<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>K</mml:mi></mml:math>-theoretic analogues of factorial Schur<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>P</mml:mi></mml:math>- and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>Q</mml:mi></mml:math>-functions

Ikeda, Takeshi; Naruse, Hiroshi

doi:10.1016/j.aim.2013.04.014

Cited by 75 publications

(115 citation statements)

References 17 publications

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“…We define the shifted stable Grothendieck polynomial of y ∈ I ∞ and z ∈ F ∞ to be the power series where the sums are over compatible sequences with w ∈ H O (y) and w ∈ H Sp (z), respectively. Ikeda and Naruse [18] have defined a family of K-theoretic Schur P -functions GP λ indexed by strict partitions λ. These functions represent Schubert classes in the K-theory of torus equivariant coherent sheaves on the maximal orthogonal Grassmannian [18,Corollary 8.1].…”

Section: Stable Grothendieck Polynomialsmentioning

confidence: 99%

“…Ikeda and Naruse [18] have defined a family of K-theoretic Schur P -functions GP λ indexed by strict partitions λ. These functions represent Schubert classes in the K-theory of torus equivariant coherent sheaves on the maximal orthogonal Grassmannian [18,Corollary 8.1]. As an application of our results on symplectic Hecke insertion, we prove the following in Section 5: = λ c zλ β |λ|−l FPF (z) GP λ where the sums are over all strict partitions λ, and b yλ and c zλ are the finite numbers of increasing shifted tableaux T of shape λ whose row reading words are orthogonal Hecke words for y and symplectic Hecke words for z, respectively.…”

Section: Stable Grothendieck Polynomialsmentioning

confidence: 99%

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A symplectic refinement of shifted Hecke insertion

Marberg

2020

Journal of Combinatorial Theory, Series A

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Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials G π indexed by permutations in the basis of stable Grothendieck polynomials G λ indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on orbit closures of the orthogonal group acting on the complete flag variety. We construct a generalization of shifted Hecke insertion for maximal chains in an analogous weak order on orbit closures of the symplectic group. As an application, we identify a combinatorial rule for the expansion of "orthogonal" and "symplectic" shifted analogues of G π in Ikeda and Naruse's basis of K-theoretic Schur P -functions.

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Section: Stable Grothendieck Polynomialsmentioning

confidence: 99%

Section: Stable Grothendieck Polynomialsmentioning

confidence: 99%

A symplectic refinement of shifted Hecke insertion

Marberg

2020

Journal of Combinatorial Theory, Series A

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“…From its origin, there are geometric studies [32,33] related to Schubert calculus, and also combinatorial ones [34,35,36] as they are some classes of symmetric polynomials. However, it was shown very recently [18,19] that Grothendieck polynomials can be expressed in the determinant from (5.1) (they moreover extended the determinant representation to factorial Grothendieck polynomials [37] originally defined in terms of set-valued semi-standard tableaux). We take the determinant form (5.1) as the definition of the Grothendieck polynomials in this paper.…”

Section: Grothendieck Polynomials and Cauchy Identitymentioning

confidence: 99%

“…The formal parameter β corresponds to the K-theoretical extension. For flag varieties of type A, Schubert polynomials is the Schur polynomials itself, and it was shown recently [18,19] that Grothendieck polynomials for flag varieties of type A is expressed in the determinant form (1.2). We show the equivalence between the wavefunctions and Grothendieck polynomials by combining the quantum inverse scattering method with a matrix product representation of the wavefunctions.…”

Section: Introductionmentioning

confidence: 99%

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Vertex models, TASEP and Grothendieck polynomials

Motegi

Sakai

2013

J. Phys. A: Math. Theor.

106

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We examine the wavefunctions and their scalar products of a one-parameter family of integrable five vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on/off-shell wavefunctions of the five vertex models are represented as a certain determinant form. Up to some normalization factors, we find the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of Grothendieck polynomials.

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3d $$ \mathcal{N} $$ = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

Ueda

Yoshida

2020

J. High Energ. Phys.

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We study a correspondence between 3d $$ \mathcal{N} $$ N = 2 topologically twisted Chern-Simons-matter theories on S1× Σg and quantum K -theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter β associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with β = −1 is isomorphic to the (equivariant) small quantum K -ring of the Grassmannian, and the Frobenius algebra with β = 0 is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas to the correlation functions of supersymmetric Wilson loops in the Chern-Simons-matter theory and show that the algebra of Wilson loops is isomorphic to the Frobenius algebra with β = −1. This allows us to identify the algebra of Wilson loops with the quantum K - ring of the Grassmannian. We also show that correlation functions of Wilson loops on S1× Σg satisfy the axiom of 2d TQFT. For β = 0, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for β = 0, and the quantum cohomology of the Grassmannian. We also discuss deformations of Verlinde algebras, omega-deformations, and the K -theoretic I -functions of Grassmannians with level structures.

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K-theoretic analogues of factorial SchurP- andQ-functions

Cited by 75 publications

References 17 publications

A symplectic refinement of shifted Hecke insertion

A symplectic refinement of shifted Hecke insertion

Vertex models, TASEP and Grothendieck polynomials

3d $$ \mathcal{N} $$ = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

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