arXiv subjects

Carl Mautner

Publications and source records attributed to Carl Mautner.

14 recordsLinked to original sources

arXiv2026

Flag Space, Matroidal Schur Algebras and the Steinberg Representation

We define via generators and relations an extended version of the matroidal Schur algebras introduced in earlier work of the author with Tom Braden. In the case of projective geometries, we relate the representation theory of the resulting algebras to the structure of the modular Steinberg representation of $GL(n,q)$. In the general case we show that the extended matroidal Schur algebras have a triangular, or Reedy, decomposition and are thus quasi-hereditary. We also obtain a new description of the original matroidal Schur algebras and clarify the connection to the flag spaces of Brylawski--Varchenko.

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arXiv2024

An Intersection Matrix for Affine Hyperplane Arrangements

For a real affine hyperplane arrangement, we define an integer intersection matrix with a natural $q$-deformation related to the intersections of bounded chambers of the arrangement. By connecting the integer matrix to a bilinear form of Schechtman-Varchenko, we show that there is a closed formula for its determinant that only depends on the combinatorics of the underlying matroid. We conjecture an analogous formula for its $q$-deformation. Our work also applies more generally in the setting of affine oriented matroids. Additionally, we give a representation-theoretic interpretation of our $q$-intersection matrix using Braden-Licata-Proudfoot-Websters's hypertoric category $\mathcal{O}$ (or more generally Kowalenko-Mautner's category $\mathcal{O}$ for oriented matroid programs). This paper is part of a broader program to categorify matroidal Schur algebras defined by Braden-Mautner.

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arXiv2022

Perverse sheaves on symmetric products of the plane

For any field $k$, we give an algebraic description of the category $\mathrm{Perv}_\mathscr{S}(S^n (\mathbb{C}^2),k)$ of perverse sheaves on the $n$-fold symmetric product of the plane $S^n(\mathbb{C}^2)$ constructible with respect to its natural stratification and with coefficients in $k$. In particular, we show that it is equivalent to the category of modules over a new algebra that is closely related to the Schur algebra. As part of our description we obtain an analogue of modular Springer theory for the Hilbert scheme $\mathrm{Hilb}^n(\mathbb{C}^2)$ of $n$ points in the plane with its Hilbert-Chow morphism.

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arXiv2021

Category $\mathcal{O}$ for Oriented Matroids

We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category $\mathcal O$. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden--Licata--Proudfoot--Webster. Applying our construction to nonlinear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups.

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arXiv2017

Ringel duality for perverse sheaves on hypertoric varieties

Motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities, we study a category of perverse sheaves with coefficients in a field $k$ on any affine unimodular hypertoric variety. Our main result is that this is a highest weight category whose Ringel dual is the corresponding category for the Gale dual hypertoric variety. On the way to proving our main result, we confirm a conjecture of Finkelberg-Kubrak in the case of hypertoric varieties. We also show that our category is equivalent to representations of a combinatorially-defined algebra, recently introduced in a related paper.

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arXiv2016

Matroidal Schur Algebras

Fix a principal ideal domain $k$. In this article we associate to a (weighted) matroid $M$ a quasi-hereditary algebra $R(M)$ defined over $k$ such that matroid duality corresponds to Ringel duality of quasi-hereditary algebras. The representation theory of these algebras is related to work of Schechtman-Varchenko and Brylawski-Varchenko. In characteristic zero, our algebras are also closely related to work of Kook-Reiner-Stanton and Denham.

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arXiv2016

Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture

Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence.

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arXiv2015

Affine pavings and the enhanced nilpotent cone

We construct affine pavings of Springer-type fibers over the enhanced nilpotent cone. This resolves a question of Achar-Henderson and implies the existence of perverse parity sheaves on the enhanced nilpotent cone.

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arXiv2015

On the exotic t-structure in positive characteristic

In this paper we study Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of a connected reductive algebraic group defined over a field of positive characteristic with simply-connected derived subgroup. In particular, we show that the heart of the exotic t-structure is a graded highest weight category, and we study the tilting objects in this heart. Our main tool is the "geometric braid group action" studied by Bezrukavnikov and the second author.

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arXiv2014

Parity sheaves and tilting modules

We show that tilting modules and parity sheaves on the affine Grassmannian are related through the geometric Satake correspondence, when the characteristic is bigger than an explicit bound.

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arXiv2014

Parity Sheaves

Given a stratified variety X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of "parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case. Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig's conjecture.

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arXiv2014

A geometric Schur functor

We give geometric descriptions of the category C_k(n,d) of rational polynomial representations of GL_n over a field k of degree d for d less than or equal to n, the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category C_k(n,d) and the Schur functor.

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arXiv2012

Sheaves on nilpotent cones, Fourier transform, and a geometric Ringel duality

Given the nilpotent cone of a complex reductive Lie algebra, we consider its equivariant constructible derived category of sheaves with coefficients in an arbitrary field. This category and its subcategory of perverse sheaves play an important role in Springer theory and the theory of character sheaves. We show that the composition of the Fourier--Sato transform on the Lie algebra followed by restriction to the nilpotent cone restricts to an autoequivalence of the derived category of the nilpotent cone. In the case of $GL_n$, we show that this autoequivalence can be regarded as a geometric version of Ringel duality for the Schur algebra.

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arXiv2009

Perverse sheaves and modular representation theory

This paper is an introduction to the use of perverse sheaves with positive characteristic coefficients in modular representation theory. In the first part, we survey results relating singularities in finite and affine Schubert varieties and nilpotent cones to modular representations of reductive groups and their Weyl groups. The second part is a brief introduction to the theory of perverse sheaves with an emphasis on the case of positive characteristic and integral coefficients. In the final part, we provide some explicit examples of stalks of intersection cohomology complexes with integral or positive characteristic coefficients in nilpotent cones, mostly in type $A$. Some of these computations might be new.

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