Journal of Commutative Algebra

Commuting nilpotent matrices and Artinian algebras

Roberta Basili, Anthony Iarrobino, and Leila Khatami

Full-text: Open access

Article information

J. Commut. Algebra, Volume 2, Number 3 (2010), 295-325.

First available in Project Euclid: 18 October 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A27: Commutativity
Secondary: 05E40: Combinatorial aspects of commutative algebra 06A11: Algebraic aspects of posets 13E10: Artinian rings and modules, finite-dimensional algebras 15A21: Canonical forms, reductions, classification 16S50: Endomorphism rings; matrix rings [See also 15-XX]

Nilpotent matrix commute Artin algebra weighted poset centralizer almost rectangular partition quiver


Basili, Roberta; Iarrobino, Anthony; Khatami, Leila. Commuting nilpotent matrices and Artinian algebras. J. Commut. Algebra 2 (2010), no. 3, 295--325. doi:10.1216/JCA-2010-2-3-295.

Export citation


  • I. Assem, D. Simson and A. Skowroński, Elements of the representation theory of associative algebras 1: Techniques of representation theory, London Math. Soc. Student Texts 65, Cambridge University Press, Cambridge, New York, 2006.
  • V. Baranovsky, The variety of pairs of commuting nilpotent matrices is irreducible, Transform. Groups 6 (2001), 3-8.
  • R. Basili, On the irreducibility of commuting varieties of nilpotent matrices, J. Algebra 268 (2003), 58-80.
  • R. Basili. Baranovsky and A. Iarrobino, Pairs of commuting nilpotent matrices, and Hilbert function, J. Algebra 320 (2008), 1235-1254.% ArXiv math.AC: 0709.2304.
  • ––––, An involution on $\N_B$, the nilpotent commutator of a nilpotent Jordan matrix $B$, preprint, 2009.
  • R. Basili. Baranovsky, A. Iarrobino and L. Khatami, Note on commuting nilpotent matrices, preprint, 2009.
  • R. Basili, T. Košir and P. Oblak, Some ideas from Ljubljana, July, 2008, preprint.
  • J. Briançon, Description de $\textHilb^n \c\ x,y\$, Invent. Math. 41 (1977), 45-89.
  • T. Britz and S. Fomin, Finite posets and Ferrers shapes, Advances Math. 158. 1 (2001), 86-127.
  • E.R. Gansner, Acyclic digraphs, Young tableaux and nilpotent matrices, SIAM J. Algebraic Discrete Methods 2 (1981), 429-440.
  • T. Harima and J. Watanabe, The commutator algebra of a nilpotent matrix and an application to the theory of commutative Artinian algebras, J. Algebra 319 (2008), %6, 2545-2570.
  • A. Iarrobino, Punctual Hilbert schemes, Amer. Math. Soc. Memoir 10, %#188 (1977), American Mathematical Society, Providence.
  • T. Košir and P. Oblak, On pairs of commuting nilpotent matrices, Transform. Groups 14 (2009), 175-182.
  • F.H.S. Macaulay, On a method for dealing with the intersection of two plane curves, Trans. Amer. Math. Soc. 5 %(4) (1904), 385-410.
  • P. Oblak, The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix, Linear Multilinear Algebra 56 (2008), 701-711. Slightly revised in ArXiv: math.AC/0701561.
  • D.I. Panyushev, Two results on centralisers of nilpotent elements, J. Pure Appl. Algebra 212 (2008), 774-779.
  • S. Poljak, Maximum rank of powers of a matrix of given pattern, Proc. Amer. Math. Soc. 106 (1989), 1137-1144.
  • A. Premet, Nilpotent commuting varieties of reductive Lie algebras, Invent. Math. 154 (2003), 653-683.
  • H.W. Turnbull and A.C. Aitken, An introduction to the theory of canonical matrices, Dover, New York, 1961.