Tensor decompositions for SL(2) and outerplanar graphs

TL;DR

This paper presents a novel tensor decomposition method for SL(2) representations using outerplanar graphs, providing a new combinatorial approach to understanding tensor products in representation theory.

Contribution

It introduces a decomposition of tensor products of SL(2) representations into sums parametrized by outerplanar graphs, a new combinatorial framework in representation theory.

Findings

Decomposition of SL(2) tensor products into irreducibles via outerplanar graphs

Establishment of a combinatorial parametrization for these decompositions

Potential applications to graphical methods in representation theory

Abstract

The main result of this article is the decomposition of tensor products of representations of SL(2) in the sum of irreducible representations parametrized by outerplanar graphs. An outerplanar graph is a graph with the vertices 0, 1, 2, ..., m, edges of which can be drawn in the upper half-plane without intersections. I allow for a graph to have multiple edges, but don't allow loops.