Ben Elias

Diagrammatics for Soergel Categories

The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.

The two-color Soergel calculus

We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W . The (two-colored) Temperley-Lieb category is embedded inside this category as the degree 0 morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones-Wenzl projectors. When W is infinite, the parameter q of the Temperley-Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for sl(2). When W is finite, q must be specialized to an appropriate root of unity, and the negligible Jones-Wenzl projector yields the Soergel bimodule for the longest element of W .

An approach to categorification of some small quantum groups II

Abstract We categorify an idempotented form of quantum sl 2 and some of its simple representations at a prime root of unity.

Diagrammatics for Coxeter groups and their braid groups

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.

A categorification of quantum sl(2) at prime roots of unity

Abstract We categorify the Beilinson–Lusztig–MacPherson integral form of quantum sl ( 2 ) specialized at a prime root of unity.

On cubes of Frobenius extensions

In this note we explain how Lusztig’s induction and restriction functors yield categorical actions of Kac-Moody algebras on the derived category of unipotent representations. We focus on the example of finite linear groups and induction/restriction associated with split Levi subgroups, providing a derived analogue of Harish-Chandra induction/restriction as studied by Chuang-Rouquier in [5]. 2010 Mathematics Subject Classification. Primary 20C33.We study composition series of derived module categories in the sense of Angeleri Hugel, Konig & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confirms part of a conjecture of Bobinski & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of different lengths and composition factors. In other words, there is no Jordan-Holder property for composition series of derived categories of quasi-hereditary algebras.We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are

Trace decategorification of the Hecke category

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Minimal Permutation Representations of Nilpotent Groups

-periodic of period dividing 4. Our construction is based on the combinatorial notion of triangulation quivers, which arise naturally from triangulations of oriented surfaces with marked points. This class of algebras contains the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin-Shapiro-Thurston and Labardini-Fragoso to triangulations of closed surfaces with punctures, hence our construction may serve as a bridge between the modular representation theory of finite groups and the theory of cluster algebras.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.Let G be a semisimple complex Lie group. In this article, we study Geometric Invariant Theory on a flag variety G/B with respect to the action of a principal 3-dimensional simple subgroup S of G. We determine explicitly the GIT-equivalence classes of S-ample line bundles on G/B. We show that, under mild assumptions, among the GIT-classes there are chambers, in the sense of Dolgachev-Hu. The Hilbert quotients Y=X//S with respect to various chambers form a family of Mori dream spaces, canonically associated with G. We are able to determine three important cones in the Picard group of any of these quotients: the pseudo-effective, the movable and the nef cones.

Kazhdan–Lusztig Conjectures and Shadows of Hodge Theory

We compute the trace decategorification of the Hecke category for an arbitrary Coxeter group. More generally, we introduce the notion of a strictly object-adapted cellular category and calculate the trace for such categories.

The Hodge theory of Soergel bimodules

A minimal permutation representation of a finite group G is a faithful G-set with the smallest possible size. We study the structure of such representations and show that for certain groups they may be obtained by a greedy construction. In these situations (except when central involutions intervene) all minimal permutation representations have the same set of orbit sizes. Using the same ideas, we also show that if the size d(G) of a minimal faithful G-set is at least c|G| for some c > 0, then d(G) = |G|/m + O(1) for an integer m, with the implied constant depending on c.