The zrank conjecture and restricted Cauchy matrices

Guo Guang Yan, Arthur L.B. Yang, Joan J. Zhou

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

The rank of a skew partition λ/μ, denoted rank(λ/μ), is the smallest number r such that λ/μ is a disjoint union of r border strips. Let sλ/μ(1t) denote the skew Schur function sλ/μ evaluated at x1 = ⋯ = xt = 1, xi = 0 for i > t. The zrank of λ/μ, denoted zrank(λ/μ), is the exponent of the largest power of t dividing sλ/μ(1t). Stanley conjectured that rank(λ/μ) = zrank(λ/μ). We show the equivalence between the validity of the zrank conjecture and the nonsingularity of restricted Cauchy matrices. In support of Stanley's conjecture we give affirmative answers for some special cases.

Original languageEnglish (US)
Pages (from-to)371-385
Number of pages15
JournalLinear Algebra and Its Applications
Volume411
Issue number1-3
DOIs
StatePublished - Dec 1 2005
Externally publishedYes

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Keywords

  • Border strip decomposition
  • Interval sets
  • Outside decomposition
  • Rank
  • Reduced code
  • Restricted Cauchy matrix
  • Snakes
  • Zrank

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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