# Note on the decomposition of states

## TL;DR

This paper presents a precise decomposition formula for the state polytope and Hilbert-Mumford index of reducible varieties, simplifying previous proofs through convex geometry and character decomposition.

## Contribution

It introduces a new, simplified proof of the state polytope decomposition for reducible varieties using convex geometry and character decomposition.

## Key findings

- Derived a sharp decomposition formula for state polytopes
- Simplified the proof of state polytope decomposition
- Enhanced understanding of the geometric structure of reducible varieties

## Abstract

We derive a sharp decomposition formula for the state polytope of the Hilbert point and the Hilbert-Mumford index of reducible varieties by using the decomposition of characters and basic convex geometry. This proof captures the essence of the decomposition of the state polytopes in general, and considerably simplifies an earlier proof by the author and Jaekwang Kim which uses a careful analysis of initial ideals of reducible varieties.