Positivity in equivariant Schubert calculus

Graham, W. A. G.

doi:10.1215/s0012-7094-01-10935-6

Cited by 96 publications

(89 citation statements)

References 10 publications

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“…Noting that h ∈ D, it must be true that Im(α i (h)) ≥ 0 for all i. Also, taking imaginary parts in (30) gives 0 = k i=1 n i Im(α i (h)). Again using the fact that all n i are strictly positive or all n i are strictly negative, it follows that Im(α i (h)) = 0 for all i.…”

Section: Definitions and Characterizationsmentioning

confidence: 99%

Complex adjoint orbits in Lie theory and geometry

Crooks

2019

Expositiones Mathematicae

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This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which adjoint orbits feature prominently (ex. hyperkähler geometry, Landau-Ginzburg models, and the theory of symplectic singularities). Also included are a few arguments and observations that, to the author's knowledge, have not yet appeared in the research literature.

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Section: Definitions and Characterizationsmentioning

confidence: 99%

Complex adjoint orbits in Lie theory and geometry

Crooks

2019

Expositiones Mathematicae

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“…[11]. There is an equivariant version of positivity [15,30] which states that (quantum) equivariant multiplication of B´stable Schubert classes yields structure constants which are polynomials in positive simple roots with (weakly) negative coefficients. Both the ordinary and equivariant positivity statements hold for the coefficients of the Chevalley formula (9).…”

Section: Consider the Expansionsmentioning

confidence: 99%

Equivariant quantum cohomology of the odd symplectic Grassmannian

Mihalcea

Shifler

2018

Math. Z.

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The odd symplectic Grassmannian IG :" IGpk, 2n`1q parametrizes k dimensional subspaces of C 2n`1 which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IG is itself a smooth Schubert variety in the submaximal isotropic Grassmannian IGpk, 2n`2q. We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case k " 2, and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.

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“…A class in Λ is non-negative if it can be written as a polynomial with non-negative integer coefficients in the classes c T (β) for β ∈ ∆. Graham has proved that all equivariant structure constants C w u,v are non-negative classes [Gra01]. For β ∈ ∆, we let ω β denote the corresponding fundamental weight.…”

Section: Equivariant Cohomologymentioning

confidence: 99%

“…The (small) equivariant quantum K-theory ring QK T (X) of Givental [Giv00] is a common generalization of the main cohomology theories considered in Schubert calculus, including K-theory, equivariant cohomology, and quantum cohomology. Equivariant quantum Ktheory is the most general theory for which the associated Schubert structure constants have positivity properties that are either known [Gra01,Mih06b,Bri02,AGM11,AC15] or conjectured [LM06,LP07,BM11]. In this paper, we prove a Chevalley formula that combinatorially determines the ring QK T (X) when X is a cominuscule variety, that is, a Grassmann variety Gr(m, n) of type A, a Lagrangian Grassmannian LG(n, 2n), a maximal orthogonal Grassmannian OG(n, 2n), a quadric hypersurface Q n , or one of two exceptional varieties called the Cayley plane E 6 /P 6 and the Freudenthal variety E 7 /P 7 .…”

Section: Introductionmentioning

confidence: 99%

A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties

Buch¹,

Chaput

Mihalcea

et al. 2018

Alg. Geom.

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We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum K-theory ring of any cominuscule flag variety G/P . We also prove that multiplication with divisor classes determines the equivariant quantum Ktheory of arbitrary flag varieties. These results prove a conjecture of Gorbounov and Korff concerning the equivariant quantum K-theory of Grassmannians of Lie type A.

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Positivity in equivariant Schubert calculus

Cited by 96 publications

References 10 publications

Complex adjoint orbits in Lie theory and geometry

Complex adjoint orbits in Lie theory and geometry

Equivariant quantum cohomology of the odd symplectic Grassmannian

A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties

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