Graded identities with involution for the algebra of upper triangular matrices

Abstract

Let $F$ be a field of characteristic zero. We prove that if a group grading on $UT_m(F)$ admits a graded involution then this grading is a coarsening of a $\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor}$-grading on $UT_m(F)$ and the graded involution is equivalent to the reflection or symplectic involution on $UT_m(F)$. A finite basis for the $(\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor},\ast)$-identities is exhibited for the reflection and symplectic involutions and the asymptotic growth of the $(\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor},\ast)$-codimensions is determined. As a consequence we prove that for any $G$-grading on $UT_m(F)$ and any graded involution the $(G,\ast)$-exponent is $m$ if $m$ is even and either $m$ or $m+1$ if $m$ is odd. For the algebra $UT_3(F)$ there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical $\mathbb{Z}$-grading and the $\mathbb{Z}_2$-grading induced by $(0,1,0)$. We determine a basis for the $(\mathbb{Z}_2,\ast)$-identities and prove that the exponent is $3$. Hence we conclude that the ordinary $\ast$-exponent for $UT_3(F)$ is $3$.