Balanced systems of primitive idempotents in matrix algebras

The article develops the concept of balanced $t$-systems of idempotents in associative semisimple finite-dimensional algebras over the field of complex numbers $\mathbb C$ this was introduced by the author as a generalization of the concept of combinatorial $t$-schemes, which in this context corresponds to the case of commutative algebras. Balanced 2-systems are considered consisting of $v$ primitive idempotents in the matrix algebra $\mathrm M_n(\mathbb C)$, known as $(v,n)$-systems. It is proved that $(n+1,n)$-systems are unique and it is shown that there are no $(n+s,n)$-systems with $n>s^2-s$ or $s>n^2-n$. The $(q+1,n)$-systems having 2-transitive automorphism subgroup $PSL(2,q)$, $q$ odd, are classified. The (4,2)- and (6,3)-systems are classified. A balanced basis is constructed in the algebras $\mathrm M_n$, $n=2,3$. Connections are established between conference matrices and $(2n,n)$-systems, and between suitable matrices and $(m^2,\dfrac{m^2\pm m}2)$-systems.