Journal of Algebraic Combinatorics

, Volume 2, Issue 2, pp 111–121 | Cite as

Combinatorics of Maximal Minors

  • David Bernstein
  • Andrei Zelevinsky


We continue the study of the Newton polytope ∏m,n of the product of all maximal minors of an m × n-matrix of indeterminates. The vertices of ∏m,n are encoded by coherent matching fields Λ = (Λσ), where σ runs over all m-element subsets of columns, and each Λσ is a bijection σ → [m]. We show that coherent matching fields satisfy some axioms analogous to the basis exchange axiom in the matroid theory. Their analysis implies that maximal minors form a universal Gröbner basis for the ideal generated by them in the polynomial ring. We study also another way of encoding vertices of ∏m,n for mn by means of “generalized permutations”, which are bijections between (nm + 1)–element subsets of columns and (nm + 1)–element submultisets of rows.

matching field Newton polytope maximal minor 


  1. 1.
    B. Sturmfels and A. Zelevinsky, “Maximal minors and their leading terms,” Advances in Math, 98 (1993), 65–112.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • David Bernstein
    • 1
  • Andrei Zelevinsky
    • 1
  1. 1.Northeastern UniversityBoston

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