Quantum hooks and the Plücker coordinate mirror - Elana Kalashnikov

Title: Quantum hooks and the Plücker coordinate mirror

Abstract: There is a natural map from the symmetric polynomial ring in r_1 variables to the quantum cohomology ring of a type A flag variety Fl(n,r_1,..r_k), given by evaluating Schur polynomials in the Chern roots of the first tautological bundle. I’ll explain how for a large class of Schur polynomials, the result is a Schubert class that can be obtained by dividing the partition into a quantum-hook and smaller partitions. Surprisingly, this is the key result proving a mirror theorem for type A flag varieties. A function W is a mirror of a Fano variety X if enumerative information of X can be determined from W: for example, the Jacobi ring of W should be  the quantum cohomology ring of X. Mirrors for Fano toric varieties are well-understood; and more recently Plücker coordinate mirrors have been proposed for a variety of homogeneous spaces. We use quantum hooks to prove that the Plücker coordinate mirror of the flag variety computes quantum cohomology relations. This is joint work with Linda Chen.