Elliptic supersymmetric integrable model and multivariable elliptic functions

Motegi, Kohei

doi:10.1093/ptep/ptx159

Cited by 4 publications

(7 citation statements)

References 49 publications

(70 reference statements)

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“…We apply his technique to the elliptic Felderhof model and obtain the determinant formula for the scalar products. Together with our results on the correspondence between the wavefunctions and the elliptic Schur functions [44,45] obtained by the Izergin-Korepin analysis on the wavefunctions, we derive the Cauchy formula for the elliptic Schur functions. This paper is organized as follows.…”

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confidence: 72%

“…The scalar products can be evaluated in another way as a sum over products of the wavefunctions. For the case of the elliptic Felderhof models, the wavefunctions are expressed as a deformed elliptic Vandermonde determinant and elliptic Schur functions, which can be shown for example by the recently-developed Izergin-Korepin technique to analyze the wavefunctions [43,44,45]. By combining this way of evaluation with the direct evaluation which gives the determinant formula, we obtained the Cauchy formula for the elliptic Schur functions.…”

Section: Resultsmentioning

confidence: 99%

“…This characterization lead Izergin [42] to found the determinant representation (Izergin-Korepin determinant) of the domain wall boundary partition functions of the U q (sl 2 ) six-vertex model. We recently extended the Izergin-Korepin technique to be able to analyze the wavefunctions [44,45,43]. We apply this technique here.…”

Section: Appendixmentioning

confidence: 99%

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Scalar products of the elliptic Felderhof model and elliptic Cauchy formula

Motegi

2018

Journal of Geometry and Physics

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We analyze the scalar products of the elliptic Felderhof model introduced by Foda-Wheeler-Zuparic as an elliptic extension of the trigonometric face-type Felderhof model by Deguchi-Akutsu. We derive the determinant formula for the scalar products by applying the Izergin-Korepin technique developed by Wheeler to investigate the scalar products of integrable lattice models. By combining the determinant formula for the scalar products with the recently-developed Izergin-Korepin technique to analyze the wavefunctions, we derive a Cauchy formula for elliptic Schur functions. *

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confidence: 72%

Section: Resultsmentioning

confidence: 99%

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Scalar products of the elliptic Felderhof model and elliptic Cauchy formula

Motegi

2018

Journal of Geometry and Physics

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“…A similar proof is given for the wavefunction of the Felderhof model in the next section. Details for the XXZ-type models will appear elsewhere [17].…”

Section: Xxz-type Modelsmentioning

confidence: 99%

Partition functions of integrable lattice models and combinatorics of symmetric polynomials

Motegi,

Sakai,

Watanabe

2015

Preprint

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We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In particular, we examine the correspondence between the wavefunctions of the XXZ type, Felderhof type and the boson type integrable models and symmetric polynomials such as the Schur, Grothendieck and symplectic Schur functions. We also give a brief report of our work on generalizing the correspondence between the Felderhof models and factorial Schur and symplectic Schur functions.

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“…We evaluate the explicit representations of the partition functions using elliptic Pfaffians, and we get two Pfaffian representations based on two versions of the Izergin-Korepin analysis. The Izergin-Korepin analysis for various types of partition functions of trigonometric models [7,8,9,11,12,53] and a closely related functional equation approach have been extended to elliptic models, and have been used to compute the DWBPF, wavefunctions and scalar products of elliptic integrable models [29,30,31,54,55,56,57,58,59,60,61,62,63,64] in recent years. As a corollary of the two elliptic Pfaffian representations of the same partition functions by the elliptic Izergin-Korepin analysis, we get an identity between the two elliptic Pfaffians.This paper is organized as follows.…”

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confidence: 99%

Elliptic free-fermion model with OS boundary and elliptic Pfaffians

Motegi

2018

Lett Math Phys

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We introduce and study a class of partition functions of an elliptic free-fermionic face model. We study the partition functions with a triangular boundary using the off-diagonal K-matrix at the boundary (OS boundary), which was introduced by Kuperberg as a class of variants of the domain wall boundary partition functions. We find explicit forms of the partition functions with OS boundary using elliptic Pfaffians. We find two expressions based on two versions of Korepin's method, and we obtain an identity between two elliptic Pfaffians as a corollary. * E-mail: kmoteg0@kaiyodai.ac.jp [18,19,20,21,22,23,24,25,26], in which more simplified factorized representations of the partition functions were found, even for elliptic models.Studying elliptic generalizations of the DWBPF is interesting, and it is particularly interesting to find determinant and Pfaffian representations. In particular, finding representations using Pfaffians of a matrix whose matrix entries are elliptic functions is interesting, since there are only a few studies on elliptic Pfaffians. For example, Rosengren [27] introduced a family of elliptic Pfaffians and showed that the partition functions of the Andrews-Baxter-Forrester (ABF) model [28] at the supersymmetric point are expressed as a sum of two elliptic Pfaffians. We mention that expressions of the DWBPF of the ABF model which hold in generic parameters are derived in [29,30,31], a factorized expression at the free-fermion point is derived in [23,24,25]. and a single determinant representation was recently derived in [32].As for the properties of elliptic Pfaffians, Okada [33], Rosengren [34,35] and Rains [36] discovered several elliptic generalizations of the Pfaffian counterpart [37] of the Cauchy determinant formulas. The properties of elliptic determinants have been extensively studied. For example, several generalizations of the Cauchy determinant formula [38] have been discovered [39,40,41,42]. On the other hand, there are only a few results on elliptic Pfaffians by Okada, Rosengren and Rains.In this paper, we study partition functions of an elliptic free-fermionic face model with a triangular boundary, and show that they can be explicitly expressed using elliptic Pfaffians. The face model we treat can be regarded as degenerations of the ABF model [28], Okado-Deguchi-Martin (elliptic Perk-Schultz) model [43,44] and Foda-Wheeler-Zuparic (elliptic Felderhof) model [23], which are face-type counterparts of the elliptic vertex models [45,46], and are elliptic analogues of the trigonometric models of the U q (sl 2 ) six-vertex model, Perk-Schultz model and the Felderhof free-fermion model [47,48,49,50,51,52]. In this paper, we treat a fundamental example of the variations of the DWBPF introduced by Kuperberg [12]. Kuperberg introduced a class of partition functions of the U q (sl 2 ) six-vertex model with a triangular boundary using an off-diagonal boundary K-matrix at the boundary, and showed that they have explicit expressions using Pfaffians. He called this boundary condition the OS boun...

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Elliptic supersymmetric integrable model and multivariable elliptic functions

Cited by 4 publications

References 49 publications

Scalar products of the elliptic Felderhof model and elliptic Cauchy formula

Scalar products of the elliptic Felderhof model and elliptic Cauchy formula

Partition functions of integrable lattice models and combinatorics of symmetric polynomials

Elliptic free-fermion model with OS boundary and elliptic Pfaffians

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