On combinatorics of quiver component formulas

TL;DR

This paper offers an alternative combinatorial proof of the component formula for quiver varieties, confirming the nonnegativity of quiver coefficients and extending results to type BCD-Schubert polynomials.

Contribution

It provides a new combinatorial proof of the component formula, linking it to Schubert polynomial splitting formulas and extending these results to types B, C, D.

Findings

Confirmed nonnegativity of quiver coefficients.

Established combinatorial proof via Grobner basis substitution.

Extended splitting formulas to type BCD-Schubert polynomials.

Abstract

Buch and Fulton conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver variety. 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 Grobner degeneration. 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.