Decomposition of the Symmetric Powers

TL;DR

This paper provides a detailed decomposition of symmetric powers of a tensor product of three 2-dimensional complex vector spaces into irreducible modules of a triple sl_2 algebra, including multiplicities.

Contribution

It presents an explicit method to decompose symmetric powers of b^2 b^2 b^2 into irreducible sl_2b b b modules, determining their multiplicities.

Findings

Explicit decomposition formulas for symmetric powers.

Determination of multiplicities of irreducible components.

Enhanced understanding of tensor product symmetries.

Abstract

A decomposition of any symmetric power of $\Bbb C^2\otimes\Bbb C^2\otimes\Bbb C^2$ into irreducible $sl_2(\Bbb C)\oplus sl_2(\Bbb C)\oplus sl_2(\Bbb C)$-submodules are presented. Namely, the multiplicities of irreducible summands in the symmetric power are determined.