Past Representation Theory Seminar

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.