Schubert puzzles and integrability II: multiplying motivic Segre classes

TL;DR

This paper extends previous work on Schubert puzzles by providing cohomological interpretations of matrix entries at finite q, relating them to motivic Segre classes and quiver varieties, enhancing understanding of flag variety products.

Contribution

It introduces a cohomological framework for motivic Segre classes using quantum integrability and R-matrix calculations, bridging puzzles, K-theory, and quiver varieties.

Findings

Cohomological interpretations of matrix entries at finite q.

Connection between Schubert puzzles and motivic Segre classes.

Explanation of puzzle computations via Lagrangian convolutions.

Abstract

In Schubert Puzzles and Integrability I we proved several "puzzle rules" for computing products of Schubert classes in K-theory (and sometimes equivariant K-theory) of d-step flag varieties. The principal tool was "quantum integrability", in several variants of the Yang--Baxter equation; this let us recognize the Schubert structure constants as q->0 limits of certain matrix entries in products of R- (and other) matrices of quantized affine algebra representations. In the present work we give direct cohomological interpretations of those same matrix entries but at finite q: they compute products of "motivic Segre classes", closely related to K-theoretic Maulik--Okounkov stable classes living on the cotangent bundles of the flag varieties. Without q->0, we avoid some divergences that blocked fuller understanding of d=3,4. The puzzle computations are then explained (in cohomology onlyin this work, not K-theory) in terms of Lagrangian convolutions between Nakajima quiver varieties. More specifically, the conormal bundle to the diagonal inclusion of a flag variety factors through a quiver variety that is not a cotangent bundle, and it is on that intermediate quiver variety that the R-matrix calculation occurs.