On balanced systems of idempotents

By definition, a balanced basis of an associative semisimple finite-dimensional algebra over the field of complex numbers $\mathbb C$ is a system of idempotents $\{e_i\}$ such that it forms a linear basis and the $\operatorname{Tr}e_i$ and $\operatorname{Tr}e_ie_j$ are independent of $i$, $j$, $i\ne j$, where $\operatorname{Tr}$ is the trace of the regular representation of the algebra. In the present paper balanced bases are constructed in the matrix algebra $\mathrm M_{p^n}(\mathbb C)$, where $p$ is an odd prime. For matrix algebras such bases have so far been known only in the cases $\mathrm M_2(\mathbb C)$ and $\mathrm M_3(\mathbb C)$. It is proved that there are no balanced bases of certain ranks having a regular elementary Abelian 2-group of automorphisms in the algebras $\mathrm M_{2^n}(\mathbb C)$, $n>1$. In addition, the balanced 1-systems of $n+1$ idempotents of rank $r$ in the algebra $\mathrm M_{rn}(\mathbb C)$ are classified.