# Generic states and stability

## TL;DR

This paper introduces the concept of the generic state polytope, extending prior work on generic initial ideals and Gr"obner fans, and explores its implications for stability in geometric invariant theory.

## Contribution

It defines the generic state polytope, proves its existence, and shows it always contains the trivial character, generalizing previous results on Gr"obner fans.

## Key findings

- Generic state polytope always contains the trivial character.
- In GIT, every point is semistable with respect to a general maximal torus.
- Provides equations for determining the worst one parameter subgroup.

## Abstract

We define the notion of the generic state polytope, analogous to the generic initial ideal and prove its existence: This greatly generalizes the work of R\"omer and Schmitz who proved the existence of generic Gr\"ober fans. We also show that a generic state polytope always contains the trivial character: Equivalently, in any GIT quotient problem of semisimple group representations, every point is semistable with respect to a {\it general} maximal torus. Also, we revisit Kempf's proof of the existence of the worst one parameter subgroup (1-ps) and describe the equations for determining the worst 1-ps.