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 order, geometry and subwords

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 Presentation for descent algebras

A semigroup approach to descent algebras for wreath products

Homology of regular semigroups and quivers of descent algebras of wreath products

**0-Hecke Algebras, J-trivial monoids
and generalizations**

0-Hecke algebra of Coxeter groups 2005

Representation of J-trivial monoids with applications to 0-hecke algebras

**Probability Theory and Representations of Semigroups**

Semigroups and Ring Theoretic Methods in Probability

Semigroups, Rings and Markov Chains

Diaconis: Introduction to Markov Chains

Diagonalizing Random Walks in Left Regular Bands with Applications to Markov Chains

Random Walks in Hyperplane Arrangements

**Semigroups, Monoids and Their Algebras**

On the irreducible representations of finite semigroups

The quiver of a left regular band

Quiver of a Monoid with a basic algebra

Mobius functions and representations of semigroups 1

Mobius functions and representations of semigroups 2

Hopf Algebras, Coxeter groups and Left Regular Bands.pdf

Representation, theory, semigroup radicals and Automata and Formal Language Theory

Synchronizing automata and representation theory

Synchronizing groups and representation theory

Transformation monoids and representations

Presentations and representations of the Rook Monoid

Representations of the q-Rook Monoid

Finite Posets and their representation algebras

A class of Semigroups of Finite Representation Type