—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Fall 2014

Wednesdays, 3:40-4:30, BVM Hall 11th floor seminar room

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Organizers:  Stephen Doty, Tony Giaquinto, Aaron Lauve, Peter Tingley

Aaron Lauve (Chicago)
Hopf structure of ring of k-Schur functions
Stephen Doty (Chicago)
Schur-Weyl duality over finite fields
Stuart Martin (Cambridge, UK)
New solutions to old problems in symmetric group cohomology
Mitja Mastnak
Semitransitive collections of matrices
No talk (fall break)

ALGECOM 11(at dePaul, all day)
Emily Peters (Loyola)
***Tuesday!*** Georgia Benkart (Colloquium)
5 Chris Drupieski (DePaul)
Gus Schrader

No talk (Thanksgiving)
Alex Weekes (Toronto)


IES (the Institute for Environmental Sustainability) and BVM Hall are located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map).

Parking is available on-campus for $7 in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue.


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.