An affine approach to Peterson comparison
Linda Chen, Elizabeth Mili\'cevi\'c, and Jennifer Morse
TL;DR
This paper explores an affine approach to Peterson comparison formulas, establishing equivalences between different quantum cohomology comparisons through combinatorial and duality methods.
Contribution
It demonstrates the equivalence of two Peterson comparison formulas using an affine perspective and duality, connecting combinatorial and geometric approaches.
Findings
Two Peterson comparison formulas are shown to be equivalent.
The affine approach unifies different quantum cohomology comparisons.
Postnikov's strange duality is key to establishing the equivalence.
Abstract
The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov's strange duality isomorphism.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Combinatorial Mathematics · Homotopy and Cohomology in Algebraic Topology