
Connections to Schubert Calculus Learning Seminar
Symmetrized Matroid Classes
Expressing combinatorial invariants of matroids as intersection numbers on algebraic varieties has become a popular tool in algebraic combinatorics. Several conjectured inequalities among combinatorial data can be traced back to positivity results for nef divisors in algebraic geometry in this way. As a special instance of this phenomenon we consider intersection numbers of matroid Chow classes with a certain set of nef divisors on the permutohedral variety. During this talk we will argue that by symmetrizing the matroid classes, one can cleanly extract the combinatorial data contained in them as the valuative data of the matroid. As a result we can reduce the problem of computing matroidal mixed Eulerian numbers to computing the g-invariant of a matroid and computing much simpler usual mixed Eulerian numbers. To arrive at this observation we present several different sets of divisors on the permutohedral variety which naturally arise in a geometric way.