Positivity in the Grothendieck group of complex flag varieties

By M. Brion. 2001

127 citations recorded by the source corpus.

Abstract We prove a conjecture of A.S. Buch concerning the structure constants of the Grothendieck ring of a flag variety with respect to its basis of Schubert structure sheaves. For this, we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs. Equivalently, the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of Schubert varieties.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1b477c8d21ad7926dd353b0a552e6d35d202d0da

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux (2006, background): “These statistics also appear in the Pieri-Chevalley formula in the K-theory of the flag variety (see [PR], [GR], [ Br ] and [LP]).”
A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties (2016, background): “Equivalently, θ0(Ou) is the class of the ideal sheaf of the boundary ∂Xu ⊂ Xu by Brion’s identity (1) together with the following lemma.”
Finiteness of cominuscule quantum K-theory (2010, background): “The Kleiman-Bertini theorem generalizes a result of Brion asserting that rational singularities are preserved when a subvariety of a homogeneous space is intersected with a general Schubert variety [1].”
Singularities of Generalized Richardson Varieties (2010, methodology): “We refer the reader to [6], [7], [8] and [9] for more information about higher cohomology of line bundles on Richardson varieties.”
Lectures on the Geometry of Flag Varieties (2004, background): “In this setting, the structure constants of the cohomology ring (a fortiori, those of the Grothendieck ring) are yet combinatorially elusive, and Buch’s conjecture was proved in [6] by purely algebro-geometric methods.”
A Littlewood-Richardson rule for theK-theory of Grassmannians (2000, unknown): “(1) After this paper was submitted, M. Brion gave a geometric proof that the K-theoretic structure constants of any flag variety G/P have alternating signs [ 27 ].”
Projections of Richardson Varieties (2010, background): “Brion [6] showed that Richardson varieties have rational resolutions in characteristic zero.”