Growth for Algebras Satisfying Polynomial Identities. (21st November 2012) - Regev, Amitai
Abstract : The n th codimension c n ( A ) of a PI algebra A measures how many identities of degree n the algebra A satisfies. Growth for PI algebras is the rate of growth of c n ( A ) as n goes to infinity. Since in most cases there is no hope in finding nice closed formula for c n ( A ), we study its asymptotics. We review here such results about c n ( A ), when A is an associative PI algebra. We start with the exponential bound on c n ( A ) then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of c n ( A ) . The representation theory of the symmetric group S n is an important tool in this theory.