A combinatorial divisibility question from noncommutative algebra

TL;DR

This paper proposes a conjecture linking divisibility properties of Kostka numbers to noncommutative algebra problems, supported by connections to Schubert calculus and partial proofs in specific cases.

Contribution

It introduces a new conjecture connecting combinatorial Kostka numbers with noncommutative algebra, providing partial proofs and motivating evidence through Schubert calculus.

Findings

Conjecture relating Kostka numbers and noncommutative algebra divisibility.

Partial proof of the conjecture in specific cases.

Connection established via Schubert calculus.

Abstract

We present a general conjecture on the divisibility of a certain expression in terms of Kostka numbers and their close variants. This conjecture is closely related to a variant of the period-index problem of noncommutative algebra, with partial implications in both directions. We present a description of the connection between these two problems via Schubert calculus as motivation and evidence for the conjecture before turning to a proof of the conjecture in a family of cases.