Past Representation Theory Seminar

We describe a method to find all toric noncommutative crepant resolutions

of a 3-dimensional toric singularity. We discuss how this method generalizes

to higher dimensions and how we can construct analogons of dimer models.

The A-identities were first studied (although implicitly) around 1955 by  Kostant. Their more systematic study was started some 10 years ago by Regev. Later on Henke and Regev studied these identities in the Grassmann algebra.
An A-monomial of degree n is an even permutation of the noncommutative variables x_1 to x_n; an A-polynomial of degree n is a linear combination of such monomials in the free associative algebra.
Henke and Regev proposed two conjectures concerning the A-identities satisfied by the Grassmann algebra, and the minimal degree of an A-identity for the matrix algebras. I shall discuss these two conjectures. The first turns out to be true while the second fails.

A profinite group is the inverse limit of an inverse system of finite groups. While such groups are set-wise `big', the inverse system gives

profinite groups a close relationship with finite groups - a conduit through which important results can flow.

Our goal is to construct a modular representation theory for profinite groups. We show how several foundational results (about relative

projectivity, vertices, sources) from the established theory for finite groups can pass through an inverse system, to the limit.

We introduce the idea of transfer of gradings via derived equivalences and we apply it to construct positive gradings on a basic

Brauer tree algebra corresponding to an arbitrary Brauer tree T. We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra, whose tree is a star. To transfer gradings via derived equivalence we use tilting

complexes constructed by taking Green's walk around T. We also prove that there is a unique grading on an arbitrary Brauer tree algebra, up to graded Morita equivalence and rescaling.

We define Brauer characters for Brauer algebras which share

many of the features of Brauer characters defined for finite groups.

Since notions such as conjugacy classes and orders of elements are not a

priori meaningful for Brauer algebras, we show which structure replaces

the conjugacy classes and determine eigenvalues associated to these.

In 1979 Cohen, Moore, and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the

mod~$p$ Moore space $\Omega P^m(p^r)$ for primes $p>2$ and used the results to find the best possible exponent for the homotopy groups of spheres and for Moore spaces at such primes. The corresponding problems for $p=2$ are still open. In this talk we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod $2$ Moore space. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod $2$ Moore space or to an improvement in the known bounds for the exponent of the $2$-torsion in the homotopy groups of spheres.

We consider the decomposition problem for Lie powers of finite-dimensional modules for a cyclic p-group C over a field K of prime characteristic p. That is, given a finite-dimensional KC-module V and a positive integer n we would like to be able to decompose the n-th Lie power $L^n(V)$ as a direct sum of indecomposable KC-modules, describing which isomorphism types of indecomposable KC-modules occur in such a decomposition and with what multiplicity. By a theorem of R. M. Bryant and M. Schocker the problem reduces to the case $n= p^m$, for $m \geq 1$. In this talk I will discuss some conjectured recursive descriptions of such Lie powers up to isomorphism.

The Lie representation, a representation of dimension (n-1)! for the symmetric group on n letters, occurs within many contexts.

The purpose of this expository lecture is to describe some connections, concrete computations, as well as open problems concerning this

representation. Their common connection is via Dehn twists of Riemann surfaces together with their homological implications. Some topics

will include

(1) the cohomology ring of pure braid groups,

(2) the structure of homotopy string links and their invariants as developed by Milnor and Habegger-Lin,

(3) the infinitesimal braid relations as occurring in Vassiliev invariants of pure braids,

(4) complexity of algorithms for factoring complex polynomials, and

(5) certain groups of natural transformations.

Hochschild cohomology governs deformations of algebras, and its graded Lie

structure plays a critical role. We explore this structure for a finite

group G acting on an algebra S by automorphisms. We capture the group

together with its action with the natural semi-direct product, S#G,

known as the "skew group algebra" or "smash product algebra". For example,

when G acts linearly on a complex vector space V, it induces an action on

the symmetric algebra S(V), a polynomial ring. The semi-direct product

S(V)#G is a surrogate for the ring of invariant polynomials on V; it

serves as the coordinate ring for the orbifold arising from the action of

G on V. Deformations of this skew group algebra S(V)#G play a prominent

role in representation theory. Such deformations include graded Hecke

algebras (originally defined independently by Drinfeld and by Lusztig),

symplectic reflection algebras (investigated by Etingof and Ginzburg in

the study of orbifolds), and rational Cherednik algebras (introduced to

solve Macdonald's inner product conjectures). We explore the graded Lie

structure (or Gerstenhaber bracket) of the Hochschild cohomology of skew

group algebras with an eye toward deformation theory. For abelian groups

acting linearly, this structure can be described in terms of inner

products of group characters. (Joint work with Sarah Witherspoon.)

The transformation algebra of an algebra A is the subalgebra of the

algebra of linear endomorphisms of A generated by all left and right

multiplications with elements in A. It was introduced by Albert as a part

of an effort to create a unified structure theory for non-associative

algebras.

One problem with the transformation algebra is that it is a very crude

invariant for general algebras. In the talk, I shall suggest a way to

compensate for this and show that by adding certain information, the

transformation algebra can be used to give a complete picture of the

category of unital division algebras of fixed (finite) dimension over a

field.