On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field

TL;DR

This paper investigates the polynomial identities of upper triangular matrices with a fine group grading and homogeneous involution, providing explicit bases, asymptotic codimension behavior, and exponents over arbitrary fields.

Contribution

It offers a comprehensive computation of polynomial identities, bases of free algebras, and asymptotic invariants for graded upper triangular matrices with involution over any field.

Findings

Computed polynomial identities and bases for the algebra

Analyzed asymptotic behavior of codimension sequences

Determined exponents for graded involution algebras

Abstract

We study the graded polynomial identities with a homogeneous involution on the algebra of upper triangular matrices endowed with a fine group grading. We compute their polynomial identities and a basis of the relatively free algebra, considering an arbitrary base field. We obtain the asymptotic behaviour of the codimension sequence when the characteristic of the base field is zero. As a consequence, we compute the exponent and the second exponent of the same algebra endowed with any group grading and any homogeneous involution.