6533b832fe1ef96bd129ad96

RESEARCH PRODUCT

A Star-Variety With Almost Polynomial Growth

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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 G 2 is the algebra constructed in Giambruno and Mishchenko (preprint), we next prove that M and G 2 are the only two finite dimensional algebras with involution generating ∗ -varieties with almost polynomial growth.