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

Abstract

There is a striking similarity between Macdonald's reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety Permn as computed by Klyachko. Toward understanding this better, we undertake an in-depth study of a q-deformation of the Sn-invariant part of the rational cohomology ring of Permn, which we call the q-Klyachko algebra. We uncover intimate links between expansions in the basis of squarefree monomials in this algebra and various notions in algebraic combinatorics, thereby connecting seemingly unrelated results by finding a common ground to study them. Our main results are as follows.• A q-analog of divided symmetrization (q-DS) using Yang-Baxter elements in the Hecke algebra.It is a linear form that picks up coefficients in the squarefree basis. • A relation between q-DS and the ideal of quasisymmetric polynomials involving work of Aval-Bergeron-Bergeron. • A family of polynomials in q with nonnegative integral coefficients that specialize to Postnikov's mixed Eulerian numbers when q = 1. We refer to these new polynomials as remixed Eulerian numbers. For q > 0, their normalized versions occur as probabilities in the internal diffusion limited aggregation (IDLA) stochastic process. • A lift of Macdonald's reduced word identity in the q-Klyachko algebra.• The Schubert expansion of the Chow class of the standard split Deligne-Lusztig variety in type A, when q is a prime power.

Remixed Eulerian numbers

“…The reader will note the similarity between the perspectives that follow: they are indeed easily seen to be equivalent. In contrast, the fact that any of these definitions is equivalent to Definition 1.2 is not obvious, and this was a key result in [17].…”

“…Some of these properties were already given in [17], while the others will be proved at various locations in the text: (1) is [17,Proposition 5.4]. ( 2) is Lemma 5.1.…”

“…This article studies a large family of polynomials, the remixed Eulerian numbers, which were introduced in a previous work of the authors [17]. The terminology follows that of Postnikov [19, §16] where mixed Eulerian numbers were introduced.…”

“…For (4), we refer to Section 6. (5) follows from the sum rule [17,Proposition 5.4], as explained in Remark 2.1. Finally, (7) is part of [17,Theorem 4.8].…”

“…Outline of the article. We recall some alternative definitions of A c (q) in §2; these have already appeared in [17]. The last one is probabilistic in nature, and is used in §3 to give a first combinatorial interpretation of A c (q).…”