A flag refinement of the $h^{\ast }$-formula for the hypersimplices

The hypersimplex ${\mathrm{\Delta }}_{k,n}$ 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 $h^{\ast }$-polynomial of ${\mathrm{\Delta }}_{k,n}$ 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 $\text{hyp}$. We also give a representation-theoretic interpretation of its flag $f$-numbers in terms of Young permutation modules.