Tonal partition algebras: fundamental and geometrical aspects of representation theory

Abstract

For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[\delta]$. We construct modules $\{ \Delta_{\underline{\mu}} \}_{\underline{\mu}}$ for $P^l_n$ over $\mathbb{Z}[\delta]$, and hence over any integral domain containing $\mathbb{Z}[\delta]$ that is a $\mathbb{Z}[\delta]$-algebra (such as $\mathbb{C}[\delta]$), that pass to a complete set of irreducible modules over the field of fractions. We show that $P^l_n$ is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer [6]. (The aim is to investigate the non-semisimple structure of the tonal partition algebras over suitable quotient fields of the natural ground ring, from a geometric perspective.) Using a `geometrical' index set for the $\Delta$-modules, we give an order with respect to which the decomposition matrix over $\mathbb{C}$ (with $\delta \in \mathbb{C}^{\times}$) is upper-unitriangular. We establish several crucial properties of the $\Delta$-modules. These include a tower property, with respect to $n$, in the sense of Green [20, \S 6] and Cox $\textit{ et al}$ [8]; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.