Vanishing of Littlewood-Richardson polynomials is in P

TL;DR

This paper extends the polynomial-time decidability of the vanishing problem from Littlewood-Richardson coefficients to the more general Littlewood-Richardson polynomials, using polyhedral and combinatorial methods.

Contribution

It introduces a polytope construction and combines existing theorems and algorithms to prove the polynomial-time solvability for Littlewood-Richardson polynomials.

Findings

Vanishing of Littlewood-Richardson polynomials is in P.

Polytope construction using edge-labeled tableau rule.

Combines saturation theorem, reading order independence, and linear programming algorithms.

Abstract

J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algorithm for combinatorial linear programming.