It has been said that the ÔproperÕ realisation of a group always involves its action by symmetries on some space. This principle manifests everywhere from geometry, to classical analysis, to number theory. Unfortunately, most symmetries are too hard to study directly, so we must usually linearise them. The study of linearised group actions is called representation theory.
In the late 1960's and early 1970's, Langlands proposed a far-reaching collection of conjectures unifying algebraic, geometric, and analytic perspectives on representation theory. Much deep work has been done in the pursuit of these conjectures, but, until recently, the fundamental objects of harmonic analysis, group characters, have not been put to their full use. In this talk, we will discuss the application of character formulas to questions about stable distributions on reductive, p-adic groups.
The study of elementary abelian groups is a central topic in the representation theory of finite groups. For example, if G is a finite group and k is a field of characteristic p, then a theorem of Chouinard tells us that a finitely-generated kG-module is projective if and only if it is projective when restricted to any elementary abelian subgroup of G. Hence detecting the projectivity of a module in this case reduces to understanding the behavior of modules for elementary abelian groups. A major problem in studying the modules for an elementary abelian group is that its group algebra almost always has wild representation type, that is, its modules are impossible to classify. Hence we are forced to restrict our attention to certain classes of these modules. In this talk I will introduce an interesting class of modules called modules of constant Jordan type, giving some motivation for the subject and a brief introduction to the algebraic geometry used in the theory. I will then discuss some conjectures about which Jordan types are realizable as the Jordan type of a finitely-generated module for an elementary abelian group.
Modules of Constant Jordan Type I
Apr. 22, 2010 2pm (MATH 350)
Lie Theory
Shawn Baland (Aberdeen)
X
The study of elementary abelian groups is a central topic in the representation theory of finite groups. For example, if G is a finite group and k is a field of characteristic p, then a theorem of Chouinard tells us that a finitely-generated kG-module is projective if and only if it is projective when restricted to any elementary abelian subgroup of G. Hence detecting the projectivity of a module in this case reduces to understanding the behavior of modules for elementary abelian groups. A major problem in studying the modules for an elementary abelian group is that its group algebra almost always has wild representation type, that is, its modules are impossible to classify. Hence we are forced to restrict our attention to certain classes of these modules. In this talk I will introduce an interesting class of modules called modules of constant Jordan type, giving some motivation for the subject and a brief introduction to the algebraic geometry used in the theory. I will then discuss some conjectures about which Jordan types are realizable as the Jordan type of a finitely-generated module for an elementary abelian group.
Modules of Constant Jordan Type II
Aug. 31, 2010 2pm (MATH 350)
Lie Theory
Nat Thiem (CU)
X
At a recent AIM workshop, the participants found a remarkable connection between two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are Hopf isomorphic. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras. This talk will be in two parts: (1) I will give an introduction to combinatorial Hopf algebras with our favorite examples (2) I will describe supercharacters and how they connect up to one of our favorite Hopf algebras.
