**Problem of the Week**

**BUGCAT**

**Zassenhaus Conference**

**Hilton Memorial Lecture**

**BingAWM**

**Math Club**

You are here: Homepage » Seminars - Academic year 2023-24 » Combinatorics Seminar » Lou Billera (Cornell)

seminars:comb:abstract.200611bil

I describe a new isomorphism invariant of matroids that is a quasisymmetric function. This invariant

- defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients,
- is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid,
- is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight,
- behaves simply under matroid duality,
- has a simple expansion in terms of P-partition enumerators, and
- is a valuation on decompositions of matroid base polytopes.

This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. From work of Lafforgue, the lack of such a decomposition implies the matroid has only a finite number of vector representations up to projective equivalence.

This is joint work with Ning Jia and Victor Reiner.

The paper is accessible at http://arxiv.org/abs/math.CO/0606646.

seminars/comb/abstract.200611bil.txt · Last modified: 2020/01/29 14:03 (external edit)

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Noncommercial-Share Alike 3.0 Unported