# CMU Algebra and Combinatorics Seminar

## Fall 2014

### Meeting Times

Wednesdays, 4:00–5:00pm, in Pearce 328.

### Abstracts

Speaker: Meera Mainkar (Central Michigan University)
Title: Anosov Lie algebras and algebraic units in number fields
Abstract: In the theory of dynamical systems, Anosov diffeomorphisms are an important class of dynamically interesting diffeomorphisms of Riemannian manifolds. The study of Anosov diffeomorphisms on nilmanifolds (compact quotients of simply connected nilpotent Lie groups) leads to interesting algebraic and arithmetic problems related to Lie algebras. In particular, this relates to very special algebraic units in number fields. We study the classification of Anosov Lie algebras by studying the properties of these algebraic units.

Speaker: Ben Salisbury (Central Michigan University)
Title: The Gindikin-Karpelevich formula and combinatorics of crystals
Abstract: The Gindikin-Karpelevich formula computes the constant of proportionality for the intertwining integral between two induced spherical representation of a $p$-adic reductive group $G$. The right-hand side of this formula is a product over positive roots (with respect to the Langlands dual of $G$) which may also be interpreted as a sum over a crystal graph. The original interpretation as a sum is due to Brubaker-Bump-Friedberg and Bump-Nakasuji where $G = \mathrm{GL}_{r+1}$, in which vertices of the crystal graph are parametrized by paths to a specific vector in the graph according to a pattern prescribed by a reduced expression of the longest element of the Weyl group of $G$. I will explain this rule, explain how it may be translated into a statistic on the Young tableaux realization of the same crystal graph, and discuss its generalization to the affine Kac-Moody setting where the notion of the longest element of the Weyl group no longer makes sense. This is joint work with Kyu-Hwan Lee, Seok-Jin Kang, and Hansol Ryu.

Speaker: Sunil Chebolu (Illinois State University)
Title: Group Algebras and Circulant Bipartite Graphs
Abstract: Let $p$ be a prime number. We characterise the group algebras over fields in which every non-trivial unit has order $p$. This characterization and some related results gave answers to some basic questions about circulant bipartite graphs. I will discuss these results which are part of a recent joint work (arXiv:1404.4096) with Keir Lockridge and Gail Yamskulna. (slides)

Speaker: Travis Scrimshaw (University of California, Davis)
Title: Crystal Structure For Rigged Configurations and the Filling Map
Abstract: Rigged configurations are combinatorial objects which arise from statistical mechanics, and they are in bijection with classically highest weight elements in a tensor product of the vector representation of finite dimensional non-exceptional affine crystals called Kirillov-Reshetikhin crystals. Schilling has shown there exists a classical crystal structure on rigged configurations in simply-laced affine type. I will present a classical crystal structure on rigged configurations for all affine types using virtual crystals. Furthermore I will present an extension of the so-called "filling map," introduced in recent work by Okado, Sakamoto, and Schilling for affine type D, to all non-exceptional affine types. This is joint work with Anne Schilling. (slides)

Speaker: Andrzej Dudek (Western Michigan University)
Title: On Ramsey-type problems for sequences and permutations
Abstract: Ramsey theory can loosely be described as the study of structure which is preserved under finite decomposition. A classical Ramsey theorem states that in any $r$-coloring of the edges of a sufficiently large complete graph, one will always find a monochromatic complete subgraph. In this talk, we discuss analogous results for sequences and permutations. In particular, we study the behavior of the following function $f(r,X)$, which is the length of the shortest sequence $Y$ such that any $r$-coloring of the entries of $Y$ yields a monochromatic subsequence that also preserves the order of $X$.

Speaker: Pin-Hung Kao (Central Michigan University)
Title: On the Combinatorial Sieve
Abstract: The ubiquitous inclusion-exclusion principle in combinatorics is as old as Grecian number theory. It was first presented by Eratosthenes in his famous sieve which Legendre later formalized. We will discuss precisely how the inclusion-exclusion principle was utilized in both formulations. We also will see the deficiency of this naive application of the inclusion-exclusion principle as well as a remedy to it by Brun. Lastly, we will see some applications of such sieves in number theory and combinatorics.

Speaker: Gabriel Feinberg (Haverford College)
Title: Homogeneous Representations of Khovanov-Lauda-Rouquier Algebras
Abstract: The Khovonov-Lauda-Rouquier (KLR) algebra arose out of attempts to categorify quantum groups. Kleshchev and Ram proved a result reducing the representation theory of these algebras to the study of irreducible cuspidal representations. In finite types, these cuspidal representations are part of a larger class of homogeneous representations, which are related to fully commutative elements of Coxeter groups. For KLR algebras of types $A_n$ and $D_n$, we classify and enumerate these homogeneous representations.

Speaker: Jordan Webster (Mid Michigan Community College)
Title: Hadamard Difference Sets in Groups with Odd Factors
Abstract: A $(v,k,\lambda)$ difference set $D$ in a group $G$ of order $v$ is a $k$-subset of $G$ such that each group element other than the identity appears $\lambda$ times in the multiset $\{d_{1}d_{2}^{-1} : d_{1}, d_{2} \in D\}$. A Hadamard difference set has parameters $(4m^{2}, 2m^{2}-m, m^{2}-m)$ for some $m\in \mathbf{Z}$. We attempt to find difference sets in groups of the form $C_{2^{r}}^{(2)}\times C_{3}^{(2)}$ and generalize the idea to similar groups. The approach will utilize character values and rational idempotents.

Speaker: Ben Salisbury (Central Michigan University)
Title: Some common constructions from representation theory
Abstract: In this expository talk, we discuss some of the basic features of a semisimple Lie algebra over $\mathbf{C}$ and its representations. The purpose of which is to elucidate some of the terms and concepts that have arisen in various talks throughout this semester (and to come later this semester and next). As time permits, we will also discuss quivers and their representations, the quantum group and crystal bases, and some combinatorics. The examples given will mainly involve the special linear Lie algebra $sl_n(\mathbf{C})$.

Speaker: Bruce E. Sagan (Michigan State University)
Title: Factoring rook polynomials
Abstract: A board $B$ is a subset of the squares of an $n\times n$ chess board. Let $r_k(B)$ denote the number of ways of placing $k$ nonattacking rooks on $B$ which means that every row and column has at most one rook. Various specializations of $r_k(B)$ count permutations, derangements, and set partitions. One nice set of boards are the Ferrers boards $B=(b_1,\dots,b_n)$ where the $b_j$ are a weakly increasing sequence of nonnegative integers and the corresponding board is obtained by choosing the lowest $b_j$ squares in column $j$ of the chess board for $1\le j\le n$. In a landmark paper, Goldman, Joichi, and White showed that if $B$ is a Ferrers board then an appropriately chosen generating function for the $r_k(B)$ factors over the integers. They also gave various applications of this result, such as a new proof of a theorem of Foata and Schützenberger. In the first half of this lecture we will provide an introduction to these beautiful results. The second half will be devoted to recent research concerning a generalization of rook placements where the rows of a board are grouped into levels and one can have at most one rook in any level or any column. This part is joint work with Kenneth Barrese, Nicholas Loehr and Jeffrey Remmel.

Title: Combinatorics of Link Diagrams and Volume
Abstract: In this talk, we will examine how the structure of the diagrams of certain families of hyperbolic semi-adequate links can be used to produce bounds on the hyperbolic volume of the link complement. In particular, we will see that these volume bounds can be expressed in terms of two diagrammatic quantities: the twist number of the link diagram and a count of the number of certain alternating tangles in the link diagram. The proof of this result, which will be sketched if time allows, is combinatorial in nature and relies on the connection between link diagrams and certain associated graphs.

Speaker: Jonathan Axtell (Iowa State University)
Title: Schur superalgebras and spin polynomial functors
Abstract: We discuss the categories $\mathrm{Pol}^I_d$, $\mathrm{Pol}^{II}_d$ of strict polynomial functors defined on vector superspaces over any field of characteristic not equal 2. These categories are related to polynomial representations of the supergroups $\mathrm{GL}(m | n)$, $Q(n)$ respectively. In particular, there is an equivalence between $\mathrm{Pol}^I_d$, $\mathrm{Pol}^{II}_d$ and the category of finite dimensional supermodules over the Schur superalgebras $S(m|n, d)$, $Q(n,d)$ respectively provided $m$, $n \geq d$. We also discuss some aspects of Sergeev duality from the viewpoint of the category $\mathrm{Pol}^{II}_d$.

Speaker: Ben Salisbury (Central Michigan University)
Title: Some (more) common constructions from representation theory
Abstract: In this (follow–up) expository talk, we discuss some of the basic features of a semisimple Lie algebra over $\mathbf{C}$ and its representations. The purpose of which is to elucidate some of the terms and concepts that have arisen in various talks throughout this semester (and to come later this semester and next). As time permits, we will also discuss the quantum group, crystal bases, and some combinatorics. The examples given will mainly involve the special linear Lie algebra $sl_n(\mathbf{C})$. And this time I promise we will actually discuss the Lie algebra material, now that the quiver representation material has been covered.

Speaker: Robert Molina (Alma College)
Title: Lights Out!
Abstract: Lights out is a handheld electronic puzzle by Tiger Toys. It consists of a 5×5 grid of buttons that have lights in them. Pressing a button changes the state (on to off, or vice versa) of the light in that button and in the lights of any vertically or horizontally adjacent buttons. A puzzle consists of a configuration of illuminated buttons and the goal is to turn all lights off using a minimum number of button pushes. We will investigate the mathematics associated with solving this puzzle and related puzzles. The primary tool we will use is linear algebra over the field GF(2), but we will also explore connections with graph theory and Fibonacci polynomials.