Degrees of Irreducible Characters and Dimensions of Hadamard Algebras

The notion of Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classical Hadamard matrix corresponding to the case of commutative algebras. The algebras admitting a Hadamard decomposition are referred to as Hadamard algebras. We study the conjecture claiming that, if a Hadamard algebra is not simple and has an irreducible character of degree $m\ge 2$, then the dimension of the algebra is not less than $2m^2$. The validity of this conjecture is confirmed for the first two values $m=2$ and $m=4$ (here $m$ must be even). Moreover, we prove a result (which is weaker than the conjecture) in which $2m^2$ is replaced by $m^2+2m$.