Algebra Seminar

A fundamental phenomenon in the representation theory of finite and compact groups is that irreducible characters tend to take smaller values on elements that are far from central. Character estimates of exponential type (that is, bounds of the form |chi(g)|<chi(1)^(1-epsilon)) are particularly useful for probabilistic applications, such as bounding the mixing time of random walks supported on conjugacy classes.

In 1981, Diaconis and Shahshahani established sharp estimates for irreducible characters of the symmetric group S_n, evaluated at a transposition t = (i j). As an application, they proved that roughly n*log(n) random transpositions are required to mix a deck of n playing cards. This was extended in 2007 by Muller--Schlage-Puchta to to arbitrary permutations in S_n. Exponential character bounds for finite simple groups were subsequently developed through a series of works by Bezrukavnikov, Liebeck, Shalev, Larsen, Guralnick, Tiep, and others. 

In this talk, Itay Glazer (Technion) will present recent progress on exponential character estimates for compact Lie groups.

This is based on joint work in progress with Nir Avni, Peter Keevash, and Noam Lifshitz.

to follow

Koszul algebras are positively graded algebras with very amenable homological properties. Typical examples include the polynomial ring over a field or the exterior and symmetric algebras of a vector space. A category is called Koszul if it has a grading with which it is equivalent to the category of graded modules over a Koszul algebra. A famous example (due to Soergel) is the principal block of category $\mathcal{O}$ for a semisimple Lie algebra. Koszulity is a very nice property, but often very difficult to check. In this talk, Thorsten Heidersdorf (Newcastle University) will give a criterion that allows to check Koszulity in case the category is a graded semi-infinite highest weight category (which is a structure that appears often in representation theory). This is joint work with Jonas Nehme and Catharina Stroppel.

to follow