2013/08/11 by Robert M. Guralnick, Guralnick, Robert M., Nham V. Ngo +1 · 1 citation
Mathematics · #14L30 #20G05 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:14L30 #msc:20G05
paper · pdf · doi:10.48550/arxiv.1308.2420
8 pages
arxiv created 2013/08/11 · openalex publication_date 2013/08/11 · arxiv updated 2013/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \Nn be the set of nilpotent n by n matrices over an algebraically closed field k. For each r≥ 2, let Cr(\Nn) be the variety consisting of all pairwise commuting r-tuples of nilpotent matrices. It is well-kown that C2(\Nn) is irreducible for every n. We study in this note the reducibility of Cr(\Nn) for various values of n and r. In particular it will be shown that the reducibility of Cr(\mathfrakgln), the variety of commuting r-tuples of n by n matrices, implies that of Cr(\Nn) under certain condition. Then we prove that Cr(\Nn) is reducible for all n, r≥ 4. The ingredients of this result are also useful for getting a new lower bound of the dimensions of Cr(\Nn) and Cr(\mathfrakgln). Finally, we investigate values of n for which the variety C3(\Nn) of nilpotent commuting triples is reducible.