ELA, Volume 13, pp. 274-343, November 2005, abstract.

Commuting Triples of Matrices

Yongho Han

The variety C(3,n) of commuting triples of n-by-n 
matrices over the complex numbers is shown to be 
irreducible for n = 7. It had been proved that C(3,n) 
is reducible for n at least 30, but irreducible for
n at most 6. Guralnick and Omladic have conjectured 
that it is reducible for n > 7.