# A Littlewood-Richardson rule for dual stable Grothendieck polynomials

@article{Galashin2017ALR,
title={A Littlewood-Richardson rule for dual stable Grothendieck polynomials},
author={Pavel Galashin},
journal={J. Comb. Theory, Ser. A},
year={2017},
volume={151},
pages={23-35}
}
• Pavel Galashin
• Published 31 December 2014
• Mathematics, Computer Science
• J. Comb. Theory, Ser. A
Abstract For a given skew shape, we build a crystal graph on the set of all reverse plane partitions that have this shape. As a consequence, we get a simple extension of the Littlewood–Richardson rule for the expansion of the corresponding dual stable Grothendieck polynomial in terms of Schur polynomials.

#### Figures and Topics from this paper

A dual approach to structure constants for K-theory of Grassmannians
• Mathematics
• Discrete Mathematics & Theoretical Computer Science
• 2020
International audience 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 ordinaryExpand
Positive specializations of symmetric Grothendieck polynomials
It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes theExpand
Dualities in Refined Grothendieck Polynomials
We give new proofs of the two types of duality for Grothendieck polynomials. Our proofs extend to proofs of these dualities for the refined Grothendieck polynomials. The second of these dualities wasExpand
Structure constants for K-theory of Grassmannians, revisited
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 2016
This work combinatorially proves that the appropriate K-theoretic analogy is through the expansion of skew reverse plane partitions into the basis of polynomials which are Hopf-dual to stable Grothendieck polynmials. Expand
Refined canonical stable Grothendieck polynomials and their duals
• Mathematics
• 2021
In this paper we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of GrothendieckExpand
CRYSTAL STRUCTURES FOR SYMMETRIC GROTHENDIECK POLYNOMIALS
• Mathematics
• 2018
The symmetric Grothendieck polynomials representing Schubert classes in the K theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type A n crystalExpand
Hook formulas for skew shapes I. q-analogues and bijections
• Mathematics, Computer Science
• J. Comb. Theory, Ser. A
• 2018
An algebraic and a combinatorial proof of Naruse's formula for the number of standard Young tableaux of skew shapes is given, by using factorial Schur functions and a generalization of the Hillman–Grassl correspondence. Expand
Refined dual Grothendieck polynomials, integrability, and the Schur measure
• Mathematics
• 2020
We construct a vertex model whose partition function is a refined dual Grothendieck polynomial, where the states are interpreted as nonintersecting lattice paths. Using this, we show refined dualExpand
Crystal Analysis of type C Stanley Symmetric Functions
• Mathematics, Computer Science
• Electron. J. Comb.
• 2017
This work provides a crystal theoretic explanation of this fact and gives an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements. Expand
Crystal structures for canonical Grothendieck functions
• Mathematics
• 2019
We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dualExpand

#### References

SHOWING 1-10 OF 16 REFERENCES
A Littlewood-Richardson rule for theK-theory of Grassmannians
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 relateExpand
Refined Dual Stable Grothendieck Polynomials and Generalized Bender-Knuth Involutions
• Mathematics, Computer Science
• Electron. J. Comb.
• 2016
The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian, and it is proved that this generalization still defines symmetric functions. Expand
A simple proof of the Littlewood-Richardson rule and applications
• Computer Science, Mathematics
• Discret. Math.
• 1998
Abstract We present a simple proof of the Littlewood-Richardson rule using a sign-reversing involution, and show that a similar involution provides a combinatorial proof of the SXP algorithm of ChenExpand
Combinatorial Aspects of the K-Theory of Grassmannians
Abstract. In this paper we study Grothendieck polynomials indexed by Grassmannian permutations, which are representatives for the classes corresponding to the structure sheaves of Schubert varietiesExpand
Young Tableaux: With Applications to Representation Theory and Geometry
Part I. Calculus Of Tableux: 1. Bumping and sliding 2. Words: the plactic monoid 3. Increasing sequences: proofs of the claims 4. The Robinson-Schensted-Knuth Correspondence 5. TheExpand
The Yang-Baxter equation, symmetric functions, and Schubert polynomials
• Mathematics, Computer Science
• Discret. Math.
• 1996
An approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation is presented. Expand
Combinatorial Expansions in K-Theoretic Bases
• Mathematics, Computer Science
• Electron. J. Comb.
• 2012
The combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$. Expand
A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras
In the representation theory of the group GLn(C), an important tool are the Young tableaux. The irreducible representations are in one-to-one correspondence with the shapes of these tableaux. Let TExpand
A local characterization of simply-laced crystals
We provide a simple list of axioms that characterize the crystal graphs of integrable highest weight modules for simply-laced quantum Kac-Moody algebras.
Combinatorial Hopf algebras and K-homology of Grassmanians
• Mathematics
• 2007
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoreticExpand