Katharina Jochemko: Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity
Time: Wed 2019-10-09 10.15 - 11.00
Lecturer: Katharina Jochemko
Abstract: Generalized permutahedra form a combinatorially rich class of polytopes
that naturally appear in many areas of mathematics such as
combinatorics, geometry, optimization and statistics. They comprise many
important classes of polytopes, for example, matroid polytopes. We study
functions on generalized permutahedra that behave linearly with respect
to dilation and taking Minkowski sums. We give a complete classification
of all positive, translation-invariant Minkowski linear functionals on
permutahedra that are invariant under permutations of the coordinates:
they form a simplicial cone and we explicitly describe the generators.
We apply our results to prove that the linear coefficients of Ehrhart
polynomials of generalized permutahedra are nonnegative, verifying
conjectures of De Loera-Haws-Koeppe (2009) and Castillo-Liu
(2018) in this case. This is joint work with Mohan Ravichandran.