Department of Mathematical Sciences
Discrete Math Seminar
Thursday, February 19th, 2026
12:00PM-12:50PM
Olin Hall 218
Speaker: Bill Martin, WPI
Title: A medley of dualities in the world of association schemes
Abstract: In this talk, we explore the compatibility among various concepts of duality related to association schemes. An association scheme is essentially equivalent to a commutative algebra of $n\times n$ matrices closed also under the entrywise product and containing identities for both the ordinary and the entrywise product. We begin with the formal duality of these \emph{Bose-Mesner algebras}, which becomes explicit in the presence of an abelian group acting transitively on the vertices. Link invariants also give us a notion of duality, called \emph{Type II duality}, which we mention briefly. Dual pairs of association schemes show up in the theory of error-correcting codes. Bounding the efficiency of a linear error-correcting code often involves the dual code; in the non-linear case, we can instead use linear programming --- and yet another notion of duality comes into play. There is another, less explicit duality between homotopy and algebraic varieties that I will mention briefly. In the last part of the talk, I will introduce some third-order tensors called \emph{scaffolds} and I will show how Bose-Mesner duality is connected to the topological duality of outer planar graphs.