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We introduce generalized Demazure operators for the equivariant oriented cohomology of the flag variety, which have specializations to various Demazure operators and Demazure–Lusztig operators in both equivariant cohomology and equivariant K-theory. In the context of the geometric basis of the equivariant oriented cohomology given by certain Bott–Samelson classes, we use these operators to obtain formulas for the structure constants arising in different bases. Specializing to divided difference operators and Demazure operators in singular cohomology and K-theory, we recover the formulas for structure constants of Schubert classes obtained in Goldin and Knutson (Pure Appl Math Q 17(4):1345–1385, 2021). Two specific specializations result in formulas for the the structure constants for cohomological and K-theoretic stable bases as well; as a corollary we reproduce a formula for the structure constants of the Segre–Schwartz–MacPherson basis previously obtained by Su (Math Zeitschrift 298:193–213, 2021). Our methods involve the study of the formal affine Demazure algebra, providing a purely algebraic proof of these results.
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This content will become publicly available on January 1, 2026
Hook formulae from Segre–MacPherson classes
Nakada’s colored hook formula is a vast generalization of many important formulae in combinatorics, such as the classical hook length formula and the Peterson’s formula for the number of reduced expressions of minuscule Weyl group elements. In this paper, we use cohomological properties of Segre–MacPherson classes of Schubert cells and varieties to prove a generalization of a cohomological version of Nakada’s formula, in terms of smoothness properties of Schubert varieties. A key ingredient in the proof is the study of a decorated version of the Bruhat graph. Weights of the paths in this graph give the terms in the generalized Nakada’s formula, and the summation over all paths is equal to the equivariant multiplicity of the Chern–Schwartz–MacPherson class of a Richardson variety. Among the applications we mention an algorithm to calculate structure constants of multiplications of Segre–MacPherson classes of Schubert cells, and a skew version of Nakada–Peterson’s formula.
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- Award ID(s):
- 2152294
- PAR ID:
- 10621064
- Publisher / Repository:
- Comb. Consort. (TCC)
- Date Published:
- Journal Name:
- Algebraic Combinatorics
- Volume:
- 8
- Issue:
- 3
- ISSN:
- 2589-5486
- Page Range / eLocation ID:
- 655 to 685
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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