A rim-hook rule for quiver flag varieties

Abstract

The Abelian/non-Abelian correspondence for cohomology (Martin in Symplectic quotients by a nonabelian group and by its maximal torus. arXiv:math/0001002 [math.SG], 2000; Ellingsrud–Strømme in On the chow ring of a geometric quotient, 1989) gives a novel description of the cohomology ring of the Grassmannian. We show that the natural generalization of this result to small quantum cohomology applies to Fano quiver flag varieties. Quiver flag varieties are generalisations of type A flag varieties. As a corollary, we see that the Gu–Sharpe mirror to a Fano quiver flag variety computes its quantum cohomology. The second focus of the paper is on applying this description to computations inside the classical and quantum cohomology rings. 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 to prove a rim-hook removal rule for the cohomology and quantum cohomology (in the Fano case) of quiver flag varieties. This result is new even in the flag case. This gives an effective way of computing products in the (quantum) cohomology ring, reducing computations to that in the cohomology ring of the Grassmannian.