# CMU Algebra and Combinatorics Seminar

## Spring 2015

### Meeting Times

Wednesdays, 4–5pm, in Pearce 126.

### Schedule

 Date Speaker Affiliation Title 1/21 Yi Su University of Michigan Electrical Networks and Electrical Lie Theory 1/28 Sivaram Narayan Central Michigan University When Leading Imply All, Mixed Matrices, and Koteljanskii Inequalities 2/4 Pete Vermeire Central Michigan University Betti Tables of Reducible Algebraic Curves 2/11 2/18 Peter Tingley Loyola University of Chicago Crystals, PBW bases, and related combinatorics 2/25 3/4 3/25 Ryan Kaliszewski Drexel University TBD 4/1 C.-Y. Jean Chan Central Michigan University TBD **4/3** William Fulton University of Michigan TBD 4/8 4/15 Sidney Graham Central Michigan University TBD 4/22 Joshua Hallam Michigan State University TBD 4/29 Carolina Benedetti Michigan State University TBD

### Abstracts

Speaker: Yi Su (University of Michigan)
Title: Electrical Networks and Electrical Lie Theory
Abstract: Curtis-Ingerman-Morrow studied the space of circular planar electrical networks. Lam and Pylyavskyy introduced an electrical Lie algebra such that the positive part of the corresponding electrical Lie group $E_{A_{2n}}$ acts on circular planar electrical networks via some natural combinatorial operations. Inspired by these operations Lam gave a cell decomposition of the space of electrical networks, which is the analogue of type A Bruhat decomposition. In this talk, I will introduce mirror symmetric circular planar electrical networks, and present some analogous results about this new class of networks and its relation to type B electrical Lie theory. If time permitting, I will talk about the structure of electrical Lie algebras of type C and D.

Speaker: Sivaram Narayan (Central Michigan University)
Title: When Leading Imply All, Mixed Matrices, and Koteljanskii Inequalities
Abstract: An $n$-by-$n$ real matrix $A$ enjoys the leading implies all (LIA) property, if, whenever $D$ is a diagonal matrix such that $A+D$ has positive leading principal minors, then all principal minors of $A$ are positive. Symmetric and $Z$-matrices are known to have this property. We discuss a new class of matrices called mixed matrices that both unifies and generalizes these two classes and their special diagonal equivalences by also having the LIA property. Nested implies all (NIA) property is also enjoyed by this new class. We also give an inductive characterization of the LIA property.

It is natural to ask what other properties of M-matrices and positive definite matrices are enjoyed by mixed matrices as well. We show that mixed P-matrices satisfy a broad family of determinantal inequalities, the Koteljanskii inequalities, previously known for those two classes. In the process, other properties of mixed matrices are developed, and consequences of the Koteljanskii inequalities are given.

This is a joint work with Charles R. Johnson.

Speaker: Pete Vermeire (Central Michigan University)
Title: Betti Tables of Reducible Algebraic Curves
Abstract: We will discuss the problem of finding free resolutions of the ideal of a projective variety. In particular, we will discuss the case where the variety in question is the union of lines which can be deformed to a smooth curve. This allows us to attempt a purely combinatorial description of the generators and syzygies of the ideal in terms of the dual graph of the configuration of lines. This was part of an REU project from two summers ago.

Speaker: Peter Tingley (Loyola University of Chicago)
Title: Crystals, PBW bases, and related combinatorics
Abstract: Kashiwara's crystals are combinatorial objects used to study complex simple Lie groups and their highest weight representations. A "crystal" consists of an underlying set, which roughly indexes a basis for a representation, along with combinatorial operations related to the Chevalley generators. Here we are interested in types A and D, where the crystals are realized using Young tableaux and Kashiwara-Nakashima tableaux respectively. However the combinatorics is not always enough, and one would like to understand an actual basis corresponding to the crystal. This can be done using Lusztig's theory of PBW bases, and so there is a unique structure-preserving bijection between the PBW basis and the combinatorially constructed crystal. We describe that bijection explicitly. In type A the answer was to some extent known to experts, but in type D we believe it is new. I will explain as much of this as I can without assuming any prior knowledge of crystals or PBW bases, mainly through examples. This is joint work with John Claxton, Ben Salisbury and Adam Shultze.

Speaker: Ryan Kaliszewski (Drexel University)
Title: TBD
Abstract: Coming soon …

Speaker: C.-Y. Jean Chan (Central Michigan University)
Title: TBD
Abstract: Coming soon …

Speaker: William Fulton (University of Michigan)
Title: TBD
NOTE THE SPECIAL DAY AND TIME!!!
Abstract: Coming soon …

Speaker: Sidney Graham (Central Michigan University)
Title: TBD
Abstract: Coming soon …

Speaker: Joshua Hallam (Michigan State University)
Title: TBD
Abstract: Coming soon …

Speaker: Carolina Benedetti (Michigan State University)
Title: TBD
Abstract: Coming soon …