This semester in addition to inivited talks by outside speakers we will have some talks by local speakers about affine Grassmannians in geometric representation theory.
This semester in addition to inivited talks by outside speakers we will have some more expository talks by local speakers, loosely organized around Springer Theory.
This semester we will be running a reading seminar on days when there is no outside speaker. The goal will be to understand the recent proof of the positivity of coefficients of Kazhdan-Lustig polynomials by Elias and Williamson.
Bibliography:
Talks:
Equivariant cohomology, flag varieties and Bott-Samelson varieties.
k-Vexillary permutations, Stanley symmetric functions, and generalized Specht modules.
Hard Lefschetz, Hodge-Riemann bilinear relations, and the Decomposition Theorem
Soergel bimodules, part II. Special room: LGRT 1322
Double Mirković-Vilonen Cycles and the Naito-Sagaki-Saito Crystal
Kazhdan-Lusztig Conjectures, Soergel bimodules, and Hodge theory
Double Mirković-Vilonen Cycles and the Naito-Sagaki-Saito Crystal Postponed to Spring
Categorifications and representations of rational Cherednik algebras
Vertex operators and braid groups from Heisenberg categorification
This semester we will run an informal seminar on Double Affine Hecke Algebras (DAHA's) led by Alexei Oblomkov. We will also have a few outside speakers on other topics.
Here is some background material on DAHA's:
This is a short book taken from a lecture course. It overlaps substantially with the previous notes. Tom Braden has two copies and can loan one of his copies out.
This is a nice survey article about the more general class of symplectic reflection algebras, but it spends a lot of time on the Cherednik/DAHA case.
Talks:
Affine friends of the symmetric group: buzzwords introduction
Geometric constructions of representations of S_n (after Springer)
Title : Bruhat order and hyperplane arrangements
Abstract: We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincar\'e polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth.
The geometric theory of representations of affine Hecke algebras.
Title : Pieces of the exotic nilpotent cone
Abstract: The exotic nilpotent cone was introduced by Kato. Besides establishing many of its basic properties, Kato proved an exotic Springer correspodence using the cone (involving the Weyl group of type C) and he used the cone to study Hecke algebras with unequal parameters.
This talk will focus on the geometry of the cone, especially its connection with nilpotent orbits in characteristic two. This is joint work with Pramod Achar and Anthony Henderson.
Title : Parity Sheaves
Title : Quiver grassmannians, quiver varieties and the preprojective algebra
Nakajima quiver varieties are certain moduli spaces that are useful in geometric representation theory. For instance, they can be used to give a geometric realization of the crystal for any irreducible highest weight representation of a symmetric Kac-Moody algebra. It that story, certain lagrangian sub-varieties that play a crucial role. We present an alternative description of these lagrangian sub-varieties as varieties of invariant subspaces of a fixed representation of a path algebra. We find this ``grassmannian-type" description advantageous, partly because it makes certain symmetries more evident. This talk is based on joint work with Alistair Savage, and also incorporates work of George Lusztig and of Ian Shipman.
Title : Cluster-tilted algebras
Abstract: Cluster-tilted algebras are endomorphism algebras of tilting objects in cluster categories. They can also be described as trivial extensions of tilted algebras and as (a special case of) Jacobian algebras of quivers with potentials. In this talk I will mainly explain the relation to the tilted algebras, but also touch the other interpretations.
Title : Rank functions on quiver representations
Abstract: A rank function assigns a nonnegative integer to any quiver representation, generalizing the notion of the rank of a linear map. They have algebraic properties similar to the classical rank function (additive with respect to direct sum, multiplicative with respect to tensor product, invariant under duality). We will discuss how to construct a "global rank function" on any quiver (with explicit examples) and how to derive more rank functions combinatorially. These can be applied to study the structure of the representation ring of a quiver. Geometrically, rank functions are not generally semicontinuous on quiver representation spaces, but we will discuss a proof that utilizes quiver Grassmannians to show that they are at least constructible.
Title: A categorical approach to Parshin reciprocity laws on algebraic surfaces
Abstract: I will outline an intrinsic proof of Parshin reciprocity laws for two-dimensional tame symbols on an algebraic surface, which generalizes the proof of residue formula on algebraic curves by Tate and the proof of Weil reciprocity laws by Arbarello, De Concini and Kac. The key ingredient is to interpret the 2-dimensional tame symbol as the "commutator" of certain central extension of a group by a Picard groupoid. This is a joint work with D. Osipov.
Title : A Geometric Proof of a Modular Schur-Weyl Theorem
Abstract: Modular representation theory is the study of representation theory over fields of positive characteristic. Generally, modular representation theory is more complicated than its characteristic zero counterpart. Recently, a number of theorems have appeared giving geometric descriptions of categories of modular representations. A version of the geometric Satake theorem due to Mirkovic-Vilonen implies that the modular representation theory of the general linear group is encoded in a category of geometric objects called perverse sheaves on an affine Grassmannian. On the other hand, a version of Springer theory due to Juteau relates the modular representation theory of the symmetric group with the geometry of the nilpotent cone in gl_n. This talk will explain how these two pieces fit together to give a geometric explanation for connections between the modular representation theory of the general linear group and that of the symmetric group.
Title : Lusztig's conjectures on modular representations
Abstract: This is the first of a series of related talks where certain mechanisms will be applied to representation theory, Langlands program, algebraic geometry and hopefully to knot invariants and quantum field theory.
The representation theoretic aspect is a proof of Lusztig's conjectures which describe numerical structure of modular representation theory (with Bezrukavnikov).
The two key geometric ideas: (i) a construction of Azumaya algebras in positive characteristic as a tool for math/physics, (ii) action of the affine braid groups on coherent sheaves on cotangent bundles of flag varieties.
Title : Lusztig's conjectures on modular representations (part II)
Title : Nilpotent orbits in characteristic 2 and Springer correspondence
Abstract: Let k and F_q be an algebraically closed and a finite field of characteristic 2 respectively. Let G be an adjoint (resp. simply connected) algebraic group of type B,C or D over k, g the Lie algebra of G and g^* the dual vector space of g. We construct the Springer correspondence for g (resp. g^*) following Lusztig's method. The correspondence is a bijective map from the set A_g (resp. A_g^*) to the set of irreducible characters of the Weyl group of G, where A_g (resp. A_g^*) is the set of all pairs (c,F) with c a nilpotent G-orbit in g (resp. g^*) and F an irreducible G-equivariant local system on c (up to isomorphism). In particular, we obtain classifications of nilpotent orbits in orthogonal Lie algebras over F_q and in the duals of classical Lie algebras over k and F_q. Finally, we describe the explicit correspondence using similar combinatorics that appears in the description of generalized Springer correspondence (defined by Lusztig) for classical groups in the case of characteristic not equal 2 and unipotent case in characteristic 2.
Title : Faces of polytopes and Koszul algebras
Abstract: Given a simple Lie algebra g and a finite-dimensional simple g-module V, we study the category G of graded finite-dimensional modules of the corresponding semidirect product Lie algebra. This framework includes the truncated current Lie algebras as well as those associated to folding of complex simple Lie algebras. Given a face of the polytope formed by the weights of V, we introduce a partial order on the simple objects in G. For certain finite subsets of the affine weight lattice, we produce Koszul algebras of finite global dimension equal to the number of weights of V which are on the face. This is joint work with Vyjayanthi Chari and Tim Ridenour.
Title : Twisted geometric Satake correspondence
Title : Representation theory of hyperplane arrangements