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22
Symmetry classes of alternatingsign matrices under one
"... In a previous article [23], we derived the alternatingsign matrix (ASM) theorem from the IzerginKorepin determinant [12, 13, 19] for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternatingsign matric ..."
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In a previous article [23], we derived the alternatingsign matrix (ASM) theorem from the IzerginKorepin determinant [12, 13, 19] for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternatingsign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (halfturnsymmetric ASMs), and even QTSASMs (quarterturnsymmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [31]. We introduce several new types of ASMs: UASMs (ASMs with a Uturn side), UUASMs (two Uturn sides), OSASMs (offdiagonally symmetric ASMs), OOSASMs (offdiagonally, offantidiagonally symmetric), and UOSASMs (offdiagonally symmetric with Uturn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs (vertically and horizontally symmetric ASMs) and another new class, VHPASMs (vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally symmetric), DASASMs (diagonally, antidiagonally symmetric), and TSASMs (totally symmetric ASMs). We enumerate several of these new classes, and we provide several 2enumerations
Loops, matchings and alternatingsign matrices
 DISCR. MATH
, 2008
"... The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge ..."
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Cited by 44 (6 self)
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The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge tilings of hexagons with cut off corners.
The many faces of alternatingsign matrices
, 2008
"... I give a survey of different combinatorial forms of alternatingsign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as cornersum matrices, heightfunction matrices, threecolorings, monotone triangles, tetrahedral order ideals, square ice, gasketandbasket ..."
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Cited by 27 (0 self)
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I give a survey of different combinatorial forms of alternatingsign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as cornersum matrices, heightfunction matrices, threecolorings, monotone triangles, tetrahedral order ideals, square ice, gasketandbasket tilings and full packings of loops. (This article has been published in a conference edition of the journal Discrete Mathematics and Theoretical
Correlation functions for the open XXZ chain I
"... We consider the XXZ spin chain with diagonal boundary conditions in the framework of algebraic Bethe Ansatz. Using the explicit computation of the scalar products of Bethe states and a revisited version of the bulk inverse problem, we calculate the elementary building blocks for the correlation func ..."
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Cited by 17 (3 self)
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We consider the XXZ spin chain with diagonal boundary conditions in the framework of algebraic Bethe Ansatz. Using the explicit computation of the scalar products of Bethe states and a revisited version of the bulk inverse problem, we calculate the elementary building blocks for the correlation functions. In the limit of halfinfinite chain, they are obtained as multiple integrals of usual functions, similar to the case of periodic boundary conditions. 1
Spin Chains with NonDiagonal Boundaries and Trigonometric SOS Model with Reflecting End
, 2011
"... In this paper we consider two a priori very different problems: construction of the eigenstates of the spin chains with non parallel boundary magnetic fields and computation of the partition function for the trigonometric solidonsolid (SOS) model with one reflecting end and domain wall boundary co ..."
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In this paper we consider two a priori very different problems: construction of the eigenstates of the spin chains with non parallel boundary magnetic fields and computation of the partition function for the trigonometric solidonsolid (SOS) model with one reflecting end and domain wall boundary conditions. We show that these two problems are related through a gauge transformation (socalled vertexface transformation) and can be solved using the same dynamical reflection algebras.
The Heisenberg XXX model with general boundaries: Eigenvectors from . . .
 SIGMA
, 2013
"... We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particu ..."
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Cited by 8 (1 self)
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We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain. We prove this result for the chains with small length. We obtain also an offshell equation (i.e. satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.
Punctured plane partitions and the qdeformed KnizhnikZamolodchikov and Hirota equations
"... Abstract. We consider partial sum rules for the homogeneous limit of the solution of the qdeformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are ..."
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Cited by 6 (5 self)
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Abstract. We consider partial sum rules for the homogeneous limit of the solution of the qdeformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ 2weighted punctured cyclically symmetric transpose complement plane partitions where τ = −(q+q −1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ 2enumerations of vertically symmetric alternating sign matrices and modifications thereof. Contents
Sum Rule for the EightVertex Model on ITS COMBINATORIAL LINE
 IN KENJI IOHARA, SOPHIE MORIERGENOUD AND BERTRAND RÉMY (EDS.), SYMMETRIES, INTEGRABLE SYSTEMS AND REPRESENTATIONS, VOL
, 2013
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Refined Cauchy/Littlewood identities and sixvertex model partition functions
 II. Proofs and new conjectures. 2014. arXiv:1405.7035. ON A FAMILY OF SYMMETRIC RATIONAL FUNCTIONS 36
"... Abstract. We prove two identities of Hall–Littlewood polynomials, which appeared recently in [2]. We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition functions associated with symmetry classes of alternating sign matrices. The ..."
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Abstract. We prove two identities of Hall–Littlewood polynomials, which appeared recently in [2]. We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition functions associated with symmetry classes of alternating sign matrices. These identities generalize those already found in [2], via the introduction of additional parameters. The left hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity. The right hand side of each identity is (one of the two factors present in) the partition function of the sixvertex model on a relevant domain. 1.
The art of number guessing: where combinatorics meets physics
, 2002
"... The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge ..."
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The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge tilings of hexagons with cut off corners.