Irreducible Characters of Hadamard Algebras

The notion of Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classical Hadamard matrix, which corresponds to the case of commutative algebras. The algebras admitting Hadamard decompositions are called Hadamard algebras. A relation for the values of an irreducible character of a Hadamard algebra on the products of involutions forming an orthogonal basis of the algebra is obtained. This relation is then applied to describe the Hadamard decompositions in an algebra of dimension 8.