On the asymptotic behaviour of the graded-star-codimension sequence of upper triangular matrices

TL;DR

This paper investigates the asymptotic growth of the graded-star-codimension sequence of upper triangular matrices with group grading and involution, revealing that the growth rate depends only on the matrix size, not on grading or involution.

Contribution

It provides a detailed analysis of the asymptotic behaviour of graded-star-codimensions, showing their independence from grading and involution, depending solely on matrix size.

Findings

Asymptotic growth is independent of grading and involution.

Growth rate depends only on the size of the matrix algebra.

Graded codimension sequence shares this independence property.

Abstract

We study the algebra of upper triangular matrices endowed with a group grading and a homogeneous involution over an infinite field. We compute the asymptotic behaviour of its (graded) star-codimension sequence. It turns out that the asymptotic growth of the sequence is independent of the grading and the involution under consideration, depending solely on the size of the matrix algebra. This independence of the group grading also applies to the graded codimension sequence of the associative algebra of upper triangular matrices.