Combinatorics/Algebra Seminar
Fall 2023 Archive:

November 27

Jonathan Smith: The combinatorial approach to group and quasigroup characters.

Abstract:

This talk will provide an elementary, combinatorial introduction to the ordinary character theory of finite groups. Since it does not involve taking the traces of matrix representations, it applies equally well to finite quasigroups.

November 13

Connor Depies: Octavian algebras and gradings on the octonions.

Abstract:

The octavians constitute a subring of the octonions which forms a maximal order. However, unlike the Gaussian integers in the complex numbers or the Hurwitz integers in the quaternions, there is not one unique copy of the octavians, but seven. In this presentation, it is shown how to properly specify a unique copy of the octavians, and then this specification is connected to a new method of constructing the octonions, which we have previously called the Kingdon construction.

October 30, November 6

Justin Stevenson: Character quasigroups: A generalization of Pontryagin duality.

Abstract:

Pontryagin duality provides a powerful tool for analyzing the structure and properties of locally compact abelian groups, in particular finite abelian groups. Now a similar thing can be done to non-abelian groups via the construction of character quasigroups. In this talk I give a brief overview of the necessary concepts and results from representation theory which underpins the existence of character quasigroups. In the second half we will see how to compute character quasigroups with multiple concrete examples.

October 16, 23

Jonathan Smith: Codes, errors, and loops.

Abstract:

These two talks will present a novel introduction to the problem of reliably transmitting information through a noisy classical channel, correcting the errors that are most likely to occur during the transmission. The approach places emphasis on algebraic structure that is carried by the ball of likely errors. It brings many unexpected consequences, including:

• Methods to build linear codes for any error pattern;
• Universal syndromes, with fractal structure, for white noise error correction;
• A new approach to modular arithmetic and the Division Algorithm; and
• Non-associative loops describing domination in circulant graphs.

The talks summarize joint work with D. Choi, M. Dağlı, F. Hummer and B. Im.

October 2, 9

Connor Depies: Octonions as alternative Clifford-like algebras.

Abstract:

The octonions are the last in a series of numbers that starts with the reals, and progresses through the complex numbers and quaternions by means of the Cayley-Dickson process. The octonion algebra is not associative, but satisfies the weaker property of alternativity.

Clifford algebras are quotients of free associative or tensor algebras, and can be viewed as deformations of exterior algebras. Unlike the quaternion algebra, the octonion algebra is not a Clifford algebra, since it is not associative. However, it has many properties in common with Clifford algebras, such as anticommuting generators and an involutory conjugation anti-automorphism.

The talks will first give an introduction to Clifford algebras and Cayley-Dickson algebras, and then introduce Kingdon algebras as an alternative analog of Clifford algebras. The octonions will appear as a Kingdon algebra, and thus as a deformation of an alternative exterior-like algebra.

September 25

Mitchell Ashburn: Isomorphisms of vector-matrix algebras.

Abstract:

We begin by considering anticommutative formed algebras, that is, anticommutative algebras with a distinguished bilinear form. Typically, we will consider bilinear forms that are symmetric, invariant, and possibly non-degenerate. The best examples of such formed algebras are Lie algebras along with their Killing forms. From these formed algebras, we then construct a family of non-associative algebras called vector-matrix algebras, which generalize Zorn's vector-matrix construction of the split octonions.

Isomorphisms of these vector-matrix algebras can be reduced to specific types of isotopies between their underlying formed algebras. Furthermore, beginning with one of these isotopies of formed algebras, we can actually construct the isomorphism between their vector-matrix algebras. We can then explore the properties of these isotopies, attempting to find under what conditions two formed algebras will produce isomorphic vector-matrix algebras. Specifically, we can say a bit about anticommutative algebras along with their Killing forms, before highlighting a few examples of specific Lie algebras with bilinear forms that have such isotopies.

September 18

Anna Romanowska (Warsaw University of Technology): Sparse barycentric coordinates for polytopes.

Abstract:

Each point of a simplex is expressed as a unique convex combination of the vertices. The coefficients in the combination are the barycentric coordinates of the point. For each point in a general polytope, there may be multiple representations, so its barycentric coordinates are not necessarily unique. There are various schemes to fix particular barycentric coordinates: Gibbs, Wachspress, cartographic, etc.

In this talk, a method for producing sparse barycentric coordinates in polytopes will be discussed. It uses a purely algebraic treatment of affine spaces and convex sets, with barycentric algebras. The method is based on the close relationship between affine and barycentric coordinates of points of polytopes, and a certain decomposition of each finite-dimensional polytope into a union of simplices of the same dimension. The talk is based on recent results obtained jointly with J.D.H. Smith and A. Zamojska-Dzienio.

September 11

Jonathan Smith: Barycentric algebras and their symmetry.

Abstract:

When algebra is applied to geometry, it is desirable for the symmetries of the algebraic structure to match the symmetries (local or global) of the geometric structure. Unfortunately, the neutral elements of traditional algebra structures such as groups and rings behave in a special, symmetry-breaking way. Thus, when attempting to apply rings to convex geometry, it often becomes necessary to restrict attention to convex cones in vector spaces.

In this talk, we will discuss the modeling of general convex sets using the non-traditional algebras known as barycentric algebras, which are based on the logic structure of open and closed unit intervals. Hitherto, there has been an asymmetry in the axiomatization of barycentric algebras. We will give a general introduction to barycentric algebras accessible at the advanced undergraduate level, and then present recent results which make their definition completely symmetric.

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