Adventures in Invariant Theory

TL;DR

This paper introduces methods for enumerating and constructing invariants of group representations, with applications in quantum entanglement and phylogenetics, highlighting character-based approaches and case studies.

Contribution

It presents a novel approach using character methods to construct invariants, applied to quantum entanglement and Markov invariants in phylogenetics.

Findings

Character methods effectively construct invariants for complex group representations.

Applications demonstrate improved understanding of quantum state spaces.

Markov invariants offer a robust alternative for statistical inference in phylogenetics.

Abstract

We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualised via two case studies arising from our recent work: entanglement measures, for characterising the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.