arXiv subjects

Kohei Motegi

Publications and source records attributed to Kohei Motegi.

At least 19 recordsLinked to original sources

arXiv2026

Tetrahedral $L$-operators, tensor Schur polynomials and $q$-deformed loop elementary symmetric functions

We study three-dimensional partition functions constructed from the tetrahedral $L$-operator introduced and studied by Bazhanov-Sergeev and Kuniba-Maruyama-Okado. First, we explore the $q=0$ case, extending the authors' previous results and giving applications by a further analysis on the Zamolodchikov-Faddeev algebra. We introduce a class of partition functions which can be expressed as the tensor Schur polynomials, a class of products of Schur polynomials. As an application, we derive the shuffle formula for the Schur polynomials which is geometrically the pushforward formula by Jo\'zefiak-Pragacz-Lascoux. We also give a derivation and a unification of the Gustafson-Milne and Feh{\'e}r--N{\'e}methi--Rim{\'a}nyi identities, and introduce a family of Laurent polynomials using divided difference operators which imitates the Schubert polynomials from the perspective of our study. We also present an application to the steady state of the multispecies totally asymmetric simple exclusion process. Second, we investigate several classes of partition functions for the generic $q$ case, and determine the explicit forms as deformations of the elementary symmetric functions. One of them can be regarded as (an extension of) a $q$-deformed loop elementary symmetric functions.

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arXiv2025

Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis

We study a certain type of multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. We show that all the coefficients in the multiple commutation relations between the $L$-operator elements are given in terms of the trigonometric weight functions for the vector representation, independent of the representation of the $L$-operator. For rank one case, our proof also gives a conceptual understanding why the coefficients can also be expressed using the Izergin-Korepin determinants. As a related result, by specializing expressions for the universal nested Bethe vector by Pakuliak-Ragoucy-Slavnov, we also find a construction of the Gelfand-Tsetlin basis for the vector representation using different $L$-operator elements from the constructions by Nazarov-Tarasov or Molev. We also present corresponding results for the Yangian $Y_h(\mathfrak{gl}_N)$.

math.QA↗

arXiv2025

Bethe roots for periodic TASEP and algebraic curve

We present an algebraic method for solving the Bethe ansatz equations for the periodic totally asymmetric exclusion process (TASEP) with an arbitrary number of sites and particles. The Bethe ansatz equations are realized as an algebraic equation on a certain Riemann surface. While our Riemann surface is essentially the same as the one introduced by Prolhac, we focus on its algebraic realization as a (singular) plane curve. Through a counting argument on the Riemann surface, we establish a rigorous proof that the Bethe ansatz equation has the expected number of solutions when counted with multiplicity. Consequently, under appropriate generic conditions, the completeness of the Bethe ansatz follows. The decomposition of the Riemann surface into connected components determines how often each value of the product of Bethe roots appears. We classify the connected components and their multiplicities using a similar argument to the spectral degeneracy of the Markov matrix discussed by Golinelli-Mallick. As a result, we give an algebro-geometric characterization of Golinelli-Mallick-type spectral degeneracy of the Markov matrix. We also give explicit formulas for the number of connected components, the number of ramification points, and the total genus of the Riemann surface. These formulas recover the table of examples presented by Prolhac. Moreover, we explore applications of the special type of Bethe roots that appear in this case to partition functions of the five-vertex model. We introduce a version of the free energy and evaluate the thermodynamic limit to find the explicit form in terms of the Riemann zeta function.

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arXiv2024

Higher rank elliptic partition functions and multisymmetric elliptic functions

We introduce and investigate a class of $\mathfrak{gl}_{M+1}$ partition functions which is an extension of the one introduced by Foda-Manabe. We characterize the partition functions by a nested version of Izergin-Korepin analysis, and determine the explicit forms, for each of the rational, trigonometric and elliptic versions. The resulting multisymmetric functions can be regarded as extensions of the rational, trigonometric and elliptic weight functions.

math-ph↗

arXiv2024

Factorization of rational six vertex model partition functions

We show factorization formulas for a class of partition functions of rational six vertex model. First we show factorization formulas for partition functions under triangular boundary. Further, by combining the factorization formulas with the explicit forms of the generalized domain wall boundary partition functions by Belliard-Pimenta-Slavnov, we derive factorization formulas for partition functions under trapezoid boundary which can be viewed as a generalization of triangular boundary. We also discuss an application to emptiness formation probabilities under trapezoid boundary which admit determinant representations.

math-ph↗

arXiv2024

Gelfand-Tsetlin Bases for Elliptic Quantum Groups

We study the level-0 representations of the elliptic quantum group $U_{q,p}(\widehat{\mathfrak{gl}}_N)$. We give a classification theorem of the finite-dimensional irreducible representations of $U_{q,p}(\widehat{\mathfrak{gl}}_N)$ in terms of the theta function analogue of the Drinfeld polynomial for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. We also construct the Gelfand-Tsetlin bases for the level-0 $U_{q,p}(\widehat{\mathfrak{gl}}_N)$-modules following the work by Nazarov-Tarasov for the Yangian $Y(\mathfrak{gl}_N)$-modules. This is a construction in terms of the Drinfeld generators. For the case of tensor product of the vector representations, we give another construction of the Gelfand-Tsetlin bases in terms of the $L$-operators and make a connection between the two constructions. We also compare them with those obtained by the first author by using the $\mathfrak{S}_n$-action realized by the elliptic dynamical $R$-matrix on the standard bases. As a byproduct, we obtain an explicit formula for the partition functions of the corresponding 2-dimensional square lattice model in terms of the elliptic weight functions of type $A_{N-1}$.

math.QA↗

arXiv2024

Tetrahedron equation and Schur functions

The tetrahedron equation introduced by Zamolodchikov is a three-dimensional generalization of the Yang-Baxter equation. Several types of solutions to the tetrahedron equation that have connections to quantum groups can be viewed as $q$-oscillator valued vertex models with matrix elements of the $L$-operators given by generators of the $q$-oscillator algebra acting on the Fock space. Using one of the $q=0$-oscillator valued vertex models introduced by Bazhanov-Sergeev, we introduce a family of partition functions that admits an explicit algebraic presentation using Schur functions. Our construction is based on the three-dimensional realization of the Zamolodchikov-Faddeev algebra provided by Kuniba-Okado-Maruyama. Furthermore, we investigate an inhomogeneous generalization of the three-dimensional lattice model. We show that the inhomogeneous analog of (a certain subclass of) partition functions can be expressed as loop elementary symmetric functions.

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arXiv2024

Algebraic formulas and Geometric derivation of Source Identities

Source identities are fundamental identities between multivariable special functions. We give a geometric derivation of rational and trigonometric source identities. We also give a systematic derivation and extension of various determinant representations for source functions which appeared in previous literature as well as introducing the elliptic version of the determinants, and obtain identities between determinants. We also show several symmetrization formulas for the rational version.

math.AG↗

arXiv2023

Free-fermions and canonical Grothendieck polynomials

We give a presentation of refined (dual) canonical Grothendieck polynomials and their skew versions using free-fermions. Using this, we derive a number of identities, including the skew Cauchy identities, branching rules, expansion formulas, and integral formulas.

math.CO↗

arXiv2023

Free fermionic probability theory and K-theoretic Schubert calculus

For each of the four particle processes given by Dieker and Warren [arXiv:0707.1843], we show the $n$-step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.

math.CO↗

arXiv2022

Yang-Baxter algebra, higher rank partition functions and $K$-theoretic Gysin map for partial flag bundles

We investigate the $K$-theoretic Gysin map for type $A$ partial flag bundles from the viewpoint of integrability. We introduce several types of partition functions for one version of $q=0$ degeneration of $U_q(\widehat{sl_n})$ vertex models on rectangular grids which differ by boundary conditions and sizes, and can be viewed as Grothendieck classes of the Grothendieck group of a nonsingular variety and partial flag bundles. By deriving multiple commutation relations for the $q=0$ $U_q(\widehat{sl_n})$ Yang-Baxter algebra and combining with the description of the $K$-theoretic Gysin map for partial flag bundles using symmetrizing operators, we show that the $K$-theoretic Gysin map of the first type of partition functions on a rectangular grid is given by the second type whose boundary conditions on one side are reversed from the first type. This generalizes the author's previous result from Grassmann bundles to partial flag bundles. We also discuss the inhomogenous version of the partition functions and applications to the $K$-theoretic Gysin map.

math-ph↗

arXiv2021

Integrable models and $K$-theoretic pushforward of Grothendieck classes

We show that a multiple commutation relation of the Yang-Baxter algebra of integrable lattice models derived by Shigechi and Uchiyama can be used to connect two types of Grothendieck classes by the $K$-theoretic pushforward from the Grothendieck group of Grassmann bundles to the Grothendieck group of a nonsingular variety. Using the commutation relation, we show that two types of partition functions of an integrable five-vertex model, which can be explicitly described using skew Grothendieck polynomials, and can be viewed as Grothendieck classes, are directly connected by the $K$-theoretic pushforward. We show that special cases of the pushforward formula which correspond to the nonskew version are also special cases of the formulas derived by Buch. We also present a skew generalization of an identity for the Grothendieck polynomials by Guo and Sun, which is an extension of the one for Schur polynomials by Fehér, Némethi and Rimányi. We also show an application of the pushforward formula and derive an integration formula for the Grothendieck polynomials.

math-ph↗

arXiv2020

Refined dual Grothendieck polynomials, integrability, and the Schur measure

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.

math.CO↗

arXiv2020

Integrability approach to Feher-Nemethi-Rimanyi-Guo-Sun type identities for factorial Grothendieck polynomials

Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Fehér, Némethi and Rimányi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive an identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get a duality formula. We also discuss a $q$-deformation of the Guo-Sun identity.

math.CO↗

arXiv2020

A class of partition functions associated with $E_{τ,η}(gl_3)$ by Izergin-Korepin analysis

Recently, a class of partition functions associated with higher rank rational and trigonometric integrable models were introduced by Foda and Manabe. We use the dynamical $R$-matrix of the elliptic quantum group $E_{τ,η}(gl_3)$ to introduce an elliptic analogue of the partition functions associated with $E_{τ,η}(gl_3)$. We investigate the partition functions of Foda-Manabe type by developing a nested version of the elliptic Izergin-Korepin analysis, and present the explicit forms as symmetrization of multivariable elliptic functions. We show that special cases are essentially the elliptic weights functions introduced in the works by Rimányi-Tarasov-Varchenko, Konno, Felder-Rimányi-Varchenko.

math-ph↗

arXiv2019

Quantum inverse scattering method and generalizations of symplectic Schur functions and Whittaker functions

We introduce generalizations of type $C$ and $B$ ice models which were recently introduced by Ivanov and Brubaker-Bump-Chinta-Gunnells, and study in detail the partition functions of the models by using the quantum inverse scattering method. We compute the explicit forms of the wavefunctions and their duals by using the Izergin-Korepin technique, which can be applied to both models. For type $C$ ice, we show the wavefunctions are expressed using generalizations of the symplectic Schur functions. This gives a generalization of the correspondence by Ivanov. For type $B$ ice, we prove that the exact expressions of the wavefunctions are given by generalizations of the Whittaker functions introduced by Bump-Friedberg-Hoffstein. The special case is the correspondence conjectured by Brubaker-Bump-Chinta-Gunnells. We also show the factorized forms for the domain wall boundary partition functions for both models. As a consequence of the studies of the partition functions, we obtain dual Cauchy formulas for the generalized symplectic Schur functions and the generalized Whittaker functions.

math-ph↗

arXiv2019

A duality formula between elliptic determinants

We prove a duality formula between two elliptic determinants. We present a proof which is a variant of the Izergin-Korepin method which is a method originally introduced to analyze and compute partition functions of integrable lattice models.

math.CA↗

arXiv2018

Izergin-Korepin analysis on the wavefunctions of the $U_q(sl_2)$ six-vertex model with reflecting end

We extend the recently developed Izergin-Korepin analysis on the wavefunctions of the $U_q(sl_2)$ six-vertex model to the reflecting boundary conditions. Based on the Izergin-Korepin analysis, we determine the exact forms of the symmetric functions which represent the wavefunctions and its dual. Comparison of the symmetric functions with the coordinate Bethe ansatz wavefunctions for the open XXZ chain by Alcaraz-Barber-Batchelor-Baxter-Quispel is also made. As an application, we derive algebraic identities for the symmetric functions by combining the results with the determinant formula of the domain wall boundary partition function of the six-vertex model with reflecting end.

math-ph↗