Prehomogeneous vector spaces obtained from triangle arrangements

TL;DR

This paper introduces a method to construct new prehomogeneous vector spaces from triangle arrangements, expanding the known classes with many potentially novel examples.

Contribution

It presents a systematic construction technique for prehomogeneous vector spaces based on geometric triangle arrangements, including new examples not previously documented.

Findings

Construction of a new series of prehomogeneous vector spaces

A main theorem linking triangle arrangements to vector space construction

Identification of several potentially new prehomogeneous vector spaces

Abstract

In this paper, we construct a new series of prehomogeneous vector spaces from figures made up of triangles, called triangle arrangements. Our main theorem states that, under suitable assumptions, we are able to construct a prehomogeneous vector space obtained from a triangle arrangement by attaching two triangle arrangements corresponding to prehomogeneous vector spaces at a vertex. We also give examples of prehomogeneous vector spaces obtained from triangle arrangements. Many of them seem to be new.