Homogeneous involutions on upper triangular matrices

TL;DR

This paper characterizes when upper triangular matrix algebras over a field admit homogeneous antiautomorphisms compatible with a given group grading, linking them to the reflection involution and showing uniqueness of such maps.

Contribution

It provides necessary and sufficient conditions for the existence of homogeneous antiautomorphisms on graded upper triangular matrices and establishes their uniqueness.

Findings

Homogeneous antiautomorphisms exist if and only if the reflection involution is homogeneous.

Such antiautomorphisms are characterized by a permutation map on the grading support.

Any two homogeneous antiautomorphisms are defined by the same permutation map.

Abstract

Let $K$ be a field of characteristic different from 2 and let $G$ be a group. If the algebra $UT_n$ of $n\times n$ upper triangular matrices over $K$ is endowed with a $G$-grading $\Gamma: UT_n=\oplus_{g\in G}A_g$ we give necessary and sufficient conditions on $\Gamma$ that guarantees the existence of a homogeneous antiautomorphism on $A$, i.e., an antiautomorphism $\varphi$ satisfying $\varphi(A_g)=A_{\theta(g)}$ for some permutation $\theta$ of the support of the grading. It turns out that $UT_n$ admits a homogeneous antiautomorphism if and only if the reflection involution of $UT_n$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $UT_n$ is defined by the map $\theta$ then any other homogeneous antiautomorphism is defined by the same map $\theta$.