We consider the alternating sign matrices of the odd order that have some kind of central symmetry. Namely, we deal with matrices invariant under the half-turn, quarter-turn and flips in both diagonals. In all these cases, there are two natural structures in the centre of the matrix. For example, for the matrices invariant under the half-turn the central element is equal $\pm 1$. It was recently found that $A^+_{HT}(2m+1)/A^-_{HT}(2m+1)$=(m+1)/m. We conjecture that similar very simple relations are valid in the two remaining cases.
It was shown by Kuperberg that the partition function of the square-ice model related to the quarter-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with the quarter-turn symmetric alternating-sign matrices of odd order, and show that the partition function of this model can be also written in a similar way. This allows to prove, in particular, the conjectures by Robbins related to the enumeration of the quarter-turn symmetric alternating-sign matrices.
It was shown by Kuperberg that the partition function of the square-ice model related to half-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with half-turn symmetric alternating-sign matrices of odd order. The partition function of the model is expressed via the above mentioned factors. The contributions to the partition function of the states corresponding to the alternating-sign matrices having 1 or -1 as the central entry are found and the related enumerations are obtained.
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbin…
For each $α\in \{0,1,-1 \}$, we count diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal number of $α$'s along the diagonal and the antidiagonal, as well as DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal and the antidiagonal. In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number of cyclically symmetric plane partitions in a given box, and the number of vert…
The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.
We consider a Hamiltonian formulation of the (2n+1)-order generalization of the Pais-Uhlenbeck oscillator with distinct frequencies of oscillation. This system is invariant under time translations. However, the corresponding Noether integral of motion is unbounded from below and can be presented as a direct sum of 2n one-dimensional harmonic oscillators with an alternating sign. If this integral of motion plays a role of a Hamiltonian, a quantum theory of the Pais-Uhlenbeck oscillator faces a ghost problem. We construct an alternative canonical formulation for the system under study to avoid t…
We prove the equality of doubly refined enumerations of Alternating Sign Matrices and of Totally Symmetric Self-Complementary Plane Partitions using integral formulae originating from certain solutions of the quantum Knizhnik--Zamolodchikov equation.
Zeilberger proved the Refined Alternating Sign Matrix Theorem, which gives a product formula, first conjectured by Mills, Robbins and Rumsey, for the number of alternating sign matrices with given top row. Stroganov proved an explicit formula for the number of alternating sign matrices with given top and bottom rows. Fischer and Romik considered a different kind of "doubly-refined enumeration" where one counts alternating sign matrices with given top two rows, and obtained partial results on this enumeration. In this paper we continue the study of the doubly-refined enumeration with respect to…
We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically and off-diagonally symmetric, quarter turn symmetric as well as quasi quarter turn symmetric alternating sign matrices. Our results prove conjectures of Fischer, Duchon and Robbins.
This work as an extension of our recent paper where we have found a numerical evidence for the fact that the numbers of the states of the fully packed loop (FPL) model with fixed link-patterns coincide with the components of the ground state vector of the dense O(1) loop model for periodic boundary conditions and an even number of sites. Here we give two new conjectures related to different boundary conditions. Namely, we suggest that the numbers of the half-turn symmetric states of the FPL model with fixed link-patterns coincide with the components of the ground state vector of the dense O(1)…
Lecture notes for the proceedings of the workshop "Algebraic Combinatorics related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S., Nara, Japan.
By the Baxter's $Q_{72}$-operator method, we demonstrate the equivalent theory between the generalized $τ^{(2)}$-model (other than two special cases with a pseudovacuum state) and the $N$-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $ς$) of $U_{\sf q}(sl_2)$ with ${\sf q}^N=1$ for odd $N$ is identified with either (for $ς^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for $ς^N \neq 1$) the de…
We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r=1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are …
In this paper we study the Rotor Model of Martins and Nienhuis. After introducing spectral parameters, a combined use of integrability, polynomiality of the ground state wave function and a mapping into the fully-packed O(1)-model allows us to determine the sum rule and a family of maximally nested components for different boundary conditions. We see in this way the appearance of 3-enumerations of Alternating Sign Matrices.
We study a one-parameter family of vector-valued polynomials associated to each simple Lie algebra. When this parameter $q$ equals -1 one recovers Joseph polynomials, whereas at $q$ cubic root of unity one obtains ground state eigenvectors of some integrable models with boundary conditions depending on the Lie algebra; in particular, we find that the sum of its entries is related to numbers of Alternating Sign Matrices and/or Plane Partitions in various symmetry classes.
The integrable loop model with mixed boundary conditions based on the 1-boundary extended Temperley--Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L+3. A refined counting is also considered.
The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes 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 aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.
