Kent Spring School in Representation Theory: Programme

Melissa Sherman-Bennett: Cluster structures on braid varieties

   Abstract: Cluster algebras, introduced by Fomin and Zelevinsky, are a class of commutative rings which are completely determined by some combinatorial input called a seed. They arise in a number of contexts, from representation theory to Poisson geometry to mirror symmetry. Braid varieties for a simple complex algebraic group G are smooth affine varieties associated to any word in the Weyl group W. Special cases of braid varieties include Richardson varieties, double Bruhat cells, and double Bott-Samelson cells.  
I'll begin with an introduction to cluster algebras, focusing on the example of the homogeneous coordinate ring of the Grassmannian (due to Scott). In this particularly nice case, seeds are encoded by Postnikov's plabic graphs. Then, I'll turn to the substantially more general situation of braid varieties. I'll discuss joint work with P. Galashin, T. Lam and D. Speyer in which we show the coordinate rings of braid varieties are cluster algebras. When G=SL(n), seeds for these cluster algebras come from "3D plabic graphs", which generalize Postnikov's plabic graphs.

Hipolito Treffinger: An introduction to tau-tilting theory

   Abstract: Inspired by Fomin-Zelevinski's cluster algebras, Adachi, Iyama and Reiten introduced τ-tilting as a completion of classical tilting theory with respect to the notion of mutation and it quickly become a central topic in representation theory of finite-dimensional algebras. This course is intended to be a gentle introduction to this subject. The first lecture will be dedicated to Auslander-Reiten theory: we will give the definition of the Auslander-Reiten translation τ and we will comment on some of its properties. We will finish the lecture by defining τ-rigid and τ-tilting modules. In the second and third lecture we will study properties of τ-rigid modules and their relationship will other concepts in representation theory such as torsion pairs. Finally, in the fourth lecture we will discuss some recent results and open problems in τ-tilting theory.

Maud de Visscher: Kazhdan-Lusztig theory for centraliser algebras.

   Abstract: In this series of lecture, we will consider centraliser algebras of the action of classical groups (and their quantum analogues) on tensor spaces. The most studied and best understood is the Temperley-Lieb algebra and we will start by reviewing its representation theory. We will then turn to the Brauer, walled Brauer and partition algebra and explain how their representation theory can be studied in a uniform way by generalising the techniques used for the Temperley-Lieb algebra. We will show how their decomposition matrices can all be described by certain parabolic Kazhdan-Lusztig polynomials. These, in turn, have very nice combinatorial description in terms of oriented (generalised) Temperley-Lieb algebras.

Paul Wedrich: A Kirby color for Khovanov homology

   Abstract: The Jones polynomial of a knot can be computed relatively straightforwardly using the Temperley-Lieb algebras, which admit a diagrammatic presentation. Surprisingly, closely related diagrammatic algebras, the dotted Temperley-Lieb algebras (a.k.a. nil-blob algebras), appear when extending Khovanov's categorification of the Jones polynomial to an invariant of smooth 4-dimensional manifolds. In the first talk, I will define these algebras, assemble them into a monoidal category, and outline how their representation theory is related to Khovanov homology. In the second talk, I will introduce a certain completion of the dotted Temperley-Lieb category and a diagrammatic calculus for it. This completion contains a special object, the eponymous Kirby color for Khovanov homology, and I will discuss its handle-slide property. The third talk will give an introduction to Khovanov homology and its annular version. In the final talk, I will outline how Khovanov homology extends to an invariant of smooth 4-manifolds and explain how the Kirby color helps in computing these invariants for 2-handlebodies. This lecture series is based on joint work with Hogancamp, Morrison, Rose and Walker. Talk 1: The dotted Temperley-Lieb category dTL and its polynomial representation. Talk 2: The Kirby object and diagrammatic calculus in the ind completion of dTL. Talk 3: Introduction to (annular) Khovanov homology. Talk 4: Khovanov homology for 4-manifolds.