2024/09/20 by Mats Boij, Boij, Mats, Anthony Iarrobino +3
Mathematics · #13E10 #14A05 #15A20 #Advanced Banach Space Theory #Advanced Topics in Algebra #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Holomorphic and Operator Theory #Primary: 15A27 #Secondary: 05A17
paper · pdf · doi:10.48550/arxiv.2409.13553
openalex publication_date 2024/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An n× n nilpotent matrix B is determined up to conjugacy by a partition PB of n, its Jordan type given by the sizes of its Jordan blocks. The Jordan type \mathfrak D(P) of a nilpotent matrix in the dense orbit of the nilpotent commutator of a given nilpotent matrix of Jordan type P is stable - has parts differing pairwise by at least two - and was determined by R. Basili. The second two authors, with B. Van Steirteghem and R. Zhao determined a rectangular table of partitions \mathfrak D-1(Q) having a given stable partition Q as the Jordan type of its maximum nilpotent commutator. They proposed a box conjecture, that would generalize the answer to stable partitions Q having ℓ parts: it was proven recently by J.~Irving, T. Košir and M. Mastnak. Using this result and also some tropical calculations, the authors here determine equations defining the loci of each partition in \mathfrak D-1(Q), when Q is stable with two parts. The equations for each locus form a complete intersection. The authors propose a conjecture generalizing their result to arbitrary stable Q.