Motivic Segre classes of Schubert cells and the connective formal group law
Abstract
We use the connective formal group law to define a one-parameter (β-)deformation of the motivic Segre classes of Schubert cells in the d-step flag variety. This β-deformation specializes to the motivic Segre classes of Schubert cells when β=1 and to the Segre-Schwartz-MacPherson classes of Schubert cells when β=0 (though the β-deformations are not simply the β-homogenizations of the motivic Segre classes). We define rational function representatives for the β-deformed classes in the d=1 case in terms of a solvable lattice model, and we prove a combinatorial formula for the structure constants in the β-deformed basis in the d=1 case using Knutson-Tao puzzles. Unexpectedly, the structure constants are built out of entries of R-matrices for the multi-parameter quantum group of type â2 (which suggests an action of multi-parameter quantum groups on equivariant connective K-rings of Nakajima quiver varieties, with their convolution algebra structures). We show that our β-deformations arise from canonical elements in a quotient of the torus-equivariant algebraic cobordism ring of the cotangent bundle of the flag variety by proving that the canonical elements satisfy a GKM type condition (in future work, we will show that these canonical elements also lie in the torus-equivariant connective K-ring, in the sense of Karpenko--Merkurjev, of this cotangent bundle).
Related
- Two-row Schur times Schubert via parallel classes of positroid varieties
- Motivic classes of Fano schemes of lines on intersections of two quadrics
- The Segre invariant and Brill-Noether theory on Hirzebruch surfaces
- Segre determinants and MHV gravity amplitudes
- Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant
BibTeX
@misc{gandhi2026motivic,
title = {{Motivic Segre classes of Schubert cells and the connective formal group law}},
author = {Raj Gandhi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.26556},
eprint = {2605.26556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.26556},
}
RIS
TY - UNPB
TI - Motivic Segre classes of Schubert cells and the connective formal group law
AU - Raj Gandhi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2605.26556
UR - https://arxiv.org/abs/2605.26556
AB - We use the connective formal group law to define a one-parameter (β-)deformation of the motivic Segre classes of Schubert cells in the d-step flag variety. This β-deformation specializes to the motivic Segre classes of Schubert cells when β=1 and to the Segre-Schwartz-MacPherson classes of Schubert cells when β=0 (though the β-deformations are not simply the β-homogenizations of the motivic Segre classes). We define rational function representatives for the β-deformed classes in the d=1 case in terms of a solvable lattice model, and we prove a combinatorial formula for the structure constants in the β-deformed basis in the d=1 case using Knutson-Tao puzzles. Unexpectedly, the structure constants are built out of entries of R-matrices for the multi-parameter quantum group of type â_2 (which suggests an action of multi-parameter quantum groups on equivariant connective K-rings of Nakajima quiver varieties, with their convolution algebra structures). We show that our β-deformations arise from canonical elements in a quotient of the torus-equivariant algebraic cobordism ring of the cotangent bundle of the flag variety by proving that the canonical elements satisfy a GKM type condition (in future work, we will show that these canonical elements also lie in the torus-equivariant connective K-ring, in the sense of Karpenko--Merkurjev, of this cotangent bundle).
ER -