Prime ideals in the quantum grassmannian

S. Launois, T. H. Lenagan, L. Rigal

Research output: Contribution to journalArticlepeer-review

Abstract

We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a natural torus action of H = (k*)(n) on the quantum grassmannian O-q(G(m,n)(k)) and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the nonnegative cells in recent studies concerning the totally nonnegative grassmannian. Finally, we use the cell decomposition to establish that the quantum grassmannian satisfies normal separation and catenarity.

Original languageEnglish
Pages (from-to)697-725
Number of pages29
JournalSelecta Mathematica (New Series)
Volume13
Issue number4
DOIs
Publication statusPublished - May 2008

Keywords

  • quantum matrices
  • quantum grassmannian
  • quantum Schubert variety
  • quantum Schubert cell
  • prime spectrum
  • total positivity
  • ALGEBRAS
  • RINGS
  • DETERMINANTS
  • MATRICES
  • SPECTRA
  • CELLS

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