Integrability approach to Fehér-Némethi-Rimányi-Guo-Sun type identities for factorial Grothendieck polynomials

Motegi, Kohei

doi:10.1016/j.nuclphysb.2020.114998

Cited by 8 publications

(5 citation statements)

References 28 publications

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“…We show an analog of identities for Schur and Grothendieck polynomials by Fehér-Némethi-Rimányi [FNR12] and Guo-Sun [GS19]. We remark that a proof of the Guo-Sun identity based on quantum integrability was given in [Mot20], and we apply a similar argument to derive our identity. Theorem 3.24.…”

Section: Proposition 34 ([Ms13 Ms14]) the Model M λ Is Integrable Wit...mentioning

confidence: 78%

“…In parallel to Theorem 3.24, we derive a Fehér-Némethi-Rimányi identity [FNR12] for refined Grothendieck functions. We give both a simple proof using the Fehér-Némethi-Rimányi identity and an alternative proof using the lattice model and the Yang-Baxter algebra similar to [Mot20] (although with a different Yang-Baxter algebra).We will show by giving an explicit example showing that our identity is a different than the one by Guo and Sun for factorial Grothendieck polynomials [GS19].…”

Section: Refined Grothendieck Polynomialsmentioning

confidence: 99%

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Refined dual Grothendieck polynomials, integrability, and the Schur measure

Motegi,

Scrimshaw

2020

Preprint

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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 dual Grothendieck polynomials are multi-Schur functions and give a number of identities, including a Littlewood and Cauchy(-Littlewood) identity. We then refine Yeliussizov's connection between dual Grothendieck polynomials and the last passage percolation (LPP) stochastic process discussed by Johansson. By refining algebraic techniques of Johansson, we show Jacobi-Trudi formulas for skew refined dual Grothendieck polynomials conjectured by Grinberg and recover a relation between LPP and the Schur process due to Baik and Rains. Lastly, we extend our vertex model techniques to show some identities for refined Grothendieck polynomials, including a Jacobi-Trudi formula.

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Section: Proposition 34 ([Ms13 Ms14]) the Model M λ Is Integrable Wit...mentioning

confidence: 78%

Section: Refined Grothendieck Polynomialsmentioning

confidence: 99%

Refined dual Grothendieck polynomials, integrability, and the Schur measure

Motegi,

Scrimshaw

2020

Preprint

Self Cite

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“…5.7]). Now if we rotate the semidual model D w by 180 degrees and extend from the endpoints in the (rotated) shape Λ w by diagonal lines (which go to the northwest), we obtain precisely a state in the uncolored model S λw (see also [GK17,MS13,Mot20]). Thus we can apply the natural bijection Θ between marked states, where we allow only tiles to be marked, and set-valued tableaux via marked Gelfand-Tsetlin patterns from [BSW20, Sec.…”

Section: −→ mentioning

confidence: 99%

Double Grothendieck polynomials and colored lattice models

Buciumas,

Scrimshaw

2020

Preprint

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We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt. We then construct a new model that we call the semidual version model for vexillary permutations. We use our semidual model and the five-vertex model of Motegi and Sakai to given a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.

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“…Then the corresponding probability will be proportional to the d-dimensional Grothendieck polynomial 𝑔 𝜌 of variables 𝑥 (ℓ) 𝑖 . As it was pointed out by one of the referees, a proof can also be given using an analogue of Gelfand-Tsetlin patterns and conditioning as in [MS20] for 𝑑 = 2. Corollary 7.2 (Single point distribution formula).…”

Section: Note That We Have φ(𝑊 ) = (𝐺 ((𝑛mentioning

confidence: 99%

MacMahon’s statistics on higher-dimensional partitions

Amanov¹,

Yeliussizov²

2023

Forum of Mathematics, Sigma

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We study some combinatorial properties of higher-dimensional partitions which generalize plane partitions. We present a natural bijection between d-dimensional partitions and d-dimensional arrays of nonnegative integers. This bijection has a number of important applications. We introduce a statistic on d-dimensional partitions, called the corner-hook volume, whose generating function has the formula of MacMahon’s conjecture. We obtain multivariable formulas whose specializations give analogues of various formulas known for plane partitions. We also introduce higher-dimensional analogues of dual stable Grothendieck polynomials which are quasisymmetric functions and whose specializations enumerate higher-dimensional partitions of a given shape. Finally, we show probabilistic connections with a directed last passage percolation model in $\mathbb {Z}^d$ .

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Integrability approach to Fehér-Némethi-Rimányi-Guo-Sun type identities for factorial Grothendieck polynomials

Cited by 8 publications

References 28 publications

Refined dual Grothendieck polynomials, integrability, and the Schur measure

Refined dual Grothendieck polynomials, integrability, and the Schur measure

Double Grothendieck polynomials and colored lattice models

MacMahon’s statistics on higher-dimensional partitions

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