Totally nonnegative cells and matrix Poisson varieties

K. R. Goodearl, S. Launois, T. H. Lenagan

Research output: Contribution to journalArticlepeer-review

Abstract

We describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices, that is, the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by A. Postnikov. In order to do this, we relate the totally nonnegative cells to torus orbits of symplectic leaves of the Poisson varieties of complex matrices. In particular, we describe the minors that vanish on a torus orbit of symplectic leaves, we prove that such families of minors are exactly the admissible families, and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices.

Original languageEnglish
Pages (from-to)779-826
Number of pages48
JournalAdvances in Mathematics
Volume226
Issue number1
DOIs
Publication statusPublished - 15 Jan 2011

Keywords

  • Totally nonnegative matrices
  • Totally nonnegative cells
  • Admissible families of minors
  • Symplectic leaves
  • Torus orbits
  • INVARIANT PRIME IDEALS
  • AFFINE SPACES
  • ALGEBRAS

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