Sums of Schubert structure constants with bounded Coxeter length

TL;DR

This paper provides a conceptual proof for the polynomial nature of sums of Schubert structure constants with bounded inversions, extending the result to all classical Lie types.

Contribution

It offers a new, conceptual proof of Pak-Robichaux's theorem and generalizes the result to all classical Lie types.

Findings

Sum of Schubert structure constants with bounded inversions are polynomial.

The proof computes the leading term of these sums.

Extension of results to all classical Lie types.

Abstract

Pak-Robichaux recently introduced a signed puzzle rule for Schubert structure constants, which they use to show that sums $\gamma_k(n)$ of these constants with a bounded number of inversions are polynomial. We give a different, conceptual proof of their theorem. Our argument computes the lead term of $\gamma_k(n)$ and extends to all classical Lie types.