Schubert puzzles and integrability II: multiplying motivic Segre classes

In Schubert Puzzles and Integrability I we proved several “puzzle rules” for computing products of Schubert classes in K K -theory (and sometimes equivariant K K -theory) of d 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 q\to 0 limits of certain matrix entries in products of R R - (and other) matrices of U q ( g [ z ± ] ) {\mathcal U}_q(\mathfrak {g}[z^\pm ]) -representations. In the present work we give direct cohomological interpretations of those same matrix entries but at finite q q : they compute products of “motivic Segre classes”, closely related to K K -theoretic Maulik–Okounkov stable classes living on the cotangent bundles of the flag varieties. Without q → 0 q\to 0 , we avoid some divergences that blocked fuller understanding of d = 3 , 4 d=3,4 . The puzzle computations are then explained (in cohomology only in this work, not K 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 R -matrix calculation occurs.