Combinatorics/Algebra Seminar
Spring 2016 Archive:

April 25

Christopher French (Grinnell College): Noncommutative association schemes of rank 4.

Abstract:

It is well-known that any finite association scheme of rank less than 6 must be commutative. But are there infinite noncommutative association schemes with rank less than 6? One way to approach this question is to strip the points from association schemes, thus isolating their underlying algebraic structure; what one obtains is a hypergroup.

It is not difficult to show that a hypergroup of rank 3 must be commutative. However, up to isomorphism, there are 5 noncommutative hypergroups of rank 4. One can then seek to "put the points back", and realize these hypergroups as association schemes. In this talk, we will consider several of these examples. We show that in some cases, the hypergroup cannot be realized even as an infinite association scheme, but in other cases, it can. We will thus give some constructions of noncommutative association schemes of rank 4.

April 11, 18

Jeremiah Goertz: E₈: The unique even positive-definite unimodular integral lattice of smallest rank.

March 28, April 4

Jonathan Smith: Fields, vector spaces, and quasigroups.

Abstract:

Using some ideas from the theory of quasigroups, we give a rigorous new definition of a field which admits precisely the "classical" fields and the field of order one. We then give a rigorous new definition of a linear combination which restricts appropriately to vector spaces over "classical" fields, and ensures that any set is a vector space over the field of order one.

This particular approach to the field of order one is motivated by the desire to integrate groups with Hopf algebras, and designs with their q-analogues.

March 21

Stefanie Wang: Linear quasigroups and representations of the free group on two generators.

Abstract:

A quasigroup  (A,*)  is said to be Z-linear if it there is an additive abelian group structure  (A, +, 0)  with automorphisms  R  and  L  such that

x*y = xR + yL.

This talk will provide an introduction to the study of Z-linear quasigroups and their abelian group representations of the free group on two generators. Through the course of the talk, we shall see how group representation theory and character theory do not completely classify Z-linear quasigroups.

March 14

Spring Break: no seminar.

March 7

Sung-Yell Song: The q-analogues of certain block designs and their existence.

Abstract:

I will introduce q-analogue t-designs, and discuss analogues of some of the results from the previous talk for projective spaces PG(F_q).

February 29

Sung-Yell Song: Directed strongly regular graphs constructed by using the flags and antiflags of partial geometries.

Abstract:

A structure (or incidence structure) consists of two finite sets of objects, called points and blocks (or lines), with an incidence relation between them. An incident point-block pair is called a flag, and a non-incident point-block pair is an antiflag. The flags and antiflags of various incidence structures (such as block designs, projective planes and Moore geometries) have been used in the construction of other structures (such as association schemes, coherent algebras, directed strongly regular graphs and block designs) for the last several decades.

In this talk, I will discuss how we can obtain certain families of directed strongly regular graphs by using flags and antiflags of partial geometries and balanced incomplete block designs. (This talk is based on work with Andries Brouwer and Oktay Olmez.)

February 15, 22

William DeMeo: Permutability in diamonds.

Abstract:

Among the oldest open questions in universal algebra is the following: Which finite lattices are congruence lattices of finite algebras? We consider a related problem about a special class of lattice, namely, the "diamonds" M_n, which are height-two modular lattices with n atoms. In 1990, Péter Pálfy and Jan Saxl used deep results from group theory to show that if a finite algebra has a congruence lattice isomorphic to M_n with n>3, and if at least three of the nontrivial congruences pairwise permute, then n - 1 is a power of a prime. In doing so, they also proved that if a G-set has M_n (n > 3) as a congruence lattice and at least 3 atoms pairwise permute, then all congruences permute.

Pálfy and Saxl leave open the following question: If a finite algebra has congruence lattice M_n for some n > 3, and if at least 3 atoms pairwise permute, does it follow that all congruences permute?

In this talk, we present some background required to understand the Pálfy-Saxl question, and then cover a little bit of tame congruence theory that helps us solve the "easy" (weakly abelian) case of the problem. We conclude with some alternative strategies that we hope lead to a solution in the harder (strongly abelian) case.

February 8

Department meeting: no seminar.

February 1

Jonas Hartwig: Noncommutative division rings of fractions and their invariants.

Abstract: This will be an introductory talk about Ore domains, which are noncommutative rings from which one can construct a divison ring, an analog of the field of fractions. The goal is to explicitly compute some subrings of invariants in some such division rings, with respect to some finite group.

January 25

Jonathan Smith: Approximate Latin squares.

Abstract: Approximate Latin squares provide the next step along a natural progression that starts with probability distributions and proceeds through doubly stochastic matrices. Let  n  be a positive integer. The space (simplex) of probability distributions on an  n-element set forms a convex polytope of dimension  (n - 1), while the space of doubly stochastic (n x n)-matrices forms a convex polytope of dimension  (n - 1)². Then the space of approximate Latin squares of order  n  forms a convex polytope of dimension  (n - 1)³. Permutation matrices are (the only) extreme points of the convex polytope of doubly stochastic matrices, and (exact) Latin squares are extreme points of the convex polytope of approximate Latin squares.

The talk is based on joint work with Bokhee Im and Hwa-Young Lee.

January 18

King Holiday: no seminar.

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