Trace identities and Invariant theory

• Grant number: 23/01635-1
• Support Opportunities: Scholarships in Brazil - Master
• Start date: May 01, 2023
• End date: February 28, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Gabriel Guzatti Vilas Boas
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 18/23690-6 - Structures, representations, and applications of algebraic systems, AP.TEM

Abstract

In this research project towards MSc degree in Mathematics we study certain classical topics of the theory of algebras with polynomial identities. We shall start with a brief overview of polynomial identities, PI algebras, varieties of algebras, T-ideals, linearization, homogeneous and multilinear structure of relatively free algebras, representation theory of the symmetric and of the general linear groups. In this part of the project we will greatly benefit from the years of additional studies during the student's undergraduate research initiation funded by this Foundation. Afterwards we shall study the generic matrices and their fundamental properties. Such a study will lead us in a natural way to the notion of an action of a group on the $d$-tuples of generic matrices, and as a consequence, to the notion of an invariant of an action. We shall record, from the point of view of Invariant theory, the classical theorem due to Newton concerning ring of symmetric polynomials and the Newton's formulae that relate the elementary symmetric functions to the power sums. Afterwards we shall make a brief detour towards some notions concerning central simple algebras and the Skolem and Noether's theorem. Then we shall study the action of the general linear group, by conjugation, on $d$-tuples of generic matrices, and the induced action on the polynomial ring in the entries of these generic matrices. Following Procesi's fundamental works we shall study the description of the invariants of such an action. In this way we shall get the first and the second fundamental theorems of Invariant theory of matrices. As a consequence we shall obtain the description of the trace identities sartisfied by the matrix algebra of order $n$. Finally we shall study applications to the identities with trace and involution, and we shall see the theorem due to Dubnov, Ivanov, Nagata, and Higman, as well as the lower and upper bounds for the nilpotence index, obtained respectively by Kuzmin and by Razmyslov.