MAT/02 – Algebra
The main target of the research is the study of the polynomial identities satisfied by an algebra over a field of characteristic zero, in particular the study of the T-ideals of the free associative algebra via combinatorial methods pertaining to the representation theory of the symmetric and general linear groups. The asymptotic calculation of the degrees of the irreducible representations of the symmetric group in characteristic zero is well known, and an analysis of the cocharacter decomposition of an algebra into irreducible characters for the symmetric group allows to obtain asymptotic evaluations that determine invariants of the corresponding varieties. In particular in order to get information about the polynomial identities satisfied by an algebra, one attaches to a T-ideal of polynomial identities some invariants such as the sequence of codimensions, the sequence of cocharacters and the sequence of colengths and through the study of their asymptotic behavior one obtains classification results of the corresponding varieties.
Keywords: Algebras with polynomial identities. Codimensions. Cocharacters. Colengths . Growth of varieties of associative and non-associative algebras.