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 UT_n of n× n upper triangular matrices over K is endowed with a G -grading Γ : UT_n=⊕ _g∈ GA_g , we give necessary and sufficient conditions on Γ that guarantees the existence of a homogeneous antiautomorphism on A , i.e., an antiautomorphism φ satisfying φ (A_g)=A_θ (g) for some permutation θ 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 θ , then any other homogeneous antiautomorphism is defined by the same map θ .