On varieties of commuting nilpotent matrices
Nham V. Ngo, Klemen \v{S}ivic

TL;DR
This paper investigates the geometric structure of varieties of commuting nilpotent matrices, establishing conditions for their irreducibility or reducibility across different dimensions and matrix sizes.
Contribution
It proves that the variety of commuting nilpotent matrices is reducible for all dimensions and sizes greater than or equal to 4, and identifies cases of irreducibility for small sizes.
Findings
N(d,n) is reducible for all d,n ≥ 4
N(3,n) is irreducible for n ≤ 6
N(d,n) is irreducible for small sizes, reducible for larger ones
Abstract
Let be the variety of all -tuples of commuting nilpotent matrices. It is well-known that is irreducible if , if or if and . On the other hand is known to be reducible for . We study in this paper the reducibility of for various values of and . In particular, we prove that is reducible for all . In the case , we show that it is irreducible for .
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Finite Group Theory Research
