A Littlewood-Richardson rule for theK-theory of Grassmannians

By Anders Skovsted Buch. 2000

307 citations recorded by the source corpus.

We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra of stable Grothendieck polynomials, which is a K-theory parallel of the ring of symmetric functions.

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

View Paper: https://www.semanticscholar.org/paper/0e2988c812176359f3ecee91c7837a4d10fa2f0b

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Citations in the corpus

A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus (2007, methodology): “ λ its structure sheaf. The classes {[OX λ ]}⊆ K(X) form an additive Z-basis of K(X). The (K-theoretic) Schubert structure constants Cν λ,µ are deﬁned by [OX λ ]·[OXµ]= X ν⊆Λ Cν λ,µ[OXν]. Buch’s rule [Bu02a] established alternation of sign, i.e., (−1)|ν|−|λ|−|µ|Cν λ,µ ∈ N. There has been signiﬁcant ”
Combinatorial Hopf algebras and K-homology of Grassmanians (2007, unknown): “In this section we briefly describe the four combinatorial Hopf algebras, MR, QSym, NSym, and Sym which we intend to generalize (see (1.1)).”
K-classes for matroids and equivariant localization (2010, background): “This should remind the reader of Schubert calculus; we refer the reader to [7] for a guide to K(G(d, n)) which pursues this analogy.”
Duality and deformations of stable Grothendieck polynomials (2016, methodology): “Both product (1) and coproduct ∆ make Γ a commutative and cocommutative bialgebra [2].”
Gröbner geometry of vertex decompositions and of flagged tableaux (2005, methodology): “ tableau. 6 This result appears as part of Theorem 5.8. It was already known in the case of single Schubert polynomials [Wac] and in the case of Grothendieck polynomials for Grassmannian permutations [Buc]. Since the paper [KM1] already provided a geometric explanation for a combinatorial formula fo”
k-Schur Functions and Affine Schubert Calculus (2013, background): “Pieri rules are given in [113] in terms of binomial numbers and a generalization for Yamanouchi tableaux gives [27] a combinatorial rule for the structure constants.”
Puzzles and (equivariant) cohomology of Grassmannians (2001, background): “− z j, and it is unclear how to extend this to arbitrary root systems. For many of the other extensions, no satisfactory explicit combinatorial formula for the structure constants is known. 24Much as [Bu] did for K-theoretic Schubert calculus on Grassmannians. In both cases, they gave a formula dire”
K-theory of minuscule varieties (2013, methodology): “We list the sequence of intermediate tableaux obtained when a forward slide is applied to a tableau T of shape (4, 4, 3)/(3, 1), starting from the box [2, 3] in row 2 and column 3.”
K-theoretic analogues of factorial Schur P- and Q-functions (2011, background): “For type A case, the corresponding statement is true as a consequence of the explicit description of LR-coefficients given there ([2], Cor.”
Positivity in the Grothendieck group of complex flag varieties (2001, background): “[Xu] ·[Xv] = X w aw u,v [Xw] for nonnegative integers aw u,v . A generalization to the Grothendieck ring K(X) of vector bundles (or of coherent sheaves) on Xwas recently formulated by A. S. Buch, see [2]. The classes [OXw ] of structure sheaves of Schubert subvarieties form an additive basis of K(X)”
Excited Young diagrams, equivariant K -theory, and Schubert varieties (2013, unknown): “y also appeared in [WY12], where they were identiﬁed as special types of ﬂagged set-valued tableaux. Flagged set-valued tableaux, which were introduced in [KMY09], generalize both set-valued tableaux [Buc02] and ﬂagged tableaux [Wac85]. 4 WILLIAM GRAHAM AND VICTOR KREIMAN Ikeda and Naruse [IN11] use”
K-theory Schubert calculus of the affine Grassmannian (2009, background): “1. Remark 7.1 implies that G(k) λ (x) = Gλ(x) whenever the largest hook length of λis less than or equal to k, where Gλ(x) is the stable Grothendieck polynomial labeled by partitions, studied by Buch [Buc]. 7.5. K-theoretic k-Schur functions. Since {Gv(x) |v∈WI af}is a “basis” of ˆΛ(n), there is a d”
Stable Grothendieck polynomials and K-theoretic factor sequences (2005, methodology): “This formula was used in [ 3 ] to express the monomial coefficients of stable Grothendieck polynomials for partitions in terms of set-valued tableaux [see Eq. (8)].”