### Abstract

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 language | English (US) |
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Pages (from-to) | 351-371 |

Number of pages | 21 |

Journal | Journal of Algebraic Combinatorics |

Volume | 21 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2005 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics