A flag refinement of the $h^*$-formula for the hypersimplices

Source abstract

The hypersimplex $Δ_{k,n}$ is the convex hull of all 0/1-vectors of length $n$ with coordinate sum $k$. Early conjectured, and Kim proved, a combinatorial formula for the $h^*$-polynomial of $Δ_{k,n}$ in terms of hypersimplicial decorated ordered set partitions. In this paper, we refine the Early--Kim formula to the flag $h$-numbers of the alcove triangulation of $\hyp$. We also give a representation-theoretic interpretation of its flag $f$-numbers in terms of Young permutation modules.