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 $\theta$.