Integrable Combinatorics

Francesco, P. Di

doi:10.1142/9789814449243_0001

Cited by 4 publications

(4 citation statements)

References 21 publications

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“…Note that a specialization of this τ -weighted TSSCPP generating function equals the degree of the variety of strictly upper triangular 2n × 2n complex matrices of vanishing square [10]. See also [35] and Section 4.2 of [11]. We replace up steps in the nest of non-intersecting lattice paths by ones and diagonal steps by zeros to obtain a new triangular integer array we call a TSSCPP boolean triangle, which we will use in our main bijection in Section 3.…”

Section: 2mentioning

confidence: 99%

Permutation Totally Symmetric Self-Complementary Plane Partitions

Striker

2018

Ann. Comb.

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Alternating sign matrices and totally symmetric self-complementary plane partitions are equinumerous sets of objects for which no explicit bijection is known. In this paper, we identify a subset of totally symmetric self-complementary plane partitions corresponding to permutations by giving a statistic-preserving bijection to permutation matrices, which are a subset of alternating sign matrices. We use this bijection to define a new partial order on permutations, and prove this new poset contains both the Tamari lattice and the Catalan distributive lattice as subposets. We also study a new partial order on totally symmetric self-complementary plane partitions arising from this perspective and show that this is a distributive lattice related to Bruhat order when restricted to permutations. 4 be rectangular, in which case we complete the array to a rectangle by adding entries equal to zero. So we have the following n × m array satisfying t i,j ≥ t i+1,j ≥ 0 and t i,j ≥ t i,j+1 ≥ 0. t

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Section: 2mentioning

confidence: 99%

Permutation Totally Symmetric Self-Complementary Plane Partitions

Striker

2018

Ann. Comb.

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“…From a different point of view, the XX0 model is closely related to integrable combinatorics and probability, i.e. the class of integrable combinatorial lattice models [11], [12], [13]. These are twoand three-dimensional integrable lattice models such as dimer models, nonintersecting Brownian motion and plane partitions (crystal melting models), etc.…”

Section: Introductionmentioning

confidence: 99%

“…The XX0 model has diverse relations with recent topics of research in mathematical physics such as integrable combinatorics and probability, i.e. the class of integrable combinatorial lattice models [11][12][13]. These are two-and three-dimensional integrable lattice models such as random tilings and dimer models [14], nonintersecting Brownian motion [15], plane partitions and theory of symmetric functions [16,17], closely related to other topics such as alternating sign matrices [18] and topological string theory [19].…”

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confidence: 99%

Phase structure of XX0 spin chain and nonintersecting Brownian motion

Saeedian

2018

J. Stat. Mech.

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We study finite size and temperature XX0 Heisenberg spin chain in weak and strong coupling regimes. By using an elegant connection of the model to integrable combinatorics and probability, we explore and interpret a possible phase structure of the model in asymptotic limit: the limit of large inverse temperature and size. First, the partition function and free energy of the model are derived by using techniques and results from random matrix models and nonintersecting Brownian motion. We show that, in the asymptotic limit, partition function of the model, written in terms of matrix integral, is governed by the Tracy-Widom distribution. Second, the exact analytic results for the free energy, which is obtained by the asymptotic analysis of the Tracy-Widom distribution, indicate a completely new and sophisticated phase structure of the model. This phase structure consists of second-and third-order phase transitions. Finally, to shed light on our new results, we provide a possible new interpretation of the phase structure in terms of dynamical behaviour of magnons in the spin chain. We demonstrate distinct features of the phases with schematic spin configurations which have definite features in each region of the phase diagram.

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“…It was introduced by Toda by considering the lattice with exponential interaction [46] and thereafter many applications of the Toda lattice have been studied, thus promoting the developments of integrable systems [22], random matrix theory [3], conformal field theory [19] and so on. It should be pointed out that the Toda lattice does not only connect with physically meaningful subjects, but relates to many branches of mathematics such as integrable geometry [29,34], integrable combinatorics [17,42], integrable probability [43], representation theory [49] etc. One of them involved in this article is the interplay between orthogonal polynomials and Toda-type equations, which is a beautiful subject and one can refer to [11,33,48] for some reviews.…”

Section: Introductionmentioning

confidence: 99%

On moving frames and Toda lattices of BKP and CKP types

Wang

Chang

et al. 2018

J. Phys. A: Math. Theor.

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This paper is mainly concerned with the geometric formulations of Toda lattices of BKP and CKP types as evolutions of invariants and their explicit expressions. The theory for discrete curves in centro-affine geometry is constructed by using the method of a moving frame. With the help of orthogonal polynomial theory, we choose the appropriate evolutions for the curves and the evolutions induced on invariants are related to these Toda-type lattices. In addition, the explicit expressions for the discrete invariant curve flows, discrete moving frames and Maurer–Cartan invariants are provided.

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Integrable Combinatorics

Abstract: Abstract. We review various combinatorial problems with underlying classical or quantum integrable structures.

Cited by 4 publications

References 21 publications

Permutation Totally Symmetric Self-Complementary Plane Partitions

Permutation Totally Symmetric Self-Complementary Plane Partitions

Phase structure of XX0 spin chain and nonintersecting Brownian motion

On moving frames and Toda lattices of BKP and CKP types

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