Triple Quantum Schubert Calculus

Abstract

We prove a triple quantum Pieri formula: the product of a quantum double Schubert polynomial $\widetilde{\mathfrak{S}}_u(x;y)$ by a quantum factorial elementary polynomial $\widetilde{E}_{p,k}(x;z)$ in a second set of variables $z$ is a sum over chains in the quantum Bruhat graph, indexed by a quantum analogue of Sottile's relation, whose coefficients are monomials in the quantum parameters times polynomials in the differences $y_i-z_j$ with nonnegative integer coefficients, and which is Graham-positive at $z=y$. At $y=z=0$ it is the quantum Pieri rule of Ciocan-Fontanine and Postnikov; at $z=y$ it is an equivariant quantum Pieri rule for the complete flag variety, indexed by chains in the quantum Bruhat graph rather than, as in the rule of Huang and Li, by the data of a reduction to classical cohomology, and proved directly. A doubled Pieri formula, positive in the same senses, treats multiplication by the quantization of a product of two elementary polynomials in the same variables, which is not the product of the quantizations: it is two iterated Pieri steps with the incompatible pairs of chains omitted. More generally, the triple quantum Schubert structure constants $c_{u,v}^{w,d}(y;z)$, the coefficients of $\widetilde{\mathfrak{S}}_u(x;y)\widetilde{\mathfrak{S}}_v(x;z)$ in the basis $\{\widetilde{\mathfrak{S}}_w(x;y)\}$, are polynomials in the $y_i-z_j$ with nonnegative integer coefficients; the proof of this appears as an appendix. We extend the Peterson-Woodward comparison formula and the quantum-to-classical reduction of Leung-Li and Huang-Li to the triple constants, and we describe and prove correct the algorithms of schubmult, which compute products of arbitrary quantum double Schubert polynomials well beyond the reach of previous methods.