On Faces and Hilbert Bases of Kostka Cones

Kostka coefficients appear in the representation theory of the general linear group and enumerate semistandard Young tableaux of fixed shape and content. The $r$-Kostka cone is the real polyhedral cone generated by pairs of partitions with at most $r$ parts, written as non-increasing $r$-tuples, 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 the $d$-faces of the $r$-Kostka cone can be determined from those of the $(3d+3)$-Kostka cone, allowing us to characterize its $2$-faces and enumerate its $d$-faces for $d \leq 4$. We provide tight asymptotics for the number of $d$-faces for arbitrary $d$ and determine the maximum number of extremal rays contained in a $d$-face for $d < r$. We then make progress towards a generalization of the Gao-Kiers-Orelowitz-Yong Width Bound on initial entries of partitions $(\lambda,\mu)$ appearing in the Hilbert basis of the $\lambda_1$-Kostka cone. We show that at least $93.7\%$ of integer pairs $\lambda_1 \geq \mu_1 > 0$ appear as the initial entries of partitions $(\lambda,\mu)$ comprising a Hilbert basis element of the $r$-Kostka cone for every $r > \lambda_1$. We conclude with a conjecture about a curious $h$-vector phenomenon.