I have placed this article on the math archive at the request of Greg Kuperberg. It is a manuscript written in the late 1980's. It is essentially unchanged from the orginal so it often does not reflect what is now known about the subject
An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices in each of these symmetry classes. We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class. Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions.
We extend the author's formula (2011) of weighted counting of inversions on permutations to the one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix recently shown by Brualdi-Schroeder and the author (both 2017) independently.
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without loops, and we determine the maximum number of edges in such graphs. We also consider n by n alternating sign mat…
We investigate alternating sign matrices that are not permutation matrices, but have finite order in a general linear group. We classify all such examples of the form $P+T$, where $P$ is a permutation matrix and $T$ has four non-zero entries, forming a square with entries $1$ and $-1$ in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices.
Alternating sign matrices with a U-turn boundary (UASMs) are a recent generalization of ordinary alternating sign matrices. Here we show that variations of these matrices are in bijective correspondence with certain symplectic shifted tableaux that were recently introduced in the context of a symplectic version of Tokuyama's deformation of Weyl's denominator formula. This bijection yields a formula for the weighted enumeration of UASMs. In this connection use is made of the link between UASMs and certain square ice configuration matrices.
In alternating sign matrices the first and last nonzero entry in each row and column is specified to be +1. Such matrices always exist. We investigate a generalization by specifying independently the sign of the first and last nonzero entry in each row and column to be either a +1 or a -1. We determine necessary and sufficient conditions for such matrices to exist.
The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.
I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops.
Lascoux and Schützenberger introduced a notion of key associated to any Young tableau. More recently Lascoux defined the key of an alternating sign matrix by recursively removing all -1's in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Moreover we obtain an elegant and direct way to compute the key of any Young tableau, and discuss consequences of our result.
We prove a determinantal identity concerning Schur functions for 2-staircase diagrams lambda=(ln+l',ln,l(n-1)+l',l(n-1),...,l+l',l,l',0). When l=1 and l'=0 these functions are related to the partition function of the 6-vertex model at the combinatorial point and hence to enumerations of Alternating Sign Matrices. A consequence of our result is an identity concerning the doubly-refined enumerations of Alternating Sign Matrices.
The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.
In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition a number of left and right columns are fixed. The main result is a simple linear relation between the number of $n \times n$ alternating sign matrices where the top row as well as the left and the right column is fixed and the number of $n \times n$ alternating sign matrices where the two top rows and the bottom row is fixed. This may be seen …
We compute 2-enumerations of certain halved alternating sign matrices. In one case the enumeration equals the number of perfect matchings of a halved Aztec diamond. In the other case the enumeration equals the number of perfect matchings of a halved fortress graph. Our results prove three conjectures by Jim Propp.
For an integer $n\geq 1$, we display a poset $Φ_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices. The Hasse diagram $\widehat Φ_n$ is obtained from the $n$-cube by adding some edges. We show that the dihedral group $D_{2n}$ acts on $\widehat Φ_n$ as a group of automorphisms.
We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the unique $1$ in the top row. We prove that the expansion can be expressed as a sum over totally symmetric plane partitions and we are also able to determine the coefficients. This establishes a new connection between alternating sign matrices and a class of plane partitions, thereby complementing the fact that alternating sign matrices are equinumerous with totall…
An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinanat and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round p…
We settle a question of Bressoud concerning the existence of an explicit bijection from a class of oriented square-ice graphs to a class of tournaments. We give an algorithm constructing such a bijection.
Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. In this paper, we prove the connectivity of alternating sign triangles, which is analogous to the connectivity of alternating sign matrices.
We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, $q$-analogs (which we name $q$TNN and $q$SFL properties). %In our discussion, essential sets, introduced by Fult…
We define a bijection that transforms an alternating sign matrix A with one -1 into a pair (N,E) where N is a (so called) ``neutral'' alternating sign matrix (with one -1) and E is an integer. The bijection preserves the classical parameters of Mills, Robbins and Rumsey as well as three new parameters (including E). It translates vertical reflection of A into vertical reflection of N. A hidden symmetry allows the interchange of E with one of the remaining two new parameters. A second bijection transforms (N,E) into a configuration of lattice paths called ``mixed configuration''.
In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of $n \times n$ alternating sign matrices with a 1 at the top of the $i$-th column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(-1) matrices generalizing alternating sign matrices.
Fischer provided a new type of binomial determinant for the number of alternating sign matrices involving the third root of unity. In this paper we prove that her formula, when replacing the third root of unity by an indeterminate $q$, is actually the $(2+q+q^{-1})$-enumeration of alternating sign matrices. By evaluating a generalisation of this determinant we are able to reprove a conjecture of Mills, Robbins and Rumsey stating that the $Q$-enumeration is a product of two polynomials in $Q$. Further we provide a closed product formula for the generalised determinant in the 0-,1- 2- and 3-enum…
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self--complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating funct…
Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are either 0 or 1, by analogy with permutation matrices. In this paper, we prove a simple characterization of permutation triangles, originally conjectured by Glick. We will also prove some properties of alternating sign triangles.
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. …
Half turn symmetric alternating sign matrices (HTSASMs) are special variations of the well-known alternating sign matrices which have a long and fascinating history. HTSASMs are interesting combinatorial objects in their own right and have been the focus of recent study. Here we explore counting weighted HTSASMs with respect to a number of statistics to derive an orthogonal group version of Tokuyama's factorisation formula, which involves a deformation and expansion of Weyl's denominator formula multiplied by a general linear group character. Deformations of Weyl's original denominator formula…
A Bruhat order for Latin squares and alternating sign hypermatrices
School of Mathematical and Statistical Sciences, University of Galway（高威大学数学与统计科学学院）；Department of Mathematics and Computer Studies, Mary Immaculate College（玛丽不可瑕疵学院数学与计算机研究系）
Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to $n \times n$ alternating sign matrices when prescribing $(1,2,...,n)$ as bottom row of the array. We define monotone $(d,m)$--trapezoids as monotone triangles with $m$ rows where the $d-1$ top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row $(k_1,...,k_n)$ is given by a polynomial $α(n…
An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an {\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by fixing one of the three indices, is an ASM. Several results concerning ASHMs are shown, such as finding the maximum number of nonzeros of an $n\times n\times n$ ASHM, and properties related to Latin squares. Moreover, we investigate completion problems, in which one asks if a subhypermatrix …
We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order $n$ but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the s…
Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra
LISN (UMR CNRS 9015), Université Paris Saclay, INRIA, CNRS（巴黎萨克雷大学）；Universitat de Barcelona & Centre de Recerca Matemàtica（巴塞罗那大学）；LIGM, Université Gustave Eiffel（巴黎东马恩拉瓦莱大学）
In 2007, the first author gave an alternative proof of the refined alternating sign matrix theorem by introducing a linear equation system that determines the refined ASM numbers uniquely. Computer experiments suggest that the numbers appearing in a conjecture concerning the number of vertically symmetric alternating sign matrices with respect to the position of the first 1 in the second row of the matrix establish the solution of a linear equation system similar to the one for the ordinary refined ASM numbers. In this paper we show how our attempt to prove this fact naturally leads to a more …
We define the alternating sign matrix polytope as the convex hull of nxn alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove t…
In their work on the infinite flag variety, Lam, Lee, and Shimozono (2018) introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of Lascoux (2002). Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguis…
The p-adic valuations of a sequence of integers T(n) counting alternating sign matrices is examined for p=2 and p=3. Symmetry properties of their graphs produce a new proof of the result that characterizes the indices for which T(n) is odd.
We show that the poset of alternating sign matrices, with Bruhat order, is isomorphic to the poset of certain submodules of the dominant Verma module for the special linear Lie algebra $\frak{sl}_n$. The latter poset consists of the intersections of Verma submodules and can also be defined in terms of a Kazhdan-Lusztig cell.
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…
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 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.
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 …
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.
The weighted projection of an alternating sign matrix (ASM) was introduced by Brualdi and Dahl (2018) as a step towards characterising a generalisation of Latin squares they introduced using alternating sign hypermatrices. If $z_n = (n,\dots,2,1)$, then the weighted projection of an ASM $A$ is equal to $z_n^TA$. Brualdi and Dahl proved that the weighted projection of an $n \times n$ ASM is majorized by the vector $z_n$, and conjectured that any positive integer vector majorized by $z_n$ is the weighted projection of some ASM. The main result of this paper presents a proof of this conjecture, v…
We consider the groundstate wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at $Δ=-{1/2}$, along with the groundstate wavefunction of the corresponding O($n$) loop model at $n=1$. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes.
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.
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.
In a previous article [math.CO/9712207], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (t…
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.
