Forthcoming events in this series
Toric noncommutative crepant resolutions
Abstract
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.
A-polynomial identities in Grassmann and in matrix algebras
Abstract
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.
Modular representation theory of profinite groups
Abstract
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.
Sign sequences and decomposition numbers of symmetric groups
Graded Blocks of Group Algebras
Abstract
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.
Brauer characters for Brauer algebras
Abstract
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.
The "bottom piece" in the decomposition of $\Omega p^m(2)$
Abstract
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.
Lie powers of modules for cyclic p-groups
Abstract
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.
On the Lie representation and its' applications
Abstract
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.
Quantizing Grassmannians, Schubert cells and cluster algebras
Abstract
The quantum Grassmannians and their quantum Schubert cells are
well-known and important examples in the study of quantum groups and
quantum geometry. It has been known for some time that their
classical counterparts admit cluster algebra structures, which are
closely related to positivity properties. Recently we have shown
that in the finite-type cases quantum Grassmannians admit quantum
cluster algebra structures, as introduced by Berenstein and
Zelevinsky. We will describe these structures explicitly and also
show that they naturally induce quantum cluster algebra structures on
the quantum Schubert cells.
This is joint work with S. Launois.
Real division algebras, restricted quiver representations and Euclidean configurations
Hochschild cohomology for finite groups acting linearly and graded Hecke algebras
Abstract
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 abelian defect group conjecture for sporadic groups
Abstract
an abelian defect group D, let N be the normaliser in G of D,
and let B be the Brauer correspondent of A. Then the abelian
defect group conjecture says that the bounded derived categories
of the module categories of A and B equivalent as triangulated
categories. Although this conjecture is in the focus of intensive
studies since almost two decades now, it has only been verified
for certain cases and a general proof seems to be out of sight.
In this talk, we briefly introduce the notions to state the
abelian defect group conjecture, report on the current state
of knowledge, and on the strategies to prove it for explicit
examples. Then we show how these strategies are pursued and
combined with techniques from computational representation theory
to prove the abelian defect group conjecture for the sporadic simple
Harada-Norton group; this is joint work with Shigeo Koshitani.
The transformation algebra of a division algebra
Abstract
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.