The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials

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 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 $a_w$ 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 an identity of Klyachko in particular.

A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula

“…We then show that as polynomials in q, A q c has nonnegative integer coefficients. A much more in-depth study of these polynomials is given by the authors in [35].…”

“…Such a Dyck path in turn determines a Ferrers diagram λ(c), and computing the q-hit polynomial as a weighted sum over all nonattacking placements in the r × r board where i rooks land inside λ(c) gives A 0 i c0 r−k−i (q). We address this connection in more detail in [35].…”

“…Our results in this section have been largely obtained by algebraic manipulations, and it makes sense to inquire about an interpretation for q-volumes that is of a geometric flavor. We give one in [35], interpreting the exponents of q as statistics on an interesting polyhedral decomposition of the permutahedron.…”

“…We postpone further study of these polynomials to a separate article [35], and proceed to make several remarks. It is worth noting that the nonnegativity of the coefficients of A c (q) does not immediately follow from the unique characterization above.…”

Positivity of Peterson Schubert Calculus

“…Acknowledgements: We thank Ana Bȃlibanu for explaining to us a proof of the irreducibility of Peterson variety, and Hiraku Abe and Tatsuya Horiguchi for explaining to us their results from [AFZ20, AHKZ21,Hor21]. LM would like to thank Sean Griffin for useful discussions and for pointing out the reference [NT21]. RG thanks Julianna Tymoczko for insightful conversations about Peterson varieties.…”