On combinatorics of quiver component formulas

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Buch and Fulton [9] conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver cycle. Knutson, Miller and Shimozono [24] proved this conjecture as an immediate consequence of their "component formula". We present an alternative proof of the component formula by substituting combinatorics for Gröbner degeneration [23, 24]. We relate the component formula to the work of Buch, Kresch, Tamvakis and the author [10] where a "splitting" formula for Schubert polynomials in terms of quiver coefficients was obtained. We prove analogues of this latter result for the type BCD-Schubert polynomials of Billey and Haiman [4]. The form of these analogues indicate that it should be interesting to pursue a geometric context that explains them.

Original languageEnglish (US)
Pages (from-to)351-371
Number of pages21
JournalJournal of Algebraic Combinatorics
Issue number3
StatePublished - May 2005
Externally publishedYes


  • Component formula
  • Degeneracy loci
  • Generalized Littlewood-Richardson coefficients
  • Quiver polynomials

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics


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