Primitive Idempotents: A Crucial Role in the Representation Theory of Finite Groups and Finite-Dimensional Algebras

In this paper we present explicit formulas for primitive idempotent in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalization of the well known Frobenius-Schur relations for semisimple algebras.