Buch and Fulton  conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver cycle. Knutson, Miller and Shimozono  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  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 . The form of these analogues indicate that it should be interesting to pursue a geometric context that explains them.
- Component formula
- Degeneracy loci
- Generalized Littlewood-Richardson coefficients
- Quiver polynomials
ASJC Scopus subject areas
- Algebra and Number Theory
- Discrete Mathematics and Combinatorics