The Kostka semigroup and its Hilbert basis

TL;DR

This paper investigates the structure of the Kostka semigroup, introduces new graph-based criteria for membership, and characterizes extremal rays, providing insights into its algebraic and combinatorial properties.

Contribution

It introduces KGR graphs and conservative subtrees as criteria for Kostka semigroup membership and classifies extremal rays of the related polyhedral cone.

Findings

Membership decision is NP-complete.

Partitions in the Hilbert basis are at most r wide.

Extremal rays correspond to a subset of the Hilbert basis.

Abstract

The Kostka semigroup consists of pairs of partitions with at most r parts that have positive Kostka coefficient. For this semigroup, Hilbert basis membership is an NP-complete problem. We introduce KGR graphs and conservative subtrees, through the Gale-Ryser theorem on contingency tables, as a criterion for membership. In our main application, we show that if a partition pair is in the Hilbert basis then the partitions are at most r wide. We also classify the extremal rays of the associated polyhedral cone; these rays correspond to a (strict) subset of the Hilbert basis. In an appendix, the second and third authors show that a natural extension of our main result on the Kostka semigroup cannot be extended to the Littlewood-Richardson semigroup. This furthermore gives a counterexample to a recent speculation of P. Belkale concerning the semigroup controlling nonvanishing conformal blocks.