On Faces and Hilbert Bases of Kostka Cones

Abstract

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.