Quasi-orthogonal subalgebras of matrix algebras

TL;DR

This paper explores the existence and structure of quasi-orthogonal subalgebras within matrix algebras, demonstrating their abundance and spanning properties in specific prime power matrix spaces.

Contribution

It establishes the existence of a large family of quasi-orthogonal subalgebras in matrix algebras of prime power dimensions and shows they span the entire algebra.

Findings

Existence of p^{2kn}-1/p^{2k}-1 such subalgebras

Subalgebras are isomorphic to M_{p^{k}}

These subalgebras span M_{p^{kn}}

Abstract

We investigate pairwise quasi-orthogonal subalgebras in $M_{p^{kn}}$ which are isomorphic to $M_{p^{k}}$ for $k \ge 1$, $n \ge 2$ and a prime number $p$ with $p \ge 3$. We prove there exist $p^{2kn}-1/p^{2k}-1$ such subalgebras and they span $M_{p^{kn}}$.