Frobenius Algebras and the Symmetric Group

Abstract

The theory of groups supplies vector spaces, quantum numbers, and matrix elements for quantum mechanics. This chapter discusses finite groups, in particular point groups and symmetric group. The symmetric group is of central importance in the theory of systems with identical particles: electrons for atoms, molecules, and solids, nucleons for nuclei, and quarks for elementary particles. The symmetric group plays a major role in the theory of linear groups. The product spaces that are symmetry adapted to the symmetric group provide a basis for the irreducible representations of the general linear and unitary groups. For any finite group, there is defined a linear algebra, a finite-dimensional vector space for which the group supplies a basis. Finite group theory is presented in a nonalgebraic way as matrix representation theory.