Feb 5: Stephen Doty (Loyola): Generalized Schur algebras and their q-analogues

I will try to explain how these algebras arise naturally in the theory of algebraic groups, and a recent approach to their study via generators and relations. I will also try to explain connections to quantum groups, especially Lusztig's "modified" form of a quantized enveloping algebra. This is based on joint work with Tony Giaquinto.

Feb 19: Peter Tingley (Loyola): Elementary construction of Lusztig's canonical basis.

In this expository talk I will explain how, for finite type quantum groups, Lusztig's canonical basis can be defined (and shown to exist) in a purely algebraic way. This is roughly Lusztig's original construction, although without the perverse sheaves. Of course without this geometry one cannot see some of the nice properties of canonical bases (such as positivity), but even the properties visible through the elementary approach are remarkable.

Feb 26: Tony Giaquinto (Loyola): Orthogonal bases of irreducible symmetric group representations

I will use Schur-Weyl duality to inductively construct inside tensor space a natural orthogonal basis for each irreducible module of the symmetric group. The same construction works in the quantized case in which similar orthogonal bases of Hecke algebra representations are produced. The corresponding primitive idempotents of these bases are different than the classical Young idempotents and their q-analogues given by Wenzl.

Mar 5: Carolina Benedetti (Michigan State): Posets, Pieri operators and Positivity

In this talk we will see how Pieri-like rules for certain combinatorial Hopf algebras (CHA's) can be encoded using labelled posets. Each interval in these labelled posets give rise to a (quasi)symmetric function and our motivation is to provide a combinatorial proof for the Schur positivity of such function. In particular, we will see how the Pieri rule for dual k-schur functions can be studied using a labelled version of the affine Bruhat order.
This is joint work with N. Bergeron.

Mar 12: Kyle Petersen (DePaul): The module of affine descents of a Weyl group

Solomon's descent algebra is a subalgebra of the group algebra of a finite Coxeter group. It has a basis given by sums of elements whose descent sets are the same. The structure of this algebra can be understood geometrically via the semigroup of faces of the Coxeter complex. This perspective was noted first by Tits, and later exploited by Diaconis and others for its relation to the combinatorics of card shuffling.

In this talk I will describe an analogous story where we study a module over the descent algebra which we call the module of affine descents. The structure of the module can be understood geometrically as an action of the Coxeter complex on something called the Steinberg torus. There are potential applications to estimating the mixing time of certain card shuffling models.

March 19: Aaron Lauve (Loyola): Quasisymmetric functions: an introduction

Quasisymmetric functions were introduced by Stanley (1972) and Gessel (1984) as an enumerative tool in algebraic combinatorics. Their definition turns out to be *just* the right amount of "symmetric" for handling a whole host of combinatorial questions. (I will briefly share a few.) The ring of quasisymmetric functions mimics the more classical ring of symmetric functions in many respects (I will briefly share a few), and has proven to be useful well beyond its original intended purpose.

The main purpose of this expository talk is to highlight two these unintended uses: a representation-theoretic interpretation due to Hivert (2000), and a Hopf theoretic interpretation due to Aguiar-Bergeron-Sottile (2006). This will set the stage for my lecture later in the semester (also expository, based on the work of Kwon, 2009): Gessel's quasisymmetric functions are characters for Lie superalgebras.

March 26 (Mundelein 308): Frank Lübeck (Aachen, Germany): Kazhdan-Lusztig polynomials and conjectures by Guralnick and Wall

A conjecture by Wall (1962) says that the number of maximal subgroups of any finite group G is at most |G|-1. Guralnick (1990's) conjectured that the dimension of 1-cohomology for any finite group with respect to an absolutely irreducible module is globally bounded. I have computed Kazhdan-Lusztig polynomials which occur in a character formula by Lusztig for representations of reductive groups in their defining (prime) characteristic. It turned out that some coefficients of these polynomials have an interesting interpretation with relevance for the above mentioned conjectures. I will try to explain the connection between these topics.

April 2: Aaron Lauve (Loyola): Quasisymmetric functions: intersections with representation theory

Quasisymmetric functions were introduced by Stanley (1972) and Gessel (1984) as an enumerative tool in algebraic combinatorics. The ring of quasisymmetric functions mimics the more classical ring of symmetric functions in many respects (I will briefly share a few), and has proven to be useful well beyond its original intended purpose.

The main purpose of this expository talk is to highlight a few of these unintended uses: a representation-theoretic interpretation due to Krob and Thibon (1999); another due to Hivert (2000), and another due to Kwon (2009).

April 9: Emily Peters (Loyola): The Fuss-Catalan algebras

The Fuss-Catalan algebras are multi-colored generalizations of the Temperley-Lieb algebras. In this talk I will define them, count them, and discuss some of their bases.

April 23: Timothy O'Brian (Loyola): Some applications of algebra and geometry in applied statistics

Working with researchers in genetics and medicine, agriculture and engineering, environmental sciences and toxicology, applied statisticians often use statistical models to approximate the underlying phenomena so as to help answer important research questions. These statistical methods include hypothesis testing and interval estimation, in conjunction with linear, nonlinear and logistic regression, and techniques of experimental and optimal design. Important results in these fields are established using mathematical techniques taken from analysis, abstract and linear algebra, convexity theory, and differential geometry.

This talk underscores and illustrates some of the mathematical underpinnings of applied statistical methods. A Taylor series expansion helps connect popular (Wald, score and likelihood) methods used in estimation and testing. Cochran’s theorem – useful to justify regression and ANOVA methods – establishes the distribution of quadratic forms, and projection operators (matrices) illustrate and underscore the basic ideas behind simple linear regression. Methods from differential geometry such as curvature highlight the differences between linear and nonlinear regression, and enable one to appreciate the differences between the various confidence intervals used in Gaussian nonlinear models. Also, the measure theory basis of optimal experimental design theory and methods are discussed as is the General Equivalence Theorem. Time permitting, illustrations will be provided as will be a discussion of the relevant software algorithms.