# Algebra Seminar

Current contacts: Vasily Dolgushev, Ed Letzter or Martin Lorenz.

The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.

• Monday April 22, 2019 at 13:30, Wachman 617
Kazhdan-Lusztig polynomials of matroids

Jacob Matherne, IAS, Princeton

Kazhdan-Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the setting of matroids. In this talk, I will compare and contrast KL theory for Coxeter groups with KL theory for matroids.

I will also associate to any matroid a certain ring whose Hodge theory can conjecturally be used to establish the positivity of the KL polynomials of matroids as well as the "top-heavy conjecture" of Dowling and Wilson from 1974 (a statement on the shape of the poset which plays an analogous role to the Bruhat poset). Examples involving the geometry of hyperplane arrangements will be given throughout. This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang.

• Monday March 25, 2019 at 13:30, Wachman 617
Topics in Galois Theory, I

Martin Lorenz, Temple University

This is the first lecture in a minicourse (probably three lectures) surveying some topics in Galois Theory that are not typically covered in the graduate algebra course (Math 8011/12): inverse Galois theory, Noether's rationality problem, the Chebotarev density theorem,... The Galois Theory portion of Math 8011/12 will be sufficient background for the material presented in this minicourse; so it will be accessible to all students in my current Math 8012 class. No proofs will be given; the goal is to describe some research directions that are of current interest.

• Monday March 18, 2019 at 13:30, Wachman 617
Classifying Actions of $T_n \otimes T_n$ on Path Algebras of Quivers

Delaney Aydel, Temple University

Let $T_n$ denote the $n$th Taft algebra. We fully classify inner-faithful actions of $T_n \otimes T_n$ on four-vertex Schurian quivers as extensions of the actions of $\mathbb{Z}_n \times \mathbb{Z}_n$. One example will be presented in full, with the remaining results briefly given.

• Monday March 11, 2019 at 13:30, Wachman 617
Catalan Functions and k-Schur functions

Anna Pun, Drexel University

Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase (n-1, ..., 1,0) (of which there are Catalan many) and a composition weight of length n. They include the Schur functions, the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture.

A key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are k-Schur functions and proved that graded k-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded k-Schur functions and resolved the Schur positivity and k-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are k-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems. This is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers. Here are the links to the papers on ArXiv: https://arxiv.org/abs/1804.03701, https://arxiv.org/abs/1811.02490

• Monday February 25, 2019 at 13:30, Wachman 617
Prime Torsion of the Brauer Group of an Elliptic Curve

Charlotte Ure, Michigan State University

The Brauer group of an elliptic curve $E$ is an important invariant with intimate connections to cohomology and rational points. Elements of this group can be described as Morita equivalence classes of central simple algebras over the function field. The Merkurjev-Suslin theorem implies that these classes can be written as tensor product of symbol (or cyclic) algebras. In this talk, I will describe an algorithm to calculate generators and relations of the $q$-torsion ($q$ a prime) of the Brauer group of $E$ in terms of these tensor products over any field of characteristic different from $2$,$3$, and $q$, containing a primitive $q$-th root of unity. This is work in progress.

• Monday February 18, 2019 at 13:30, Wachman 617
On the Replacement Property for $PSL(2,p)$

Aidan Lorenz, Temple University

The replacement property (or Steinitz Exchange Lemma) for vector spaces has a natural analog for finite groups and their generating sets. For the special case of the groups $PSL(2,p)$, where $p$ is a prime larger than 5, first partial results concerning the replacement property were published by Benjamin Nachman. The main goal of this talk is to outline the methods involved in providing a complete answer for $PSL(2,p)$ (which was accomplished during the Summer of 2018). This talk is based on a paper in preparation joint with Baran Zadeoglu and David Cueto Noval.

• Monday February 11, 2019 at 13:30, Wachman 617
Detecting free objects in associative algebras, II

Edward Letzter, Temple University

A discussion of more recent results, and still-open questions, on free subalgebras and free multiplicative subsemigroups of associative algebras.

• Monday February 4, 2019 at 13:30, Wachman 617
Detecting free objects in associative algebras: A survey

Edward Letzter, Temple University

In the 1970s, Lichtman asked whether or not the multiplicative group of units of a noncommutative division algebra contains a free subgroup and Makar-Limanov asked whether or not a finitely generated infinite dimensional noncommutative division algebra must contain a free subalgebra. These questions are still open in general, even if many important special cases have been resolved, and have recently received renewed attention. (These questions can be considered in analogy to the Tits Alternative for linear groups as well as Gromov's Theorem on groups with polynomial growth.) My talks will survey both older and newer results.

• Monday January 28, 2019 at 13:30, Wachman 617
Paschke Categories, K-homology and the Riemann-Roch Transformation

Khashayar Sartipi, University of Illinois at Chicago

For a separable C^*-algebra A, we introduce an exact C^*-category called the Paschke Category of A, which is completely functorial in A, and show that its K-theory groups are isomorphic to the topological K-homology groups of the C^*-algebra A. Then we use the Dolbeault complex and ideas from the classical methods in Kasparov K-theory to construct an acyclic chain complex in this category, which in turn, induces a Riemann-Roch transformation in the homotopy category of spectra, from the algebraic K-theory spectrum of a complex manifold X, to its topological K-homology spectrum. This talk is based on the preprint https://arxiv.org/abs/1810.11951