Homogeneous involutions on upper triangular matrices

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 $\theta$.