Motivic Segre classes of Schubert cells and the connective formal group law
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913162592256000 |
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| author | Gandhi, Raj |
| author_facet | Gandhi, Raj |
| contents | 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 $β=0$. 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. The proof of the puzzle formula involves intertwiners for representations of the multi-parameter quantum group of type $\widehat{a}_2$. We show that our $β$-deformations can be viewed as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle of the flag variety by proving that the canonical elements satisfy a GKM type condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26556 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Motivic Segre classes of Schubert cells and the connective formal group law Gandhi, Raj Combinatorics Algebraic Topology Representation Theory 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 $β=0$. 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. The proof of the puzzle formula involves intertwiners for representations of the multi-parameter quantum group of type $\widehat{a}_2$. We show that our $β$-deformations can be viewed as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle of the flag variety by proving that the canonical elements satisfy a GKM type condition. |
| title | Motivic Segre classes of Schubert cells and the connective formal group law |
| topic | Combinatorics Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2605.26556 |