Positivity of Peterson Schubert Calculus
| Autor: | Goldin, Rebecca, Mihalcea, Leonardo, Singh, Rahul |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | The Peterson variety is a subvariety of the flag manifold $G/B$ equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert classes indexed by arbitrary Coxeter elements are dual (up to an intersection multiplicity) to the fundamental classes of Peterson cell closures. Dividing these classes by the intersection multiplicities yields a $\mathbb Z$-basis for the equivariant cohomology of the Peterson variety. We prove several properties of this basis, including a Graham positivity property for its structure constants, and stability with respect to inclusion in a larger Peterson variety. We also find formulae for intersection multiplicities with Peterson classes. This explains geometrically, in arbitrary Lie type, recent positivity statements proved in type A by Goldin and Gorbutt. Comment: To appear in Advances in Mathematics. 31 pages, 1 table. Modifications: - Expanded introduction; - Improved explanation of the equivariant naturality of Peterson varieties; - Modified Theorem 1.1 to make explicit the positive integrality of pairing; - Added corollary and remark to make explicit extension to integral cohomology (Cor 3.8, Rmk 3.9); - Adjusted exposition for clarity |
| Databáze: | arXiv |
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