Tonal partition algebras: fundamental and geometrical aspects of representation theory

TL;DR

This paper introduces tonal partition algebras, constructs their modules, and explores their semisimplicity, geometric structure, and representation theory properties, including decomposition matrices and highest weight category features.

Contribution

It defines tonal partition algebras over integers, constructs modules, and analyzes their semisimple and geometric properties, extending classical representation theory frameworks.

Findings

Tonal partition algebras are semisimple over their field of fractions.

Decomposition matrices are upper-unitriangular with a geometric index set.

Modules exhibit tower, contravariant forms, and highest weight category properties.

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.