A Star-Variety With Almost Polynomial Growth

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.