Homogeneous involutions on upper triangular matrices

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