SEMINAR: Groups and Combinatorics Seminar, Decomposing tensor products over fields of small characteristic

Abstract:

One motivation for this talk comes from representation theory: decomposing a tensor product of irreducible (or indecomposable) representations as a sum of smaller degree irreducible (or indecomposable) representations. Other motivations come from quantum mechanics and Frobenius algebras.

Consider an $r imes r$ matrix $K_r$ over a field $F$ with 1s on the main diagonal and first upper diagonal (positions $(i,i)$ and $(i,i+1)$) and zeros elsewhere. The tensor product $K_r times K_s$ is a unipotent matrix whose Jordan canonical form is determined by some partition of $rs$. We will show that this partition enjoys surprising symmetries: duality, periodicity, regularity. Our original motivation was to study this partition when the characteristic $p$ of $F$ is small (i.e. $p<r+s-1$). The large characteristic case ($p e r+s-1$) was solved recently by Iima and Iwamatsu.

This is joint work with Cheryl E. Praeger and Binzhou Xia.