Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
157 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Tonal partition algebras: fundamental and geometrical aspects of representation theory (1912.01898v1)

Published 4 Dec 2019 in math.RT

Abstract: For $l,n \in \mathbb{N}$ we define tonal partition algebra $Pl_n$ over $\mathbb{Z}[\delta]$. We construct modules ${ \Delta_{\underline{\mu}} }_{\underline{\mu}}$ for $Pl_n$ 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 $Pl_n$ 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.

Summary

We haven't generated a summary for this paper yet.