Geometry of varieties of mutually orthogonal matrices

A. E. Guterman ; S. A. Zhilina ; K. D. Mukhanov

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVII, Tome 534 (2024), pp. 5-34

Résumé

For the ring of square matrices $\mathrm{Mat}_n(\Bbbk)$ of order $n$ over a field $\Bbbk$, one can construct the orthogonality graph $\operatorname{O}(\mathrm{Mat}_n(\Bbbk))$, whose vertices are the zero divisors of the ring $\mathrm{Mat}_n(\Bbbk)$. Two vertices $A$ and $B$ are connected by an edge if $AB=BA=0$. The notion of the distance between two elements of the ring naturally implies that one can consider the set $\operatorname{O}^d_n$ of pairs of elements lying within the distance at most $d$. It is proved that such sets form affine algebraic varieties, a decomposition of these varieties into irreducible components is provided, and their dimensions are calculated. The paper also describes the sets that are defined similarly for the ring of upper triangular matrices and suggests generalizations of these results to arbitrary finite-dimensional algebras.