Some remarks on varieties of pairs of commuting upper triangular matrices and an interpretation of commuting varieties

TL;DR

This paper investigates the structure of varieties of pairs of commuting upper triangular matrices, revealing bounds on when these varieties are complete intersections and providing geometric interpretations involving Grassmannians.

Contribution

It establishes a finite bound for the non-complete intersection property and introduces a geometric map relating commuting matrices to Grassmannian subvarieties.

Findings

Existence of a finite bound m < 18 for non-complete intersection cases.

Characterization of the variety of pairs of commuting strictly upper triangular matrices.

Definition of a natural map linking commuting matrices to Grassmannian subvarieties.

Abstract

It is known that the variety of pairs of n x n commuting upper triangular matrices isn't a complete intersection for infinitely many values of n; we show that there exists m such that this happens if and only if n > m. We also show that m < 18 and that it could be found by determining the dimension of the variety of pairs of commuting strictly upper triangular matrices. Then we define a natural map from the variety of pairs of commuting n x n matrices onto a subvariety defined by linear equations of the grassmannian of subspaces of codimension 2 of a vector space of dimension n x n.