A positive formula for type $A$ Peterson Schubert calculus
| Autor: | Goldin, Rebecca, Gorbutt, Brent |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | Peterson varieties are special nilpotent Hessenberg varieties that have appeared in the study of quantum cohomology, representation theory, and combinatorics. In type $A$, the Peterson variety $Y$ is a subvariety of the complete flag variety $Fl(n; \mathbb C)$, and is invariant under the action of a subgroup $S\cong \mathbb C^*$ of $T$, where $T$ is the standard (noncompact) torus acting on $Fl(n; \mathbb C)$. Using the Peterson Schubert basis introduced by Harada and Tymoczko obtained by restricting a specific set of Schubert classes from $H_T^*(Fl(n; \mathbb C))$ to $H_S^*(Y)$, we describe the product structure of the equivariant cohomology $H_{S}^*(Y)$. In particular, we show that the product is manifestly positive in an appropriate sense by providing an explicit positive combinatorial formula for its structure constants. Our method requires a new combinatorial identity of binomial coefficients that generalizes Vandermonde's identity. Comment: 48 pages, significant edits for clarity |
| Databáze: | arXiv |
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