Functional Relations in Solvable Lattice Models I: Functional Relations and Representation Theory

Kuniba, Atsuo; Наканиши, Томоки; Suzuki, Junji

doi:10.1142/s0217751x94002119

Cited by 300 publications

(482 citation statements)

References 54 publications

(189 reference statements)

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“…They were recently combined into a universal bilinear form [10], [11]. The bilinear functional relations have the most simple closed form for the models of the A k−1 -series and representations corresponding to rectangular Young diagrams.…”

Section: Introductionmentioning

confidence: 99%

Quantum Integrable Models and Discrete Classical Hirota Equations

Krichever

Lipan

Wiegmann

et al. 1997

Communications in Mathematical Physics

246

355

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Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota's equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for A k−1 -type models appear as discrete time equations of motions for zeros of classical τ -functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained.

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Section: Introductionmentioning

confidence: 99%

Quantum Integrable Models and Discrete Classical Hirota Equations

Krichever

Lipan

Wiegmann

et al. 1997

Communications in Mathematical Physics

246

355

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“…The T -systems appear in the solution of exactly solvable models in statistical mechanics, in the Bethe ansatz of generalized Heisenberg quantum spin chains based on representations of Yangians of each simple Lie algebra [18,21]. The transfer matrices of the model satisfy a recursion relation in the highest g-weight of the Y (g)-modules corresponding to the auxiliary space.…”

Section: 2mentioning

confidence: 99%

“…The T -systems in general are related to the so-called Y -systems via a birational transformation [21]. The following is a quantum version of this transformation.…”

Section: 2mentioning

confidence: 99%

“…The T -systems [18,21] satisfied by the transfer matrices of the generalized Heisenberg model or the q-characters of quantum affine algebras [16] can be considered as discrete dynamical systems with special initial conditions. More generally, the equations of these systems can be shown [6] to be mutations in an infinite-rank cluster algebra [13].…”

Section: Introductionmentioning

confidence: 99%

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The Solution of the Quantum A 1 T-System for Arbitrary Boundary

Francesco

Kedem

2012

Commun. Math. Phys.

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Abstract. We solve the quantum version of the A 1 T -system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infiniterank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A 1 Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T -system and the quantum lattice Liouville equation, which is the quantized Y -system.

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“…Definitions, initial data, and cluster algebra connection The so-called T -systems are 2 + 1-dimensional discrete integrable systems of evolution equations in a discrete time variable k ∈ Z. They were introduced in the context of integrable quantum spin chains, as a system of equations satisfied by the eigenvalues of transfer matrices of generalized Heisenberg magnets, with the symmetry of a given Lie algebra [21]. In the case of type A, the T -system equation is also known as the octahedron recurrence, and appears to be central in a number of combinatorial objects, such as the lambda-determinant and the Alternating Sign Matrices [24] [8], the puzzles for computing Littlewood-Richardson coefficients [20], generalizations of Coxeter-Conway frieze patterns [5][1] [3], and the domino tilings of the Aztec diamond [12] [25].…”

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confidence: 99%

T-Systems, Networks and Dimers

Francesco

2014

Commun. Math. Phys.

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Abstract. We study the solutions of the T-system for type A, also known as the octahedron equation, viewed as a 2+1-dimensional discrete evolution equation. These may be expressed entirely in terms of the stepped surface over which the initial data are specified, via a suitably defined flat GL n connection which embodies the integrability of this infinite rank system. By interpreting the connection as the transfer operator for a directed graph or network with weighted edges, we show that the solution at a given point is expressed as the partition function for dimers on a bipartite graph dual to the "shadow" of the point onto the initial data stepped surface. We extend the result to the case of other geometries such as that of the evaporation of a cube corner crystal, and to a reformulation of the Kenyon-Pemantle discrete hexahedron equation.

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Functional Relations in Solvable Lattice Models I: Functional Relations and Representation Theory

Cited by 300 publications

References 54 publications

Quantum Integrable Models and Discrete Classical Hirota Equations

Quantum Integrable Models and Discrete Classical Hirota Equations

The Solution of the Quantum A 1 T-System for Arbitrary Boundary

T-Systems, Networks and Dimers

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