Pairs of commuting nilpotent matrices, and Hilbert function

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dc.creatorBasili, Roberta
dc.creatorIarrobino, Anthony
dc.date2007-09-14
dc.date2008-02-09
dc.date.accessioned2026-07-07T09:19:23Z
dc.date.available2026-07-07T09:19:23Z
dc.descriptionLet K be an infinite field and denote by H(n,K) the family of pairs (A,B) of commuting nilpotent n by n matrices with entries in K. There has been substantial recent study of the connection between H(n,K) and the fibre H[n] of the punctual Hilbert scheme of the plane, over an n-fold point of the symmetric product, by V. Baranovsky, R. Basili, and A. Premet. We study the stratification of H(n,K) by the Hilbert function of the Artinian ring K[A,B]. We show that when dim_K K[A,B] = n, then the generic element of the pencil A+λB, λ\in K, has Jordan partition the maximum partition P(H) whose diagonal lengths are the Hilbert function of K[A,B]. We denote by Q(P) the maximum Jordan partition of a nilpotent A commuting with a nilpotent B of Jordan partition P. We show that the stable partitions - those such that Q(P)=P - are those whose parts differ by at least two. In characteristic zero, the latter is a special case of a result of D. Panyushev. Our result on pencils shows that Q(P) has decreasing parts. In related work, T. Kosir and P. Oblak have shown further that Q(P) is itself stable.
dc.descriptionSubstantial revision after referee and other comments. The main results are the same but there are also new results and examples. Proofs are often rewritten for clarity, more detailed, and in one case, corrected. 17 pages
dc.identifierhttps://arxiv.org/abs/0709.2304
dc.identifierhttp://arxiv.org/abs/0709.2304
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/154369
dc.subjectCommutative Algebra
dc.subjectAlgebraic Geometry
dc.subject13D40; 14C05; 15A22
dc.titlePairs of commuting nilpotent matrices, and Hilbert function
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