Past Representation Theory Seminar

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.