An efficient algorithm computing composition factors of $T(V)^{\otimes n}$

TL;DR

This paper introduces an efficient algorithm for computing the composition factors of tensor powers of free associative algebras, significantly extending computational capabilities and revealing new theoretical insights.

Contribution

The authors develop a novel algorithm that computes composition factors more efficiently and introduce PD-modules as a new framework for understanding these coefficients.

Findings

Extended the computational range by over 750 times

Reinterpreted coefficients as counting solutions to decomposition-puzzles

Developed a new representation theoretic framework called PD-modules

Abstract

We present an algorithm that computes the composition factors of the n-th tensor power of the free associative algebra on a vector space. The composition factors admit a description in terms of certain coefficients $c_{\lambda\mu}$ determining their irreducible structure. By reinterpreting these coefficients as counting the number of ways to solve certain `decomposition-puzzles' we are able to design an efficient algorithm extending the range of computation by a factor of over 750. Furthermore, by visualising the data appropriately, we gain insights into the nature of the coefficients leading to the development of a new representation theoretic framework called PD-modules.