  The rim-hook rule for quantum cohomology of the Grassmannian allows one to
reduce quantum calculations to classical calculations in the cohomology of the
Grassmannian. We use the Abelian/non-Abelian correspondence for cohomology to
prove a rim-hook removal rule for the cohomology of quiver flag varieties.
Quiver flag varieties are generalisations of type A flag varieties; this result
is new even in the flag case. This gives an effective way of computing products
in their cohomology, reducing computations to that in the cohomology ring of
the Grassmannian. We then prove a quantum rim-hook rule for Fano quiver flag
varieties (including type A flag varieties). As a corollary, we see that the
Gu--Sharpe mirror to a Fano quiver flag variety computes its quantum
cohomology.
