On the quantum product of Schubert classes

By William Fulton, Christopher Woodward. 2001

169 citations recorded by the source corpus.

We give a formula for the smallest powers of the quantum parameters q that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties G/P. We also include a complete proof of Peterson's quantum version of Chevalley's formula, also for general G/P's.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/79778aef8fb8ebc1204ab6d2bcf3cb5ae21db1ed

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Citations in the corpus

k-Schur Functions and Affine Schubert Calculus (2013, methodology): “For example in [115] a different formulation of charge is used, which comes from the Ram–Yip formula for Macdonald polynomials [137], and is related to the quantum Bruhat graph, which first arose in connection with the quantum cohomology of the flag variety [24, 49].”
Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells (2017, result): “We refer to [FW04], in the non-equivariant setting, and e.”
Classification problems and mirror duality (2005, result): “cture. Note that we do not need the whole cohomology structure: we just need to know quantum multiplication by the divisor classes, and this can be computed using Peterson’s quantum Chevalley formula [FW]. I calculated the quantum Lefschetz reduction for the cases (6,1),(8,1),(9,1) and the results a”
QUANTUM K-THEORY OF GRASSMANNIANS (2008, background): “but the reader can consult [2, 14, 17, 20, 57] and references therein.”
Quantum Cohomology of Minuscule Homogeneous Spaces (2006, methodology): “In [ FW ], Fulton and Woodward described the minimal power of q that can appear in the quantum product [X(u)] ∗ [X(v)] of two Schubert classes.”
Quantum cohomology of G/P and homology of affine Grassmannian (2007, methodology): “However, our indexing agrees with the ones in [7, 8, 13].”
A mirror symmetric construction of qHT∗(G/P)(q) (2005, unknown): “Fix t ∈ (T∨)WP and consider b0 ∈ B ∨ − ∩ U ∨ +tẇP ẇ −1 0 U ∨ + with factorization b0 = u1tẇP ẇ −1 0 u −1 2 , for u1, u2 ∈ U ∨ +.”
Finiteness of cominuscule quantum K-theory (2010, background): “Notice that dX(2) = max{d(x, y) | x, y ∈ X}, and it follows from [13] that dX(2) is the smallest degree of the quantum parameter q that occurs in the square of a point in the small quantum ring QH(X).”
Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties (2001, background): “Woodward [14] have in fact recently proved that no two Schubert classes in qH∗(G/P ) ever multiply to zero.”
Curve neighborhoods of Schubert varieties (2013, background): “The equivariant quantum Chevalley formula is the following result [10, 14, 24].”
Positroid varieties: juggling and geometry (2009, background): “y corresponding the KT-classes. 8. QUANTUM COHOMOLOGY, TORIC SCHUR FUNCTIONS, AND POSITROIDS 8.1. ModulispacesofstablerationalmapstotheGrassmannian. Forbackgroundmaterialonstablemapswereferthereaderto[FuWo04]. LetI,J∈ [n] k ,whichweassumetobe ﬁxedthroughout thissection. WenowinvestigatethevarietyS(I”
Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures (2014, unknown): “+ kand ν1 > λ1 > ν2 > λ2 > ··· > νr> λr. Bertram’s quantum Pieri formula gives the quantum multiplication by special Schubert classes: Proposition 7.2.3 (Quantum Pieri [8], see also [10, 24]). We have σk⋆σ1logqσλ= X σν+q X σµ where the ﬁrst sum is the same as the classical Pieri fo”
Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian (2005, methodology): “2, the claim follows from the Pieri-Chevalleytype formulas for the ordinary integral cohomology, which were proved by Fulton and Woodward [9], Lemma 8.1.”