Algebraic proof of explicit formulas of basic relative invariants of homogeneous cones

TL;DR

This paper provides an algebraic proof for explicit formulas of basic relative invariants of homogeneous cones, replacing previous analytic methods and offering a more algebraic approach to the problem.

Contribution

It introduces a purely algebraic proof for the formulas of invariants of homogeneous cones, improving upon prior analytic-based proofs.

Findings

Explicit formulas for basic relative invariants of homogeneous cones.

An algebraic method replaces analytic arguments in the proof.

Enhanced understanding of the structure of homogeneous cones.

Abstract

The aim of this paper is to give another proof to a result on the image of a homogeneous quadratic map which is positive with respect to a homogeneous cone, given by Graczyk and Ishi in 2014. The new proof depends on a purely algebraic method, whereas the original depends on analytic arguments. This enables us to give explicit formulas of the basic relative invariants of homogeneous cones, obtained by the previous paper [J. Lie Theory 24 (2014), 1013--1032] without analytic arguments.