Tropical Invariants for Permutation Group Actions

TL;DR

This paper studies tropical invariants under permutation group actions, showing finite generation conditions, constructing separating invariants, and applying these to embeddings of orbit spaces, with results valid over general semirings.

Contribution

It characterizes when the invariant sub-semiring is finitely generated and constructs explicit separating invariants for permutation group actions in tropical algebra.

Findings

Invariant sub-semiring is finitely generated iff the group is generated by 2-cycles.

Existence of finitely many separating invariants of bounded degree.

Invariant rational tropical functions are generated by polynomials of degree related to prime numbers.

Abstract

We consider the action of a permutation group $G$ of order $k$ on the tropical polynomial semiring in $n$ variables. We prove that the sub-semiring of invariant polynomials is finitely generated if and only if $G$ is generated by $2$-cycles. There do exist finitely many separating invariants of degree at most $\max\{n,{n\choose 2}\}$. Separating tropical invariants can be used to construct bi-Lipschitz embeddings of the orbit space ${\mathbb R}^n/G$ into Euclidean space. We also show that the invariant polynomials of degree $\leq n p_1p_2\cdots p_k$ generate the semifield of invariant rational tropical functions, where $p_1,p_2,\dots,p_k$ are the first $k$ prime numbers. Most results are also true over arbitrary semirings that are additively idempotent and multiplicatively cancellative.