Schubert 拼图与可积性 II：乘性动机 Segre 类
Schubert puzzles and integrability II: multiplying motivic Segre classes
AI总结：
本文在有限 q 下给出 Schubert 拼图规则中矩阵元的上同调解释，证明其计算动机 Segre 类乘积，并通过 Nakajima 箭图簇的拉格朗日卷积阐明拼图计算，避免了 d=3,4 的发散问题。
摘要：
在《Schubert 拼图与可积性 I》中，我们证明了若干“拼图规则”，用于计算 d 步旗簇的 K 理论（有时是等变 K 理论）中 Schubert 类的乘积。主要工具是“量子可积性”，涉及 Yang--Baxter 方程的若干变体；这使我们能够将 Schubert 结构常数识别为量子仿射代数表示的 R 矩阵（及其他矩阵）乘积中某些矩阵元在 q->0 极限下的值。在当前工作中，我们直接给出这些矩阵元在有限 q 时的上同调解释：它们计算“动机 Segre 类”的乘积，这些类与旗簇余切丛上的 K 理论 Maulik--Okounkov 稳定类密切相关。不取 q->0 极限，我们避免了在 d=3,4 情形中阻碍更充分理解的某些发散。随后，拼图计算（本文仅在上同调中，而非 K 理论中）通过 Nakajima 箭图簇之间的拉格朗日卷积得到解释。更具体地说，旗簇对角线包含的余法丛穿过一个并非余切丛的箭图簇，而 R 矩阵计算正是发生在该中间箭图簇上。
英文摘要：
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.
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