# Research Seminars: Algebra and Combinatorics

## Fall 2013

Time & Location: All talks are on Thursdays in Gibson 400D at 11:00 AM unless otherwise noted.
Organizer: Tai Huy Ha

August 29

## Topic

SpeakerInstitution

Abstract: TBA

September 5

## Topic

SpeakerInstitution

Abstract: TBA

September 12

Abstract:

September 19

## Branching through g2, Part 1

Mahir Cantulane university

Abstract:

Understanding the branching of a representation on a subgroup, or on a subalgebra is an important quest for several different areas of mathematics including algebraic geometry, invariant theory, number theory, geometric topology, as well as algebraic combinatorics.  In this (first of several) talk(s), after providing some background on our branching problem for the simple Lie algebra g2, we present some of our computations and clarify some mistakes in the literature. This is a joint with Roger Howe.

September 26

## Hyperdeterminants of Polynomials

Luke OedingAuburn university

Abstract:

Hyperdeterminants were brought into a modern light by Gelfand, Kapranov, and Zelevinsky in the 1990's. Inspired by their work, I will answer the question of what happens when you apply a hyperdeterminant to a polynomial (interpreted as a symmetric tensor).

The hyperdeterminant of a polynomial factors into several irreducible factors with multiplicities. I identify these factors along with their degrees and their multiplicities, which both have a nice combinatorial interpretation. The analogous decomposition for the mu-discriminant of polynomial is also found. The methods I use to solve this algebraic problem come from geometry of dual varieties, Segre-Veronese varieties, and Chow varieties; as well as representation theory of products of general linear groups.

October 3

## Unipotent Invariant Matrices

Michael Joycetulane university

Abstract:

The special linear group SL_n acts on n by n matrices by a twisted conjugation -- simultaneous left multiplication by g and right multiplication by g transpose.  We study this representation and its restriction to embedded copies of SL_2.  This corresponds to determining which matrices are fixed by a given unipotent matrix.  We explicitly illustrate the dependency on the Jordan type of the unipotent matrix.  Then we generalize this action to the wonderful compactification of non-degenerate symmetric (resp. skew-symmetric) matrices, and study the fixed points of a unipotent matrix having just a single Jordan block.

October 10

October 17

## "Recent" Developments in the Computation of Gröbner Bases

John PerryUniversity of southern mississippi

Abstract:

The "Lazard model" of Gröbner basis computation is to view an algorithm as the reduction to row-echelon form of a very large matrix. Faugère used this approach to develop two algorithms that revolutionized the computation of Gröbner basis algorithms, drawing attention especially for cracking cryptographic systems. The theory of one of these algorithms took a few years to clarify, though it now seems to have settled. We conclude by resurrecting an approach first described by Gritzmann and Sturmfels that seemed stillborn, suggesting how it might build on these past advances.

October 17

## On the Ubiquity of Modular Forms and Ap\'ery-like Numbers

Armin Straubuniversity of illinois at urbana-champaign

Abstract:

In the first part of this talk, we give examples from the theories of short random walks, binomial congruences, positivity of rational functions and series for $1/\pi$, in which modular forms and Ap\'ery-like numbers appear naturally (though not necessarily obviously). Each example is taken from personal research of the speaker.

The second part, which is based on joint work with Bruce C. Berndt, is motivated by the secant Dirichlet series $\psi_s(\tau) = \sum_{n = 1}^{\infty} \frac{\sec(\pi n \tau)}{n^s}$, recently introduced and studied by Lal\'{\i}n, Rodrigue and Rogers as a variation of results of Ramanujan. We review some of its properties, prove a conjecture on special values of this Dirichlet series, and put these into the context of Eichler integrals of general Eisenstein series.

October 24

## Positive Moments of Ranks and Cranks for Overpartitions

Holly SwisherOregon State University

Abstract:

In recent work, Andrews, Chan and Kim extend a result of Garvan about even rank and crank moments of partitions to positive moments. In a similar fashion we extend a result of Mao about even rank moments of overpartitions. We investigate positive Dyson-rank, $M_2$-rank, first residual crank, and second residual crank moments of overpartitions. In particular, we prove a conjecture of Mao which states that the positive Dyson-rank moments are larger than the positive $M_2$-rank moments.  We also prove some additional inequalities involving rank and crank moments of overpartitions, including an interlacing property.

October 31

## Symbolic Powers of Monomial Ideals

Tai Huy Hatulane university

Abstract:

In this talk we shall discuss the relationship between symbolic and regular powers of monomial ideals. We shall address the question of when one power contains the other, and the question of when the two powers are the same.

November 6

## Construction of Automorphic Representations

Shuichiro Takedauniversity of missouri

Abstract:

The theory of automorphic representations has been one of the central themes of contemporary mathematics. One of the major questions is a very simple question on how to construct automorphic representations in meaningful ways. One of the known technologies for contracting automorphic representations is known as the method of theta lifting, which allows one to construct cuspidal automorphic representations of one reductive group out of those on another. In this talk, we will give an overview this theory and a recent development by Gan, Qiu and the speaker.

November 14

## Topic

Speakerinstitution

Abstract: TBA

November 20

## The Two Sides of the Extended Shuffle Conjecture

Abstract:

Eugene Gorsky and Andrei Negut [1] have recently put the finishing touches to what may be viewed as the symmetric function side of the general m; n Shuffle conjecture. Earlier, Tatsuyuki Hikita  [2] gave a beautiful construction of the combinatorial side as a weighted enumeration of m,n-Parking Functions. All these developments are gravid with challenging Combinatorial problems. In this talk I will report on my findings in an effort to translate some of the contents of these remarkable publications in a language that is more accessible to the general Algebraic Combinatorial audience. In particular I have made an effort to state the resulting m,n- Shuffle Conjecture using notation that is as close as possible to the statement of the original Shuffle Conjecture. This translation would not have been possible without  the invaluable and continuous help I got from Gorsky and Negut.  As a bi-products we can now announce some new results in the area with a variety of ramifications that range from Representation Theory to Symmetric Function Theory and ultimately to Combinatorics. This is joint work with Francois Bergeron and Emily Leven.

[1] E. Gorsky and A. Negut, Refined knot invariants and Hilbert schemes, arXiv preprint arXiv:1304.3328 (2013).

[2] T. Hikita,  Affine Springer fibers of type A and combinatorics of diagonal coinvariants, arXiv preprint arXiv:1203.5878 (2012)

November 21

## Equivariant K-theory of Spherical Varieties

Soumya BanerjeeYale University

Abstract:

A spherical G-variety is a rather special kind of a homogeneous spaces (or its equivariant compactification) that is associated to a reductive group G.  Well known examples of such varieties include flag varieties and lesser known examples include the moduli spaces of complete collineations.

In this talk, we will outline a uniform approach which computes the equivariant $K_{0}$ for any spherical variety.  This is a joint work with Mahir Can.

November 28

December 5

## Asymptotic Formulas for Shifted Stacks and Unimodal Sequences

Karl Mahlburglouisiana state university

Abstract:

I will discuss recent results on the enumeration of unimodal sequences of natural numbers. This includes the combinatorial and asymptotic study of Euler's integer partitions, Auluck's generalized Ferrers diagrams, Wright's stacks, and Andrews' convex compositions. One result provides the asymptotic main term for the enumeration of shifted stacks, which answers an open problem in statistical mechanics due to Temperley.

See Spring 2014

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu