five-vertex limit of the asymmetric six-vertex model and identify its associated Yang-Baxter algebra as convolution algebra arising from the equivariant Schubert calculus of Grassmannians. We show how our method can be used to construct (Schur algebra type) quotients of the current algebra $\mathfrak{gl}_2[t]$ acting on the tensor product of copies of its evaluation representation $\mathbb{C}^2[t]$. Finally we connect it with the COHA for the $A_1$-quiver.

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arXiv2018

Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality

We describe how certain cyclotomic Nazarov-Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category O of type $D$. Furthermore we study a family of subalgebras of these endomorphism rings which exhibit similar behaviour to the family of Brauer algebras even when they are not semisimple. The translation functors on this parabolic category O are studied and proven to yield a categorification of a coideal subalgebra of the general linear Lie algebra. Finally this is put into the context of categorifying skew Howe duality for these subalgebras.

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arXiv2018

The affine VW supercategory

We define the affine VW supercategory $\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$, which arises from studying the action of the periplectic Lie superalgebra $\mathfrak{p}(n)$ on the tensor product $M\otimes V^{\otimes a}$ of an arbitrary representation $M$ with several copies of the vector representation $V$ of $\mathfrak{p}(n)$. It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in $\mathfrak{p}(n)\otimes \mathfrak{p}(n)$. When $M$ is the trivial representation, the action factors through the Brauer supercategory $\mathit{s}\mathcal{B}\mathit{r}$. Our main result is an explicit basis theorem for the morphism spaces of $\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$ and, as a consequence, of $\mathit{s}\mathcal{B}\mathit{r}$. The proof utilises the close connection with the representation theory of $\mathfrak{p}(n)$. As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation.

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arXiv2017

Cellular structures using $\textbf{U}_q$-tilting modules

We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple cases for $q$ being a root of unity and ground fields of positive characteristic. Our approach also generalizes to certain categories containing infinite-dimensional modules. As applications, we give a new semisimplicty criterion for centralizer algebras, and recover the cellularity of several known algebras (with partially new cellular bases) which all fit into our general setup.

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arXiv2016

Semisimplicity of Hecke and (walled) Brauer algebras

We show how to use Jantzen's sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\textbf{U}_q$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in\mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\textbf{A}$ and $\textbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.

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arXiv2016

On the category of finite-dimensional representations of $\OSPrn$: Part I

We study the combinatorics of the category F of finite-dimensional modules for the orthosymplectic Lie supergroup OSP(r|2n). In particular we present a positive counting formula for the dimension of the space of homomorphism between two projective modules. This refines earlier results of Gruson and Serganova. Moreover, for each block B of F we construct an algebra A(B) whose module category shares the combinatorics with B. It arises as a subquotient of a suitable limit of type D Khovanov algebras. It will turn out that A(B) is isomorphic to the endomorphism algebra of a minimal projective generator of B. This provides a direct link from F to parabolic categories O of type B or D, with maximal parabolic of type A, to the geometry of isotropic Grassmannians of types B/D and to Springer fibres of types C/D. We also indicate why F is not highest weight in general.

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arXiv2015

Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties like graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.

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arXiv2015

2-row Springer fibres and Khovanov diagram algebras for type D

We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as topological point of view. We show that the irreducible components and their pairwise intersections are iterated P^1-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type D diagram calculus labelling the irreducible components in an convenient way which relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type D setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type A to other types. The results will be connected to Brauer algebras at non-generic parameters in a subsequent paper.

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arXiv2015

G(l,k,d)-modules via groupoids

In this note we describe a seemingly new approach to the complex representation theory of the wreath product $G\wr S_d$ where $G$ is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of $G\wr S_d$. This directly implies a classification of simple modules. As an application, we get a Gelfand model for $G\wr S_d$ from the classical involutive Gelfand model for the symmetric group. We describe the Schur-Weyl duality which motivates our approach and relate it to various Schur-Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type $G(\ell,k,d)$.

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arXiv2015

Categorification of tensor product representations of sl(k) and category O

We construct categorifications of tensor products of arbitrary finite-dimensional irreducible representations of $\mathfrak{sl}_k$ with subquotient categories of the BGG category $\mathcal{O}$, generalizing previous work of Sussan and Mazorchuk-Stroppel. Using Lie theoretical methods, we prove in detail that they are tensor product categorifications according to the recent definition of Losev and Webster. As an application we deduce an equivalence of categories between certain versions of category $\mathcal{O}$ and Webster's tensor product categories. Finally we indicate how the categorifications of tensor products of the natural representation of $\mathfrak{gl}(1|1)$ fit into this framework.

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arXiv2015

Walled Brauer algebras as idempotent truncations of level 2 cyclotomic quotients

We realize (via an explicit isomorphism) the walled Brauer algebra for an arbitrary integral parameter delta as an idempotent truncation of a level two cyclotomic degenerate affine walled Brauer algebra. The latter arises naturally in Lie theory as the endomorphism ring of so-called mixed tensor products, i.e. of a parabolic Verma module tensored with some copies of the natural representation and its dual. This provides us a method to construct central elements in the walled Brauer algebras and can be applied to establish the Koszulity of the walled Brauer algebra if delta is not zero.

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arXiv2015

Koszul gradings on Brauer algebras

We show that the Brauer algebra over the complex numbers for an integral parameter delta can be equipped with a grading, in the case of delta being non-zero turning it into a graded quasi-hereditary algebra. In which case it is Morita equivalent to a Koszul algebra. This is done by realizing the Brauer algebra as an idempotent truncation of a certain level two VW-algebra for some large positive integral parameter N. The parameter delta appears then in the choice of a cyclotomic quotient. This cyclotomic VW-algebra arises naturally as an endomorphism algebra of a certain projective module in parabolic category O for an even special orthogonal Lie algebra. In particular, the graded decomposition numbers are given by the associated parabolic Kazhdan-Lusztig polynomials.

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