On Faces and Hilbert Bases of Kostka Cones

Abstract

The $r$-Kostka cone is the real polyhedral cone generated by pairs of partitions with at most $r$ parts such that the corresponding Kostka coefficient is nonzero. We provide several results showing that its faces have interesting structural and enumerative properties. We show that, for fixed $d$, the number of $d$-faces of the $r$-Kostka cone is a polynomial in $r$ with a positive integer expansion in the binomial basis, and we provide exact formulas for $d \leq 4$. We prove that the maximum number of extremal rays in a $d$-face stabilizes to some explicit constant as $r$ increases. We then work towards a generalization of the Gao-Kiers-Orelowitz-Yong Width Bound.