Y-proper graded cocharacters of upper-triangular matrices od order m graded by the m-tuple \phi=(0,0,1,...,m-2)

TL;DR

This paper computes the Y-proper graded cocharacters of upper-triangular matrix algebras with a specific Z_m-grading, providing explicit formulas for small matrix sizes, advancing understanding of graded polynomial identities.

Contribution

It introduces explicit formulas for the Y-proper graded cocharacters of UT_m(F) under a particular Z_m-grading, a novel computation for these algebraic structures.

Findings

Explicit formulas for m=2,3,4,5

Y-proper graded cocharacter sequences computed

Advances understanding of graded identities in upper-triangular matrices

Abstract

Let F be a field of characteristic 0. We consider the algebra UT_m(F) of upper triangular matrices of order m endowed with na elementar Z_m-grading induced by the m-tuple phi=(0,0,1,...,m-2), then we compute its Y-proper graded cocharacter sequence and we give the explicit formulas for the multiplicities in the case m=2,3,4,5.