Stable Grothendieck polynomials and K-theoretic factor sequences

By Anders Skovsted Buch, Andrew Kresch, Mark Shimozono, Harry Tamvakis, Alexander Yong. 2005

82 citations recorded by the source corpus.

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565–596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood–Richardson rule of Buch (Acta Math 189(1):37–78, 2002). The proof is based on a generalization of the Robinson–Schensted and Edelman–Greene insertion algorithms. Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schützenberger (C R Acad Sci Paris Ser I Math 294(13):447–450, 1982). In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665–687, 1999) for the cohomological quiver polynomials.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cb2871f1b59ba35302a97fd2c8c5f6c747e2aeca

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Citations in the corpus

K-theory of minuscule varieties (2013, methodology): “It was proved in [5] that the coefficient of Oν in the expansion of Gw is equal to (−1) |ν|−l(w) times the number of increasing tableaux T of shape ν for which the Hecke permutation w(T ) is equal to w.”
A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus (2007, methodology): “In particular, as a nonalgebraic geometric application, in forthcoming work [ThYo07+], we relate the ideas in this paper to [ BuKrShTaYo06 ] and the study of longest strictly increasing subsequences in random words.”