  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.

Combinatorics;Algebraic Geometry;Rings and Algebras