Log-concavity of matroid h-vectors and mixed Eulerian numbers

Abstract

For any matroid M , we compute the Tutte polynomial T M (x, y) using the mixed intersection numbers of certain tautological classes in the combinatorial Chow ring A • (M ) arising from Grassmannians. Using mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the h-vector of a matroid complex, improving on an old conjecture of Dawson whose proof was announced recently by Ardila, Denham and Huh.

Mixed Eulerian numbers and Peterson Schubert calculus

“…from ( 1) and (3). We remark that a similar computation was described in [10] in terms of matroids when Φ is of type A. Schubert calculus on Peterson variety was developed by Harada and Tymoczko ([20]). We call it Peterson Schubert calculus (see Section 7 for details).…”

“…Remark 5.5. A similar computation for the mixed Eulerian numbers was described in terms of matroids in [10].…”

“…Furthermore, these numbers are related to the intersection number of the toric variety associated with a weight polytope and the Schubert variety. We refer to [10,14,26,28,30].…”

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

“…It can be shown that it suffices to check the equality in Theorem 4.11 for such f . 5 Now on the one hand, S w 1 n computes certain intersection numbers a w for the permutahedral variety, as explained in [34], following [3,23]. On the other hand, work of Klyachko [24,25] mentioned in the introduction shows that these same numbers a w can be computed in the ring K 1 n H * (Perm n , Q) Sn by the formula (n−1)!×Top n (S w ).…”

“…The permutahedral variety in type A has garnered a lot of attention recently, especially since the study of its cohomology ring plays a key role in the dramatic recent resolutions of various questions concerning log-concavity of various polynomials arising in combinatorics; see for instance [5,22].…”