Equivariant Pieri rules for isotropic Grassmannians.(English)Zbl 1339.14030

Let $$V$$ be an $$N$$-dimensional complex vector space equipped with a symmetric or skew-symmetric bilinear form $$\omega$$, which can be either trivial or non-degenerate. The Grassmannians $$I G_{\omega}(m, N)$$ of classical Lie type parameterize $$m$$-dimensional isotropic vector subspaces of $$V$$. The cohomology ring of an isotropic Grassmannian $$X =I G_{\omega}(m, N)$$, or more generally of a homogeneous variety, has an additive basis of Schubert classes represented by Schubert subvarieties $$X_{\lambda}$$. One of the central problems of Schubert calculus is to find a manifestly positive formula for the structure constants of the cup product of two Schubert cohomology classes, or equivalently, for the triple intersection numbers of three Schubert subvarieties in general position. Such a positive formula, called a Littlewood-Richardson rule, has deep connections to various subjects, including geometry, combinatorics and representation theory.
An isotropic Grassmannian $$X$$ can be written as a quotient of a classical complex simple Lie group $$G$$ by a maximal parabolic subgroup $$P$$ (with two notable exceptions of Lie type $$D_n$$). Fix a choice of maximal complex torus $$T$$ and a Borel subgroup $$B$$ with $$T \subset B \subset P$$. The Schubert varieties $$X_{\lambda}$$ are closures of $$B$$-orbits, and hence are $$T$$-stable. They give a basis $$[X_{\lambda}]^T$$ for the $$T$$-equivariant cohomology $${H^*}_T (X)$$ as a $${H^*}_T (pt)$$-module. The structure coefficients $${N^{\nu}}_{\lambda,\mu}$$ in the equivariant product, $[X_{\lambda}]^T \cdot [X_{\mu}]^T=\sum\limits_{\nu}{N^{\nu}}_{\lambda,\mu} [X_{\nu}]^T,$ are homogeneous polynomials which satisfy a positivity condition conjectured by D. Peterson [Lectures on quantum cohomology of $$G/B$$, MIT (1996)] and proved by W. Graham [Duke Math. J. 109, No. 3, 599–614 (2001; Zbl 1069.14055)]. In particular, they are Graham-positive, meaning they are polynomials in the negative simple roots, with non-negative integer coefficients. These equivariant structure coefficients carry much more information than the triple intersection numbers of Schubert varieties, and are more challenging to study.
In the present paper, the authors give for the first time an equivariant Pieri rule for Grassmannians of Lie types $$B, C$$, and $$D$$, as well as a new proof of the Pieri rule in type $$A$$. Such a rule concerns products with the special Schubert classes $$[X_{p}]^T$$ , which are related to the equivariant Chern classes of the tautological quotient bundle, and generate the $$T$$ -equivariant cohomology ring. Using geometric methods, they give a manifestly positive formula for the structure coefficients $${N^{\mu}}_{\lambda,p}$$ of the equivariant multiplication $$[X_{\lambda}]^T \cdot [X_{p}]^T$$.
Reviewer: Cenap Özel (Bolu)

MSC:

 14M15 Grassmannians, Schubert varieties, flag manifolds 14N15 Classical problems, Schubert calculus 55N91 Equivariant homology and cohomology in algebraic topology

Zbl 1069.14055

Software:

Quantum Calculator
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References:

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