Algebra and Combinatorics Seminar, Spring 2013

Feb 6, Aaron Lauve (Loyola) 4:00PM: Convolution powers of the identity in graded Hopf algebras

We answer the question, "what are the eigenvalues of the kth convolution power of the identity operator acting on a graded Hopf algebra?"... well, the answer is (+1/-1). So I'll spend most of the time saying things about multiplicity and implications. Additionally, I will connect it to the classical "indicators" of Frobenius and Schur, which help classify representations of finite groups. (These were reformulated in the language of Hopf algebras by Susan Montgomery and coauthors.) This is joint work with Marcelo Aguiar.

Feb 27, Jenny Wilson (U. of Chicago) 4:00pm: FI-modules and stability phenomena of representations of the classical Weyl groups

Last year, Church, Ellenberg, and Farb developed a new framework for studying sequences of representations of the symmetric groups, using a concept they call an FI--module. I will give an overview of this theory, and describe how it generalizes to sequences of representations of the classical Weyl groups in type B/C and D. The theory of FI--modules has provided a wealth of new results by numerous authors working in algebra, geometry, and topology. I will outline some of these results, including applications to coinvariant algebras, and the cohomology of configurations spaces and hyperplane complements.

March 13: Amy Pang (Stanford) 4:00pm: Card Shuffling and other Hopf-power Markov Chains

A Hopf-power Markov chain is a random walk where the transition probabilities are given by the coproduct-then-product operator on a combinatorial Hopf algebra. Key examples include the Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards, a model of rock-breaking, and the restriction-then-induction of representations of the symmetric group. I'll give the general definition of these processes, and explain how Hopf algebra structure theory, such as the Cartier-Milnor-Moore and Poincare-Birkhoff-Witt theorems, give the stationary distributions and other information about their long term behaviour. This is a generalisation of joint work with Persi Diaconis and Arun Ram

March 20: Alex Yong (UIUC) 4:00pm: Varieties in flag manifolds and their patch ideals

This talk addresses the problem of how to analyze and discuss singularities of a variety X that ``naturally'' sits inside a flag manifold. Our three main examples are Schubert varieties, Richardson varieties and Peterson varieties. The overarching theme is to use combinatorics and commutative algebra to study the *patch ideals*, which encode local coordinates and equations of X. Thereby, we obtain formulas and conjectures about X's invariants. We will report on projects with (subsets of) Erik Insko (Florida Gulf Coast U.), Allen Knutson (Cornell), Li Li (Oakland University) and Alexander Woo (U. Idaho).

March 27: Alex Ellis (Columbia) 4:00pm: Odd symmetric functions and odd categorified quantum sl(2)

We introduce odd analogues of the symmetric functions, the nilHecke algebra, and the cohomology of Grassmannians. These algebras are used in constructing an odd categorification of quantum sl(2) and, conjecturally, odd Khovanov homology. By work of Hill and Wang, there is a relation to the categorification of Kac-Moody superalgebras as well. Joint with Mikhail Khovanov and Aaron Lauda.

April 3: Tom Halverson (Macalaster) 4:00pm: Groups, Representation Graphs, and Centralizer Algebras

For a group G and a G-module V over the complex numbers, we construct a representation graph \Gamma with vertices labeled by the irreducible G-modules and edges given by the rule for tensoring by V. We show that many of the graph-theoretic properties of \Gamma come from the representation-theoretic properties of G (or vice versa). We then construct the tensor power centralizer algebra Z of endomorphisms that commute with G on the k-fold tensor product of V. We see that the representation theory of Z is also controlled by the graph \Gamma. We will observe the examples that lead to Z being the symmetric group, the Brauer algebra, the partition algebra, and the Temperley-Lieb algebra. We then study the special case where G is a finite subgroup of SU(2), and via the McKay correspondence the graph is one of the extended affine Dynkin diagrams. A beautiful combinatorial representation theory emerges.

April 17: Dave Pennys (Toronto) 4:00pm: Planar algebras

I will give an overview of Jones' planar algebras, with attention to specific examples. I will not assume any familiarity with the subject beyond some basic linear algebra. I will then explain how planar algebras are useful in various areas of mathematics, including knot theory, graph theory, and tensor categories.

April 24: Tony Giaquinto (Loyola) 4:00pm: On the cohomology of the Weyl algebra, the quantum plane, and the q-Weyl algebra

Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler-Poincare characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.