Abstract
The problem of computing products of Schubert classes in the cohomology ring can be formulated as theproblem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials. We reformulate theproblem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to itsbasis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions forskew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch’s K-theoretic Littlewood-Richardson rule.
Highlights
The theory of symmetric functions supports Schubert calculus of the Grassmannian X = Gr(k, n) by way of the Schur function basis
The cohomology classes corresponding to Schubert varieties form an integral basis for the cohomology ring of X, which in turn is isomorphic to a certain quotient of the ring of symmetric polynomials in k variables
The development of a rich theory of tableaux enabled the computation of the structure constants Cλνμ
Summary
The realization of Schur polynomials as weight generating functions of semi-standard Young tableaux offers combinatorial tools for the study. The Poincaredual of the product [Xλ] · [Xμ] in H∗(X) is identified by the skew Schur function sμ∨/λ, and the Littlewood-Richardson numbers arise in its ordinary Schur expansion: sν/λ = This viewpoint has since given rise to many simple proofs of the Littlewood-Richardson rule For the Grassmannian variety X, Buch (2002) showed that the stable limits Gλ are generating series of set-valued tableaux and can be applied to K◦X in a way that mirrors the Schur role in cohomology. We reach the K-theoretic structure constants through the more general class of generating functions of reverse plane partitions on skew shapes, gν/λ. A series of natural bijections leads to Buch’s K-theoretic LittlewoodRichardson rule for cνλμ
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