The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials

TL;DR

This paper computes the cohomology class expansion of the permutahedral variety in Schubert classes, revealing positive structure constants linked to mixed Eulerian numbers, and provides combinatorial interpretations and invariance properties.

Contribution

It introduces a new expression for structure constants using mixed Eulerian numbers, proves their positivity, invariance, and cyclic sum rules, and offers combinatorial interpretations for vexillary permutations.

Findings

Structure constants expressed as sums of normalized mixed Eulerian numbers.

Proved positivity of structure constants for all permutations.

Established combinatorial interpretations for vexillary permutations.

Abstract

We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants $a_w$ are expressed as a sum of \emph{normalized} mixed Eulerian numbers indexed naturally by reduced words of $w$. The description implies that the $a_w$ are positive for all permutations $w\in S_n$ of length $n-1$, thereby answering a question of Harada, Horiguchi, Masuda and Park. We use the same expression to establish the invariance of $a_w$ under taking inverses and conjugation by the longest word, and subsequently establish an intriguing cyclic sum rule for the numbers. We then move toward a deeper combinatorial understanding for the $a_w$ by exploiting in addition the relation to Postnikov's divided symmetrization. Finally, we are able to give a combinatorial interpretation for $a_w$ when $w$ is vexillary, in terms of certain tableau descents. It is based in part on a relation between the numbers $a_w$ and principal specializations of Schubert polynomials. Along the way, we prove results and raise questions of independent interest about the combinatorics of permutations, Schubert polynomials and related objects. We also sketch how to extend our approach to other Lie types, highlighting in particular an identity of Klyachko.