Cocharacters of polynomial identities of upper triangular matrices

TL;DR

This paper presents an algorithm to compute the generating function of cocharacter sequences for polynomial identities of upper triangular matrices, providing explicit multiplicity formulas for specific cases.

Contribution

It introduces a simple algorithm to determine the generating function of cocharacter sequences and explicitly computes multiplicities for certain partitions and matrix sizes.

Findings

Algorithm efficiently computes generating functions

Explicit multiplicity formulas for large partitions

Results for small matrices and all partitions

Abstract

We give an easy algorithm which calculates the generating function of the cocharacter sequence of the T-ideal of the polynomial identities of the algebra of upper triangular matrices over a field of characteristic zero. Applying this algorithm we have found the explicit form of the multiplicities in two cases: (i) for the "largest" partitions; (ii) for matrices of small size and for all partitions.