Valuations from representation theory and tropical geometry

TL;DR

This paper explores the connection between valuation spaces, tropical geometry, and representation theory of reductive groups, introducing graded valuations as a modification of Payne's seminorms.

Contribution

It introduces graded valuations and demonstrates their relation to tropical geometry and representation theory, providing new examples from reductive groups.

Findings

Graded valuations relate to tropical geometry.

Examples from reductive group representations.

Modified valuation spaces extend Payne's seminorms.

Abstract

We recall the space of seminorms discussed by Payne in \cite{P} and define a slight modification, the space of graded valuations. After explaining how these spaces relate to tropical geometry, we describe examples of graded valuations which come from the representation theory of reductive groups.