Schedule of the Workshop

Abstracts:

Arkady Berenstein: Littlewood-Richardson coefficients for reflection groups 
The goal of my talk (based on a recent joint paper with Edward Richmond) is to compute the Littlewood-Richardson (LR) coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra H(G/P), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the LR coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of G. In particular, our formula gives a combinatorial proof of positivity of the LR coefficients in the cases when off-diagonal Cartan matrix entries are less than or equal to -2.

Klaus Bongartz: Indecomposables live in all smaller lengths
Using the theory of representation-finite algebras one gets that there is no gap in the lengths of the indecomposable representations of a finite dimensional algebra over an algebraically closed field. In fact, the same result holds for any k-linear abelian category occurring in representation theory as long as all simple objects have trivial endomorphism algebra k.

Vyjayanthi Chari: Truncated Weyl modules, Fusion products and Schur Positivity
Local Weyl modules are a family of indecomposable finite dimensional modules of the current algebra associated to a simple Lie algebra. The structure of these modules is now well understood. The tensor product of the local Weyl modules is not always a local Weyl module, and this leads naturally to the study of certain truncations of these modules. In this talk, we shall discuss these truncations and formulate a conjecture on their structure when the simple Lie algebra is of classical type. We shall see that the conjecture naturally implies a result of Lam, Postnikov and Pavalvsky on Schur positivity when the simple algebra is of type sl_n+1. We provide further evidence for our conjecture by showing a maximal dimension property for tensor products of irreducible representations of simple Lie algebras. Finally, we define a refinement of the sorting operation of Fomin et al for an arbitrary simple Lie algebra. This is based on joint work with Ghislian Fourier and Daisuke Sagaki.

Joseph Chuang: A dual approach to representations of symmetric groups
(Joint work with Hyohe Miyachi.) I will explain how the representation theory of symmetric groups might be organized around Ariki-Koike algebras arising as algebras of extensions. The approach is inspired by an interpretation of Frenkels level-rank duality as Koszul duality.

Corrado De Concini: The Infinitesimal index
Let G be a compact Lie group with Lie algebra g. Given a G-manifold M with a G-equivariant one form ω, we consider the zeroes M0 of the corresponding moment map. We then define a map, called infinitesimal index, of S[g*]-modules from the equivariant cohomology of M0 with compact support to the space of invariant distributions on g*. In the case in which G is a torus, N is a linear complex representation of G, M = T*N with tautological one form, we are going to explain how this is used to compute the equivariant cohomology of M0 with compact support using certain spaces of polynomials which appear in approximation theory and compute volumes of certain variable polytopes. (joint with C. Procesi and M. Vergne) Slides

Richard Dipper: Unipotent Specht modules of finite general linear groups and a conjecture of Higman.
This is joint work with B. Ackermann and Qiong Guo.
Let G=GL(n,q), q some prime power, and let U be the subgroup of unipotent lower triangular matrices of G. We decompose unipotent Specht modules associated with two part partitions as U-modules over finite fields of characteristic not dividing q completely into irreducible components. This provides a new construction and representation theoretic interpretation for the standard basis of theses modules which was obtained by Brandt, James, Lyle and myself in previous work. It is conceivable that this construction is accessible to generalization to arbitrary partitions of n. It is explained, how this new approach relates to a conjecture of Higman which states, that the number of conjugacy classes of U should be a polynomial in q with integral coefficients.

Evgeny Feigin: PBW degeneration: representations, flag varieties and combinatorics in type A
The PBW filtration on a highest weight representation of a simple Lie algebra g is induced by the standard (degree) filtration on the universal enveloping algebra of lowering operators. The associated graded space carries a structure of a representation of the degenerate Lie algebra and the degenerate Lie group. We will describe these representations for the Lie algebras of type A. We will also define the degenerate analogues of the flag varieties. We will give an explicit description of these singular varieties, construct desingularizations and derive a formula for the q-characters of the highest weight g-modules.

Peter Fiebig: On the representation theory of affine Kac-Moody algebras at the critical level.
In the talk I want to report on a joint research project with Tomoyuki Arakawa on the category O associated to an affine Kac-Moody algebra at the critical level. In his ICM address in 1990 in Kyoto, Lusztig conjectured a link between critical level representations and the representation theory of modular Lie algebras and quantum groups. Compatible with this is the Feigin-Frenkel conjecture on the characters of simple critical highest weight representations. Both conjectures motivated our approach, whose main goal is to establish a functorial connection between intersection cohomology sheaves on affine Grassmannians and the critical level category O in a way that is Koszul-dual to the results of Frenkel and Gaitsgory.

Stéphane Gaussent: Some formulas for Hall-Littlewood polynomials
The Hall-Littlewood polynomials form a basis of the symmetric polynomials that interpolates between the Schur functions and the monomial ones. Using a geometrical interpretation, I will first explain how to obtain a combinatorial formula for the components of the expansion of Hall-Littlewood polynomials in the monomial basis (a joint work with Littelmann). In type A, Macdonald has also a combinatorial formula for these polynomials. In the second part of the talk, I will recall his formula and explain how Klostermann proves that our formula is a generalization of Macdonalds one.

Anne Henke: Symmetric powers and Schur algebras of Brauer algebras
Let an orthogonal or symplectic group G act on tensor space. The endomorphism ring is known to be a Brauer algebra. Now let G act on a direct sum of all symmetric powers of the natural module. We determine the endomorphism ring and its structure. This is joint work with Steffen König.

Lidia Angeleri Huegel: Stratifications of derived categories
This is a report on joint work with Steffen König and Qunhua Liu. We consider the notion of a recollement introduced by Beilinson, Bernstein, and Deligne in 1982. Recollements of triangulated categories describe the middle term by a triangulated subcategory and a triangulated quotient category. They may be regarded as the analog of a short exact sequence. Iterated recollements are then analogues of geometric or topological stratifications, or of composition series of algebraic objects. I will discuss the question whether a derived module category admits such stratifications, and whether they are unique up to ordering and equivalence. The main result will be a positive answer in form of a Jordan Hölder theorem for derived module categories of piecewise hereditary finite dimensional algebras.

Masaki Kashiwara: Cyclotomic quiver Hecke algebras categorify highest weight modules
Kovanov-Lauda conjectured that the cyclotomic quiver Hecke algebras categorify the irreducible highest weight modules of any symmetrizable Kac-Moody Lie algebra. I shall present its affirmative proof using the categorification of the equality ad(e_i) = k_i^-1 e_i' - k_i e_i''. It is a joint work with Seok-Jin Kang.

Alexander Kleshchev: Standard and irreducible modules for Khovanov-Lauda-Rouquier algebras
We review the approach to the representation theory of affine and cyclotomic KLR algebras through standard modules. In the first part of the talk we will explain how combinatorics of Lyndon words generalizes combinatorics of Zelevinsky multisegments and allows one to construct Bernstein-Zelevinsky type classification of irreducible modules for affine KLR algebras of finite type. In the second part we will talk about Specht modules for cyclotomic Hecke algebras. The results are joint with Arun Ram and Andrew Mathas.

Toshiyuki Kobayashi: Restrictions of Verma modules to symmetric pairs and some applications to differential geometry
I will discuss a "framework" of branching problems for generalized Verma modules with respect to reductive symmetric pairs from the viewpoint of "discrete decomposability", and explain some basic results on the size of irreducible summands and multiplicities.
As an application, I plan to explain a new and simple method to obtain Cohen-Rankin operators for holomorphic automorphic forms and Juhls conformally equivariant differential operators together with their generalizations.

Henning Krause: Bousfield classes of finite group representations 
We explain the structure of the Bousfield lattice of the stable module category of a finite group. This is based on joint work with Dave Benson and Srikanth Iyengar.

Thomas Lam: Electrical networks and Lie theory
We will discuss a class of Lie algebras and Lie groups which arise in the study of electrical resistor networks. For our purposes, these networks are essentially weighted undirected graphs that satisfy certain equivalence relations, corresponding to networks which are electrically equivalent. We shall show that these equivalence relations lead to Lie algebras satisfying a deformation of Serres relation.
There are electrical analogues of total positivity, geometric crystals, and R-matrices. The ideas from representation theory can then be applied to the inverse (or Dirichlet to Neumann) problem: can the resistances of an electrical network be recovered from making external measurements?
This is joint work with Pavlo Pylyavskyy.

Qunhua Liu: On derived simplicity of symmetric algebras
Recollement, defined by Beilinson, Berstein and Deligne, can be viewed as an analogue of short exact sequence of triangulated categories. Derived module categories of rings (or algebras) are of particular interests among all triangulated categories. Following Wiedemann, we say a ring R is derived simple if its derived module category admits no nontrivial recollements in terms of derived module categories of two other rings. More precisely, depending on the level of the derived module category one considers, R is said to be derived simple with respect to D*(Mod) or D*(mod) (* ∈ {b,+,-,∅}). In general derived simplicities on different levels are not equivalent.
In this talk I will first discuss lifting and restricting recollements on different levels; and then prove that an indecomposable symmetric algebra is D^b(mod)-derived simple, and under some condition also D(Mod)-derived simple. As example, blocks of group algebras of finite groups satisfy this condition and are derived simple on all levels.

Markus Reineke: GW/QM correspondence
The tropical vertex, a group of formal automorphisms of a torus, relates two seemingly unrelated geometries: the Gromow-Witten theory of certain toric surfaces (by work of Gross-Pandharipande-Siebert), and moduli spaces of representations of certain quivers. We will review these geometries and discuss a general factorization formula in the tropical vertex relating Gromov-Witten invariants and Euler characteristics of quiver moduli. Examples and special classes admitting explicit formulas will be discussed.

Alistair Savage: The representation theory of equivariant map algebras
Suppose a finite group acts on a scheme (or algebraic variety) X and a finite-dimensional Lie algebra g. Then the space of equivariant algebraic maps from X to g is a Lie algebra under pointwise multiplication. Examples of such equivariant map algebras include (multi)current algebras, (multi)loop algebras, three point Lie algebras, and the (generalized) Onsager algebra. In previous work with Erhard Neher and Prasad Senesi, we classified all finite-dimensional irreducible representations of such Lie algebras. It turns out that they are almost all evaluation representations. In this talk, I will briefly review these results and then discuss recent work with Erhard Neher on the extensions between finite-dimensional irreducible representations. Under some additional assumptions, we are able to give an explicit description of the space of extensions, which in turn allows us to give a nice description of the blocks of the category of finite-dimensional representations. Time permitting, I will also discuss recent joint work with Ghislain Fourier, Tanusree Khandai, and Deniz Kus on local Weyl modules for equivariant map algebras. Slides

Anne Schilling: Crystal energies via the charge in types A and C
The energy function of affine crystals is an important grading used in one-dimensional configuration sums of statistical mechanical models and generalized Kostka polynomials. It is defined by the action of the affine Kashiwara crystal operators through a local combinatorial rule generalizing descents and the R-matrix.
Nakayashiki and Yamada related the energy function in type A to the charge statistic of Lascoux and Schuetzenberger. Computationally, it is much more efficient to compute charge than energy since its definition involves a recursive definition of local energy and the combinatorial R-matrix, for which not in all cases efficient algorithms exist. In this talk we relate energy to a new charge statistic in type C which comes from the Ram-Yip formula for Macdonald polynomials. This involves in particular the   generalization of parts of the Kyoto path model for perfect crystals to the nonperfect setting, which yields an isomorphism between affine highest weight crystals and tensor products of Kirillov-Reshetikhin crystals.
This is joint work with Cristian Lenart. Slides

Wolfgang Soergel: Modular Koszul-Duality
I explain how the Koszul duality formalism can be made to work for relatively small characteristic, establishing an equivalence between a geometrically and representation theoretically defined triangulated categories.

Harry Tamvakis: Giambelli formulas for isotropic Grassmannians
The classical Giambelli formula expresses any Schubert class in the cohomology ring of the Grassmannian of m-planes in complex N-space as a Jacobi-Trudi determinant in certain special Schubert classes. We will discuss an analogue of this formula for the Grassmannians in the other classical Lie types. In particular we establish a surprising relation between the cohomology of odd and even orthogonal Grassmannians. This is joint work with Anders Buch and Andrew Kresch.

Hugh Thomas: Quotient-closed subcategories of representations of quivers and sorting order on reflection groups
Let Q be a quiver without oriented cycles. We show that the quotient-closed full subcategories of rep Q which are cofinite (i.e. contain all but finitely many indecomposable representations of Q) are naturally in bijection with the elements of the Weyl group W associated to Q, and that the inclusion order on the subcategories corresponds to the "sorting order" on W introduced by Armstrong. In the Dynkin case, cofiniteness is trivially satisfied, and we obtain a bijection between the quotient-closed subcategories and the elements of W. Even the fact that these sets have the same cardinality seems to be new. The representation theory of preprojective algebras plays an important role in our analysis. This is joint work with Steffen Oppermann and Idun Reiten.