Balanced systems of primitive idempotents in matrix algebras

Abstract

The article develops the concept of balanced -systems of idempotents in associative semisimple finite-dimensional algebras over the field of complex numbers this was introduced by the author as a generalization of the concept of combinatorial -schemes, which in this context corresponds to the case of commutative algebras. Balanced 2-systems are considered consisting of primitive idempotents in the matrix algebra , known as -systems. It is proved that -systems are unique and it is shown that there are no -systems with or . The -systems having 2-transitive automorphism subgroup , odd, are classified. The (4,2)- and (6,3)-systems are classified. A balanced basis is constructed in the algebras , . Connections are established between conference matrices and -systems, and between suitable matrices and -systems.