MIT Infinite Dimensional Algebra Seminar (Spring 2014)

I will give an introduction to representation stability, a program describing stability in families of e.g. S_n-representations as n goes to infinity, via a detailed look at two applications in topology. First, I'll discuss configuration spaces of points on manifolds; representation stability for their cohomology; and differences in the stable behavior between open manifolds, closed manifolds, and smooth projective varieties. Second, I will describe representation stability for the mod-p homology of congruence subgroups, and its use in Calegari-Emerton's recent classification of stable mod-p Hecke eigenforms.

Representation stability for the mod-p homology of congruence subgroups, due to Putman, Church-Ellenberg-Farb-Nagpal, and Church-Ellenberg, was recently used in Calegari-Emerton's classification of stable mod-p Hecke eigenforms. The proof of representation stability rests on Church-Ellenberg's homological regularity theorem for FI-modules over Z, which guarantees that FI-modules have controlled resolution by "combinatorial" FI-modules (those coming from objects of Deligne's category Rep(S_t)). No background or details from last week's talk will be assumed.

We study the representation theory of quotients of shifted Yangians. These algebras of interest because they quantize slices inside affine Grassmannians. We will give some conjectures and some results concerning finite-dimensional and Verma modules for these algebras.

For any algebra A, the Hochschild cohomology is defined as the graded Ext group HH*(A)=Ext_A-bimod*(A,A). This cohomology gives one a glimpse into the deformation theory of a given algebra. As with any group of self-extensions, there is a canonical product on the Hochschild cohomology. In this talk I will describe how, given a Koszul algebra A, one can place a canonical degree 1, square 0, derivation d_Kos on the tensor product of A with its Koszul dual so that the cohomology of the resulting dg-algebra is equal to the Hochschild cohomology HH*(A) as an algebra. Some examples will also be discussed.

Following Sklyanin's proposal in the periodic case, we derive the generating function for the Gaudin model with boundary. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. We study the relevant algebraic structure for the algebraic Bethe ansatz. In the case when the boundary matrix is upper-triangular, we implement the algebraic Bethe ansatz, obtaining the eigenvalues of the generating function and the corresponding Bethe states.

I will discuss an example of a reducible characteristic variety of a simple highest weight module for sl_12. In other types it has been known since the early 1980s (thanks to work of Kashiwara and Tanisaki) that reducible characteristic varieties can occur. Interestingly the singularity under Beilinson-Bernstein localization is the same as that used by Kashiwara and Saito to demonstrate a reducible characteristic cycle for IC sheaves on the flag variety of SL_8. I will briefly discuss how the examples were found (using parity sheaves, positivity and decomposition numbers for perverse sheaves).