Mathematical Sciences: PI Algebras in Prime Characteristics

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This project is concerned with applying modular techniques to the study of P.I. algebras over base fields of arbitrary characteristic. The principal investigator will study the identities of P.I. algebras in the various degrees when the characteristic of the base field is either zero or positive. New results and techniques combine these two cases and relate them to a classical problem of Procesi about polynomials over the integers whose values on generic matrices are divisible by p. The main tools in this study are combinatorics and group representation theory. In this project, the structure theory of algebras that satisfy polynomial identities will be examined with applications to combinatorics and the representation theory of the symmetric groups and the classical linear groups. An example of an algebra that satisfies a polynomial identity is given by the ring of matrices of a given dimension over a field. Indeed often these algebras can be represented as rings of matrices. This work has important consequences for several different areas of mathematics.