The Regev conjecture and cocharacters for identities of associative algebras of PI-exponent 1 and 2

A result confirming the Regev conjecture for the codimension of associative algebras with unit which are of PI-exponent 2 is obtained. It is proved that the sequence of multiplicities of irreducible summands in proper cocharacters of algebras of PI-exponent 2 is of period 2, beginning with some index, whereas this sequence is constant for the ordinary cocharacters of the algebras of PI-exponent 1.