Abstract
The th codimension of a PI algebra measures how many identities of degree the algebra satisfies. Growth for PI algebras is the rate of growth of as goes to infinity. Since in most cases there is no hope in finding nice closed formula for , we study its asymptotics. We review here such results about , when is an associative PI algebra. We start with the exponential bound on then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of . The representation theory of the symmetric group is an important tool in this theory.
Highlights
We study algebras A satisfying polynomial identities PI algebra
A natural question arises: give a quantitative description of how many identities such algebra A satisfies? We assume that A is associative, though the general approach below can be applied to nonassociative PI algebras as well
The study of growth for PI algebra A is mostly the study of the rate of growth of the sequence cn A of its codimensions, as n goes to infinity
Summary
We study algebras A satisfying polynomial identities PI algebra. A natural question arises: give a quantitative description of how many identities such algebra A satisfies? We assume that A is associative, though the general approach below can be applied to nonassociative PI algebras as well. In order to overcome this difficulty we introduce the sequence of codimensions
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