Sum Rule for the Eight-Vertex Model on Its Combinatorial Line

Zinn-Justin, Paul

doi:10.1007/978-1-4471-4863-0_26

Cited by 20 publications

(50 citation statements)

References 41 publications

(89 reference statements)

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“…Zero-energy states. The supersymmetry singlets of the XYZ Hamiltonian (13) were studied in [11,75,37,38,98]. These investigations rely on the following existence conjecture: if the number of sites N = 2n + 1 is odd then there are exactly two linearly independent zero-energy states.…”

Section: An Example: the Xyz Chainmentioning

confidence: 99%

“…where the p n (z) are related to the polynomials found by Bazhanov and Mangazeev [11,75] for the spin−1/2 XYZ chain (we recall them in appendix B, see also [98]). The second factor in the square norm can also be written in terms of the polynomials p n (ζ):…”

Section: Conjecture 44 (Degree) With This Choice Of Normalisation Almentioning

confidence: 99%

“…This allowed to prove the early conjectures. Off-critical extensions to the XYZ model are known, too [11,75,98]. Yet, in many cases a combinatorial interpretation is still missing (see however [88,87]).…”

Section: Introductionmentioning

confidence: 99%

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Spin Chains with Dynamical Lattice Supersymmetry

Hagendorf

2013

J Stat Phys

114

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Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the N = (2, 2) superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension possess a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjectured.

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Section: An Example: the Xyz Chainmentioning

confidence: 99%

Section: Conjecture 44 (Degree) With This Choice Of Normalisation Almentioning

confidence: 99%

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Spin Chains with Dynamical Lattice Supersymmetry

Hagendorf

2013

J Stat Phys

114

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“…(63)]. A form of this result in which the odd orthogonal characters on the RHSs of (54) and (55) are replaced by certain symplectic characters was obtained previously by Zinn-Justin [48,Eq. (54)].…”

Section: 2mentioning

confidence: 87%

Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

Ayyer

Behrend

2019

Journal of Combinatorial Theory, Series A

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We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related factorizations involving sums of two Schur polynomials, and certain odd-sized sets of variables. Our results generalize the factorization identities proved by Ciucu and Krattenthaler (Advances in combinatorial mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that if, in some of the results, the partitions are taken to have rectangular or double-staircase shapes and all of the variables are set to 1, then factorization identities for numbers of certain plane partitions, alternating sign matrices and related combinatorial objects are obtained.2010 Mathematics Subject Classification. 05A15, 05E05, 05E10.

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“…A natural generalisation of the square norm of the eigenstates for the spin-one XXZ chain is the infinite-cylinder partition function (see [18] for an explanation of this interpretation) : . .…”

Section: Scalar Products and The Square Normmentioning

confidence: 99%

The nineteen-vertex model and alternating sign matrices

Hagendorf¹

2015

J. Stat. Mech.

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It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz, and both a simple component and its square norm are expressed in terms of the Izergin-Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natural polynomial normalisation scheme its square norm and the simple component coincide with generating functions for weighted enumeration of alternating sign matrices.

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Sum Rule for the Eight-Vertex Model on Its Combinatorial Line

Cited by 20 publications

References 41 publications

Spin Chains with Dynamical Lattice Supersymmetry

Spin Chains with Dynamical Lattice Supersymmetry

Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

The nineteen-vertex model and alternating sign matrices

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