A Star-Variety With Almost Polynomial Growth

Abstract

Let F be a field of characteristic zero. In this paper we construct a finite dimensional F-algebra with involution M and we study its ∗-polynomial identities; on one hand we determine a generator of the corresponding T-ideal of the free algebra with involution and on the other we give a complete description of the multilinear ∗-identities through the representation theory of the hyperoctahedral group. As an outcome of this study we show that the ∗-variety generated by M, var(M,∗) has almost polynomial growth, i.e., the sequence of ∗-codimensions of M cannot be bounded by any polynomial function but any proper ∗-subvariety of var(M,∗) has polynomial growth. If G2 is the algebra constructed in Giambruno and Mishchenko (preprint), we next prove that M and G2 are the only two finite dimensional algebras with involution generating ∗-varieties with almost polynomial growth.