Past Representation Theory Seminar

I will explain some ongoing work on understanding algebraic D-moldules via their reduction to positive characteristic. I will define the p-cycle of an algebraic D-module, explain the general results of Bitoun and Van Den Bergh; and then discuss a new construction of a class of algebraic D-modules with prescribed p-cycle.

In this talk I will give a definition of cluster algebra and state some main results.

Moreover, I will explain how the combinatorics of certain cluster algebras can be modeled in geometric terms.

Let $A$ be a finite-dimensional $K$-algebra over an algebraically closed field $K$. Denote by $\Omega_A$ the syzygy operator on the category $\mod A$ of finite-dimensional right $A$-modules, which assigns to a module $M$ in $\mod A$ the kernel $\Omega_A(M)$ of a minimal projective cover $P_A(M) \to M$ of $M$ in $\mod A$. A module $M$ in $\mod A$ is said to be periodic if $\Omega_A^n(M) \cong M$ for some $n \geq 1$. Then $A$ is said to be a periodic algebra if $A$ is periodic in the module category $\mod A^e$ of the enveloping algebra $A^e = A^{\op} \otimes_K A$. The periodic algebras $A$ are self-injective and their module categories $\mod A$ are periodic (all modules in $\mod A$ without projective direct summands are periodic). The periodicity of an algebra $A$ is related with periodicity of its Hochschild cohomology algebra $HH^{*}(A)$ and is invariant under equivalences of the derived categories $D^b(\mod A)$ of bounded complexes over $\mod A$. One of the exciting open problems in the representation theory of self-injective algebras is to determine the Morita equivalence classes of periodic algebras.

We will present the current stage of the solution of this problem and exhibit prominent classes of periodic algebras.

We investigate symmetric quotient algebras of symmetric algebras,

with an emphasis on finite group algebras over a complete discrete

valuation ring R with residue field of positive characteristic p. Using elementary methods, we show that if an

ordinary irreducible character of a finite group gives

rise to a symmetric quotient over R which is not a matrix algebra,

then the decomposition numbers of the row labelled by the character are

all divisible by p. In a different direction, we show that if is P is a finite

p-group with a cyclic normal subgroup of index p, then every ordinary irreducible character of P gives rise to a

symmetric quotient of RP. This is joint work with Shigeo Koshitani and Markus Linckelmann.

We explain how Khovanov-Lauda-Rouquier algebras in finite type A are affine cellular in the sense of Koenig and Xi. In particular this reproves finiteness of their global dimension. This is joint work with Alexander Kleshchev and Joseph Loubert.

This talk is about some recent joint work with Sarah Witherspoon. The representations of some finite dimensional Hopf algebras have curious behaviour: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. I shall describe a family of examples of such Hopf algebras and their modules, and the classification of left, right, and two-sided ideals in their stable module categories.

To a finite connected acyclic quiver Q there is associated a path algebra kQ, for an algebraically closed field k, a Coxeter group W and a preprojective algebra. We discuss a bijection between elements of the Coxeter group W and the cofinite quotient closed subcategories of mod kQ, obtained by using the preprojective algebra. This is taken from a paper with Oppermann and Thomas. We also include a related result by Mizuno in the case when Q is Dynkin.

Given a root system, the choice of a root basis divides the set of roots into the positive and the negative ones, it also yields an ordering on the set of positive roots. The set of positive roots with respect to this ordering is called a root poset. The root posets have attracted a lot of interest in the last years. The set of antichains (with a suitable ordering) in a root poset turns out to be a lattice, it is called lattice of (generalized) non-crossing partitions. As Ingalls and Thomas have shown, this lattice is isomorphic to the lattice of thick subcategories of a hereditary abelian category of Dynkin type. The isomorphism can be used in order to provide conceptual proofs of several intriguing counting results for non-crossing partitions.

Many interesting examples of singular symplectic algebraic varieties and their symplectic resolutions are built by Hamiltonian reduction. There is a corresponding construction of "quantum Hamiltonian reduction" which is of substantial interest to representation theorists. It starts from a twisted-equivariant D-module, an analogue of an algebraic vector bundle (or coherent sheaf) on a moment map fiber, and produces an object on the quantum analogue of the symplectic resolution. In order to understand how far apart the quantisation of the singular symplectic variety and its symplectic resolution can be, one wants to know "what gets killed by quantum Hamiltonian reduction?" I will give a precise answer to this question in terms of effective combinatorics. The answer has consequences for exactness of direct images, and thus for t-structures, which I will also explain. The beautiful geometry behind the combinatorics is that of a stratification of a GIT-unstable locus called the "Kirwan-Ness stratification." The lecture will not assume familiarity with D-modules, nor with any previous talks by the speaker or McGerty in this series. The new results are joint work with McGerty.

Recently, there has been a great deal of interest in the theory of modules over algebraic quantizations of so-called symplectic
resolutions. In this talk I'll discuss some new work -joint, and very much in progress- that open the door to giving a geometric description to certain categories of such modules; generalizing classical theorems of Kashiwara and Bernstein in the case of D-modules on an algebraic variety.

A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction. I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.

Using the Borel-Schur algebra, we construct explicit characteristic-free resolutions for Weyl modules for the general linear group. These resolutions provide an answer to the problem, posed in the 80's by A. Akin and D. A. Buchsbaum, of constructing finite explicit and universal resolutions of Weyl modules by direct sums of divided powers. Next we apply the Schur functor to these resolutions and prove a conjecture of Boltje and Hartmann on resolutions of co-Specht modules. This is joint work with I. Yudin.

The symmetric group S_{mn} acts naturally on the collection of set partitions of a set of size mn into n sets each of size m, and the resulting permutation character is the Foulkes character. These characters are the subject of the longstanding Foulkes Conjecture. In this talk, we define a deflation map which sends a character of the symmetric group S_{mn} to a character of S_n. The values of the images of the irreducible characters under this map are described combinatorially in a rule which generalises two well-known combinatorial rules in the representation theory of symmetric groups, the Murnaghan-Nakayama formula and Young's rule. We use this in a new algorithm for computing irreducible constituents of Foulkes characters and verify Foulkes’ Conjecture in some new cases. This is joint work with Anton Evseev (Birmingham) and Mark Wildon (Royal Holloway).

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

Central extensions of a finite group G correspond to 2-cocycles on G, which give rise to an abelian cohomology group known as the Schur

multiplier of G. Recently, the Schur multiplier was defined in a much more

general setting of a monoidal category. I will explain how to twist algebras by categorical 2-cocycles and will mention the role of

such twists the theory of quantum groups. I will then describe an approach to twisting rational Cherednik algebras by cocycles,

and will discuss possible applications of this new construction to the representation theory of these algebras.

It has been shown that the Auslander-Reiten-quiver of an indecomposable algebra contains a finite component if and only if A is representation finite. Moreover, selfinjective algebras are representation finite if and only if the tree types of the stable components are given by Dynkin Diagrams. I will present similar results for the Auslander-Reiten-quiver of a functorially finite resolving subcategory Ω. We will see that Brauer-Thrall 1 and Brauer-Thrall 1.5 can be proved for these categories with only little extra effort. Furthermore, a connection between sectional paths in A-mod and irreducible morphisms in Ω will be given. Finally, I will show how all finite Auslander-Reiten-quivers of A-mod or Ω are related to Dynkin Diagrams with a notion similar to the tree type that coincides in a finite stable component.

We introduce a deformation process of universal enveloping algebras of Borcherds-Kac-Moody algebras, which generalises quantum groups' one and yields a large class of new algebras called coloured Borcherds-Kac-Moody algebras. The direction of deformation is specified by the choice of a collection of numbers. For example, the natural numbers lead to classical enveloping algebras, while the quantum numbers lead to quantum groups. We prove, in the finite type case, that every coloured BKM algebra have representations which deform representations of semisimple Lie algebras and whose characters are given by the Weyl formula. We prove, in the finite type case, that representations of two isogenic coloured BKM algebras can be interpolated by representations of a third coloured BKM algebra. In particular, we solve conjectures of Frenkel-Hernandez about the Langland duality between representations of quantum groups. We also establish a Langlands duality between representations of classical BKM algebras, extending results of Littelmann and McGerty, and we interpret this duality in terms of quantum interpolation.

To any complex smooth variety Y with an action of a finite group G, Etingof associates a global Cherednik algebra. The usual rational Cherednik algebra corresponds to the case of Y= C^n and a finite Coxeter group G

We will define certain Verdier quotients of the singularity category of a ring R, called defect categories. The triviality of these defect

categories determine, for example, whether a commutative local ring is Gorenstein, or a complete intersection. The dimension (in the sense of Rouquier) of the defect category thus gives a measure of how close such a ring is to being Gorenstein, respectively, a complete intersection. Examples will be given. This is based on joint work with Petter Bergh and Steffen Oppermann.

In 1977, Cline Parshall, Scott and van der Kallen wrote a seminal paper `Rational and generic cohomology' which exhibited a connection between the cohomology for algebraic groups and the cohomology for finite groups of Lie type, showing that in many cases one can conclude that there is an isomorphism of cohomology through restriction from the algebraic to the finite group.

One unfortunate problem with their result is that there remain infinitely many modules for which their theory---for good reason---tells us nothing. The main result of this talk (recent work with Parshall and Scott) is to show that almost all the time, one can manipulate the simple modules for finite groups of Lie type in such a way as to recover an isomorphism of its cohomology with that of the algebraic group.

A triangulated category admits a strong generator if, roughly speaking,

every object can be built in a globally bounded number of steps starting

from a single object and taking iterated cones. The importance of

strong generators was demonstrated by Bondal and van den Bergh, who

proved that the existence of such objects often gives you a

representability theorem for cohomological functors. The importance was

further emphasised by Rouquier, who introduced the dimension of

triangulated categories, and tied this numerical invariant to the

representation dimension. In this talk I will discuss the generation

time for strong generators (the least number of cones required to build

every object in the category) and a refinement of the dimension which is

due to Orlov: the set of all integers that occur as a generation time.

After introducing the necessary terminology, I will focus on categories

occurring in representation theory and explain how to compute this

invariant for the bounded derived category of the path algebras of type

A and D, as well as the corresponding cluster categories.

By classical results of Thomason, the Grothendieck group of a

triangulated category classifies the triangulated subcategories. More

precisely, there is a bijective correspondence between the set of

triangulated subcategories and the set of subgroups of the Grothendieck

group. In this talk, we extend Thomason's results to "higher"

triangulated categories, namely the recently introduced n-angulated

categories. This is joint work with Marius Thaule.

Mirkovic-Vilonen polytopes are a combinatorial tool for studying
perfect bases for representations of semisimple Lie algebras.  They
were originally introduced using MV cycles in the affine Grassmannian,
but they are also related to the canonical basis.  I will explain how
MV polytopes can also be used to describe components of Lusztig quiver
varieties and how this allows us to generalize the theory of MV
polytopes to the affine case.

The geometric Langlands correspondence relates rank n integrable connections on a complex Riemann surface $X$ to regular holonomic D-modules on  $Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.  If we replace $X$ by a smooth irreducible curve over a finite field of characteristic p then there is a version of the geometric Langlands correspondence involving $l$-adic perverse sheaves for $l\neq p$.  In this lecture we consider the case $l=p$, proposing a $p$-adic version of the geometric Langlands correspondence relating convergent $F$-isocrystals on $X$ to arithmetic $D$-modules on $Bun_n(X)$.

The geometric Langlands correspondence relates rank n integrable connections 
on a complex Riemann surface $X$ to regular holonomic D-modules on 
$Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.  If we replace 
$X$ by a smooth irreducible curve over a finite field of characteristic p 
then there is a version of the geometric Langlands correspondence involving 
$l$-adic perverse sheaves for $l\neq p$.  In this lecture we consider the 
case $l=p$, proposing a $p$-adic version of the geometric Langlands 
correspondence relating convergent $F$-isocrystals on $X$ to arithmetic 
$D$-modules on $Bun_n(X)$.

In April 2010 Eguchi--Ooguri--Tachikawa observed a fascinating connection between the elliptic genus of a K3 surface and the largest Mathieu group. We will report on joint work with Miranda Cheng and Jeff Harvey that identifies this connection as one component of a system of surprising relationships between a family of finite groups, their representation theory, and automorphic forms of various kinds Mock modular forms, and particularly their shadows, play a key role in the analysis, and we find several of Ramanujan's mock theta functions appearing as McKay--Thompson series arising from the umbral groups.

De Concini-Kac-Procesi conjecture gives a good estimate for the dimensions of finite--dimensional non-restricted representations of quantum groups at m-th root of unity. According to De Concini, Kac and Procesi such representations can be split into families parametrized by conjugacy classes in an algebraic group G, and the dimensions of representations corresponding to a conjugacy class O are divisible by m^{dim O/2}. The talk will consist of two parts. In the first part I shall present an approach to the proof of De Concini-Kac-Procesi conjecture based on the use of q-W algebras and Bruhat decomposition in G. It turns out that properties of representations corresponding to a conjugacy class O depend on the properties of intersection of O with certain Bruhat cells. In the second part, which is more technical, I shall discuss q-W algebras and some related results in detail.

This is a sequel to Lecture I (given in the algebra seminar, Tuesday). It will be slightly more specialized. The finite Weil representation is the algebra object that governs the symmetries of the Hilbert space H =C(Z/p): The main objective of this talk is to introduce the geometric Weil representation which is an algebra-geometric (l-adic perverse

Weil sheaf) counterpart of the finite Weil representation. Then, I will explain how the geometric Weil representation is used to prove the main technical results stated in Lecture I. In the course, I will explain the Grothendieck geometrization procedure by which sets are replaced by algebraic varieties and functions by sheaf theoretic objects. This is a joint work with R. Hadani (Austin).

I'll present the work of Gaitsgory arXiv:1108.1741. In it he uses Beilinson-Drinfeld factorization techniques in order to uniformize the moduli stack of G-bundles on a curve. The main difference with the gauge theoretic technique is that the the affine Grassmannian is far from being contractible but the fibers of the map to Bun(G) are contractible.

I will discuss some of new concepts and objects of two-dimensional number theory: 

how the same object can be studied via low dimensional noncommutative theories or higher dimensional commutative ones, 

what is higher Haar measure and harmonic analysis and how they can be used in representation theory of non locally compact groups such as loop groups and Kac-Moody groups, 

how classical notions split into two different notions on surfaces on the example of adelic structures, 

what is the analogue of the double quotient of adeles on surfaces and how one

could approach automorphic functions in geometric dimension two.

The class of finite dimensional algebras over an algebraically closed field K

may be divided into two disjoint subclasses (tame and wild dichotomy).

One class

consists of the tame algebras for which the indecomposable modules

occur, in each dimension d, in a finite number of discrete and a

finite number of one-parameter families. The second class is formed by

the wild algebras whose representation theory comprises the

representation theories of all finite dimensional algebras over K.

Hence, the classification of the finite dimensional modules is

feasible only for the tame algebras. Frequently, applying deformations

and covering techniques, we may reduce the study of modules over tame

algebras to that for the corresponding simply connected tame algebras.

We shall discuss the problem concerning connection between the

tameness of simply connected algebras and the weak nonnegativity of

the associated Tits quadratic forms, raised in 1975 by Sheila Brenner.

This is based on joint work with Dave Jorgensen. Given a Gorenstein algebra,

one can define Tate-Hochschild cohomology groups. These are defined for all

degrees, non-negative as well as negative, and they agree with the usual

Hochschild cohomology groups for all degrees larger than the injective

dimension of the algebra. We prove certain duality theorems relating the

cohomology groups in positive degree to those in negative degree, in the

case where the algebra is Frobenius (for example symmetric). We explicitly

compute all Tate-Hochschild cohomology groups for certain classes of

Frobenius algebras, namely, certain quantum complete intersections.