Sept 10: Aaron Lauve (Loyola), Hopf structure of ring of k-Schur functions

The k-Schur functions have many conjecturally equivalent definitions, as well as t-variants, non commutative and quasisymmetric variants, and even torus-equivariant variants, and arise in a variety of settings, including (co)homology of the affine Grassmannian, Macdonald/Schur positivity, and more. We highlight some of these. Additionally, the ring S_k of k-Schur functions is realized as a Hopf subalgebra of the Hopf algebra S of symmetric functions.

Some have found it easier to study the graded dual of k-Schur functions, S^k, which is a quotient Hopf algebra of S (and which happens to be a generalization of Stanley symmetric functions). These two modes of study are equivalent—products being exchanges with coproducts and so on. In this talk, we show that they are in fact the same: S_k and S^k are isomorphic as Hopf algebras. We give several variants of this result, then frame it in the context of important open problems in the area. (Joint with Franco Saliola.)

Sept 17: Stephen Doty (Loyola), Schur-Weyl duality over finite fields

Classical Schur-Weyl duality is a double centralizer property for the natural commuting actions of the general linear and symmetric groups acting on a tensor power. Although originally proved over the field of complex numbers in 1927 by Issai Schur, it was known at least by 1980 that the same result holds over any infinite field. This talk will explore what happends over finite fields. It turns out that the classical result is still true, provided only that the field is big enough. A precise lower bound is available. This result is joint work with Dave Benson [Archiv der Mathematik (Basel) 93 (2009), 425-435]. I will try to give some details about the proof and make some related remarks.

Sept 24: Stuart Martin (Cambridge University), New solutions to old problems in symmetric group cohomology

Oct 1: Mitja Mastnak (St. Mary's University), Semitransitive collections of matrices

We say that a collection C of complex n-by-n matrices is semitransitive, or, more precisely, acts semitransitively on the underlying n-dimensional vector space V, if for every pair of nonzero vectors x, y in V there is an element A of C such that either Ax=y or Ay=x. The notion coincides with the notion of transitivity for groups of matrices, but not in general. Topological version of the notion can is defined in the obvious way.

Semitransitivity was introduced in 2005 by H. Rosenthal and V. Troitsky who first studied it in the context of WOT-closed algebras of Hilbert space operators. It was later studied in finite and infinite dimensional settings by many authors. A good deal of results were obtained, sometimes in line with initial conjectures but quite often not.

This will be a survey talk of some interesting results and tools used in the area. Most of the talk is accessible to students with a good working knowledge of linear algebra. I will also discuss some recent work, joint with J. Bernik, in which we relate the notion of semitransitivity to the study of prehomogeneous vector spaces.

Oct 22: Emily Peters (Loyola), Categories generated by a trivalent vertex

A fusion category is a tensor category with a finite number of simple objects and some conditions which make linear algebra work nicely (it is semisimple, k-linear and rigid). Examples of fusion categories come from quantum groups at roots of unity, and finite-depth subfactors (and that is the only mention of subfactors I will make in this talk). We are interested in describing all small trivalent categories: A trivalent category is generated by a single object and all morphisms can be draw using trivalent graphs. Why are we interested in trivalent categories? Well, first, a lot of the quantum-group and subfactor fusion categories are trivalent. Second, we have tools of linear algebra and diagrammatic combinatorics that make these calculations fun and easy. In particular, in this talk I plan to prove theorems using the method of discharging (of four-color-theorem fame).

Nov 5: Chris Drupieski (DePaul), Examples and results on strict polynomial superfunctors

Strict polynomial superfunctors are generalizations to the Z/2Z-graded world of (ordinary) strict polynomial functors. They include generalizations of the symmetric power functors, the exterior power and anti-symmetric power functors, and the divided power (or symmetric tensor) functors. In this talk I will describe some basic examples of strict polynomial superfunctors, discuss their connections to Schur superalgebras and algebraic (super)groups, and discuss (with illustrations) some of my recent results making cohomology calculations in the category of strict polynomial superfunctors.

Nov 12: Gus Schrader (Berkeley), Integrable systems from the classical reflection equation.

The phase spaces of many well-known classical integrable systems, such as the relativistic Toda system and the classical periodic XXZ spin chain, can be realized as symplectic leaves of a quasitriangular Poisson-Lie group G. The Hamiltonians of such models are given by the restriction of conjugation-invariant functions on G. In a recent work, we construct integrable systems on Poisson homogeneous spaces of the form G/K, where the subgroup K arises as the fixed points of a group automorphism satisfying the classical reflection equation. We show that the subalgebra of K-bi-invariant functions on G is Poisson commutative, and that the Hamiltonian dynamics generated by these Hamiltonians are described by Lax equations. We will also explain how to realize the classical XXZ chain with reflecting boundaries as an example of our construction.

Nov 14: Frederico Galetto (Queen's, Canada), Equivariant resolutions of De Concini-Procesi ideals

Minimal free resolutions in commutative algebra are an important tool to extract information about a module. For modules over polynomial rings with a reasonable group action, a minimal free resolution of the module inherits an action by the same group. Understanding how the group acts on a resolution leads to a refinement of classical invariants of the module. I will provide a brief overview of minimal free resolutions over polynomial rings and their equivariant counterparts. Then I will present examples of resolutions, with the action of a symmetric group, arising from certain ideals introduced by De Concini and Procesi with particular significance in representation theory, geometry and combinatorics.

Nov 19: Dennis Stanton (Minnesota), Another (q,t)-world

A well studied (q,t)-analogue of symmetric functions are the Macdonald polynomials. In this talk I will survey another (q,t)-analogue, where q is a prime power and t is an indeterminate. Analogues of facts about the symmetric group S_n are given for GL_n(F_q), including:
  (1) counting factorizations of certain elements into reflections,
  (2) combinatorial properties of appropriate binomial coefficients,
  (3) Hilbert series for invariants on polynomial rings.

Some new conjectured explicit Hilbert series of invariants are given. This is joint work with Joel Lewis and Vic Reiner.

Dec 3: Alex Weekes (Toronto), Yangians and Nakajima monomial crystals

To each simple Lie algebra is associated a Yangian: an infinite-dimensional algebra that has many familiar traits in common with the universal enveloping algebra, including a PBW theorem, Verma modules and a highest weight theory. The Yangian also quantizes certain Poisson subvarieties of the affine Grassmannian, and we may study the representations of the Yangian in this light. There turns out to be a link here with Nakajima's monomial crystals, which were originally motivated by quiver varieties. We will discuss some results and conjectures surrounding this link.