14:30
Varieties determined by their jets and invariant theory
Abstract
joint work with R Gurjar
Forthcoming events in this series
joint work with R Gurjar
For a positively graded algebra A we construct a functor from the derived
category of graded A-modules to the derived category of graded modules over
the quadratic dual A^! of A. This functor is an equivalence of certain
bounded subcategories if and only if the algebra A is Koszul. In the latter
case the functor gives the classical Koszul duality. The approach I will
talk about uses the category of linear complexes of projective A-modules.
Its advantage is that the Koszul duality functor is given in a nice and
explicit way for computational applications. The applications I am going to
discuss are Koszul dualities between certain functors on the regular block
of the category O, which lead to connections between different
categorifications of certain knot invariants. (Joint work with S.Ovsienko
and C.Stroppel.)
Lusztig discover an integral lift of the Frobenius morphism for algebraic groups in positive characteristic to quantum groups at a root of unity. We will describe how this map may be constructed via the Hall algebra realization of a quantum group.
I will describe how to build a noncommutative ring which dictates
the process of resolving certain two-dimensional quotient singularities.
Algebraically this corresponds to generalizing the preprojective algebra of
an extended Dynkin quiver to a larger class of geometrically useful
noncommutative rings. I will explain the representation theoretic properties
of these algebras, with motivation from the geometry.
For the integers $a$ and $b$ let $P(a^b)$ be all partitions of the
set $N= {1,..., ab}$ into parts of size $a.$ Further, let
$\mathbb{C}P (a^b)$ be the corresponding permutation module for the
symmetric group acting on $N.$ A conjecture of Foulkes says
that $\mathbb{C}P (a^b)$ is isomorphic to a submodule of $\mathbb{C}P
(b^a)$ for all $a$ not larger than $b.$ The conjecture goes back to
the 1950's but has remained open. Nevertheless, for some values of
$b$ there has been progress. I will discuss some proofs and further
conjectures. There is a close correspondence between the
representations of the symmetric groups and those of the general
linear groups, via Schur-Weyl duality. Foulkes' conjecture therefore
has implications for $GL$-representations. There are interesting
connections to classical invariant theory which I hope to mention.
The representation theory of symmetric groups starts with
the permutation modules. It turns out that the annihilator of a
permutation module can be described explicitly in terms of the
combinatorics of Murphy's cellular basis of the group algebra of the
symmetric group in question. In fact, we will show that the
annihilator is always a cell ideal. This is recent joint work with K.
Nyman.
Category theory is used to study structures in various branches of
mathematics, and higher-dimensional category theory is being developed to
study higher-dimensional versions of those structures. Examples include
higher homotopy theory, higher stacks and gerbes, extended TQFTs,
concurrency, type theory, and higher-dimensional representation theory. In
this talk we will present two general methods for "categorifying" things,
that is, for adding extra dimensions: enrichment and internalisation. We
will show how these have been applied to the definition and study of
2-vector spaces, with 2-representation theory in mind. This talk will be
introductory; in particular it should not be necessary to be familiar with
any category theory other than the basic idea of categories and functors.
I will discuss how one can construct nice cellular
algebras using the cohomology of Springer fibres associated with two
block nilpotent matrices (and the convolution product). Their
quasi-hereditary covers can be described via categories of highest
weight modules for the Lie algebra sl(n). The combinatorics of torus
fixed points in the Springer fibre describes decomposition
multiplicities for the corresponding highest weight categories. As a
result one gets a natural subcategory of coherent sheaves on a
resolution of the slice to the corresponding nilpotent orbit.
There are many triangulated categories that arise in the study
of group cohomology: the derived, stable or homotopy categories, for
example. In this talk I shall describe the relative cohomological
versions and the relationship with ordinary cohomology. I will explain
what we know (and what we would like to know) about these categories, and
how the representation type of certain subgroups makes a fundamental
difference.