Representation Theory Seminar

2011-2012

The seminar meets weekly, Tuesdays from 16:00-18:00 in the seminar room of the Mathematics Department on the third floor.

Organizers: First Semester- Professor Stuart Margolis. Second Semester- Professor Malka Schaps

Lectures

November 1, 2011-November 8, 2011

Title: Poset Cohomology, Leray Numbers and the Global Dimension of Left Regular Band Algebras

Speaker: Stuart Margolis, Department of Mathematics, Bar-Ilan University

Abstract: Left regular bands, semigroups satisfying the identities xx = x and xyx = xy, have arisen in a myriad of areas of mathematics over the last 15 years. This includes algebraic combinatorics, hyperplane arrangements, Coxeter groups and probability theory. The representation theory of finite left regular bands has been intensively explored as well.

The main purpose of this talk is to show that the cohomology of left regular band algebras has a natural topological meaning. We show that the n-th Ext module between a pair of simple CM modules, where C is the complex numbers and M a finite left regular band is the n-1-th reduced cohomology group of the order complex of the right order (for hyperplane monoids this is exactly the face order) of M. In particular, this gives us a very easy way to compute the quiver of CM from the Hasse diagram of the right order on M. Connections with Leray numbers are also discussed.

Applications will be given to hyperplane monoids, partially commutative left regular bands and other as well.

No previous knowledge about left regular bands or their algebras is assumed.

The talk is based on the slides here, that I stole (with permission) from my brilliant co-authors, Franco Saliola (the first slide set) and Benjamin Steinberg.

Global Dimension of Left Regular Bands

Poset Cohomology, Leray Numbers and the Global Dimension of Left Regular Bands

November 22, 2011

Title: On the Quiver of Monoids with Basic Algebras

Speaker: Stuart Margolis, Department of Mathematics, Bar-Ilan University

Abstract:

Hyperplane monoids and 0-Hecke monoids have been extensively
studied over the past years. They have the property that their algebras are
basic, that is, all their irreducible representaitons are 1-dimensional,
over any field (and in fact over any commutative semiring!).

The purpose of this talk is to completely characterize the class of finite
monoids with basic algebras over a given field. We use Hochschild-Mitchell
Cohomology to compute the Ext-quiver of these as well as a more general
class of monoid algebras.

This is joint work with Benjamin Steinberg based on the paper:

Quiver of a Monoid with a basic algebra

Upcoming Talks:

November 29 Louis Rowen, "Full Quivers and Specht's Conjecture"
December 6-13 Michael Schein, TBA
January Malka Schaps, TBA

2010-2011

During the first semester of 2010-2011תשע"א the Seminar was organized by Professor Malka Schaps. The talks were centered around an "Introduction to Affine Lie Algebras" by Dr. Crystal Hoyt and "Affine Lie algebras, the Symmetric group and Finite Algebraic Groups" by Professor Schaps. Lecture notes for these talks can be found here: http://u.math.biu.ac.il/~hoyt/Affine.htm. Dr. Michael Schein also gave a series of lectures on the topic "Modular representation theory of GL(n, F_q)".

The organizer this semester is Stuart Margolis. Please contact the organizer if you want to give a talk.

The second semester in תשע"א 2010-2011 will be a Working/Learning Seminar on the topic, "Around Coxeter".

We will begin with an introduction to the basics of Coxeter groups and their related structures: Bruhat and related partial orders, the Coxeter Complex and more generally hyperplane arrangements.

Then we will look into related monoids and algebras: The 0-Hecke monoid and algebra and Bruhat order, the Coxeter monoid structure on the Coxeter Complex, Descent Algebras and their quivers. Quivers of monoid algebras. Applications to Probability theory, Automata and Formal Language Theory, hyperplane arrangments.

Here are a list of papers according to topic:

Bruhat Order

Bruhat closure

Introduction to Bruhat order

Bruhat order, geometry and subwords

Bruhat order for monoids

Bruhat order and the Rook monoid

Descent Algebras, Hyperplane Arrangement Semigroups and Algebras

Hyperplane algebras and descent algebras

The face semigroups algebra of a hyperplane arrangment

Quiver of descent algebras

Quiver Presentation for descent algebras