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