Many combinatorial objects, such as matroids, graphs, and posets, can be realized as generalized permutahedra - a beautiful family of polytopes. This realization respects the natural multiplication of these objects as well as natural "breaking" operations. Surprisingly many of the important invariants of these objects, when viewed as functions on polytopes, satisfy an inclusion-exclusion formula with respect to subdivisions. Functions that satisfy this formula are known as valuations. In this talk, I will discuss recent work with Federico Ardila that completely describes the relationship between the algebraic structure on generalized permutahedra and valuations. Our main contribution is a new easy-to-apply method that converts simple valuations into more complicated ones. This method unifies many combinatorial and algebro-geometric valuations on matroid as well as giving new valuations on posets and building sets.
Speaker(s): Mario Sanchez (UC Berkeley)