Ottawa-Carleton joint algebra seminar (Fall 2013)

Geometric Realizations of the Basic Representation of Affine sl(n)

Geometric Realizations of the Basic Representation of Affine sl(n)

Abstract: In their 1990 paper, ten Kroode and van de Leur gave explicit realizations of the basic representation of affine gl(n) and sl(n). In particular, there exists one realization for each partition of n (with the two extreme cases being the well-known principal and homogeneous realizations). In this talk, we will give a geometric interpretation of these various realizations. For the principal realization, our construction is given in terms of the equivariant cohomology of the Hilbert scheme of points in the plane and Nakajima quiver varieties. We then obtain similar constructions for the other realizations via a certain generalization of the Hilbert scheme.
 

Classification of simple modules with finite-dimensional weight spaces for the Lie algebra of vector fields on a torus

Classification of simple modules with finite-dimensional weight spaces for the Lie algebra of vector fields on a torus

Abstract: We establish a classification of simple modules with finite-dimensional weight spaces for the Lie algebra of vector fields on a torus. This work generalizes the classical result of O.Mathieu for the Virasoro Lie algebra. There are two classes of such simple modules: (1) cuspidal modules and (2) modules of the highest weight type. Cuspidal modules are those which have a uniform bound on the dimension of their weight spaces. In order to classify cuspidal modules we construct a functor into the category of modules that admit a compatible action of the commutative algebra of functions on a torus. This is a joint work with V.Futorny.
 

The mirabolic Hecke algebra

The mirabolic Hecke algebra

Abstract:The Iwahori-Hecke algebra of the symmetric group is the convolution algebra arising from the variety of pairs of complete flags over a finite field. Considering convolution on the space of triples of two flags and a vector we obtain the mirabolic Hecke algebra, which had originally been described by Solomon. We will see a new presentation of this algebra which shows that it is a quotient of a cyclotomic Hecke algebra. This lets us recover Siegel's results about its representations, as well as proving new 'mirabolic' analogues of classical results about the Iwahori-Hecke algebra.
 

Invariants of degree 3 and torsion in the Chow ring of a versal flag variety

Invariants of degree 3 and torsion in the Chow ring of a versal flag variety

Abstract: In this talk we will discuss the connection between degree 3 cohomological invariants of a split semisimple group and the torsion in the Chow group of codimension two cycles of the corresponding versal flag variety. The talk is based on joint work with Alexander Merkurjev and Kirill Zainoulline.
 

On fully residually-R groups

On fully residually-R groups

Abstract: The notion of a (fully) residually-C group, where C is a class of groups, was introduced long time ago. Usually, C is chosen to be a class of groups with nice properties, such as finite groups, nilpotent groups, free groups, etc. For some classes C, a group G is residually-C if and only if G is fully residually-C. However, the situation is very different for the class of free groups, as was shown by Benjamin Baumslag: a group is fully residually free if and only if it is residually free and commutative transitive. In this talk, we consider the class R of finitely generated toral relatively hyperbolic groups. I will explain that groups from R are commutative transitive and generalize Baumslag's theorem to this class. This is joint work with Inna Bumagin.
 

Periods of generic torsors of groups of multiplicative type

Periods of generic torsors of groups of multiplicative type

Abstract: For a smooth commutative linear algebraic group G, its first Galois cohomology has an abelian group structure. The period of a G-torsor is then defined to be the order of the corresponding element in the first Galois cohomology group. In this talk I will show that the period of a generic G-torsor can be computed using coflasque resolutions of G.