Mixed Eulerian numbers and matroids

Expressing combinatorial data as intersection numbers on algebraic varieties has become a popular tool in algebraic combinatorics. One particularly useful setting for the study of matroids is the permutohedral variety. We describe three families of divisors naturally arising from the geometry of this variety and discuss how they intersect amongst each other and with matroid classes. As a result we deduce explicit and recursive formulas for mixed volumes of hypersimplices and their generalisation to matroids. To get a grasp on the combinatorial data contained in these intersection numbers we consider the symmetrization map on the Chow ring of the permutohedral variety. This turns out to map a (loopless) matroid class to the G-invariant of the matroid: a classical universal valuative invariant.

Parts of this talk are based on joint work with Gaku Liu and Mateusz Michalek, see arXiv:2502.04980