Tensor diagrams and cluster algebras

TL;DR

This paper demonstrates that rings of SL(V) invariants for 3-dimensional vector spaces have natural cluster algebra structures, using tensor diagrams and web bases to describe their combinatorial properties.

Contribution

It introduces a cluster algebra framework for rings of invariants in 3D, connecting classical invariant theory with modern cluster algebra techniques.

Findings

Rings of SL(V) invariants in 3D have natural cluster algebra structures.

Cluster variables include Weyl's generators.

Tensor diagrams and web bases effectively describe these structures.

Abstract

The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. We describe and explore these cluster structures using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G.Kuperberg.