Short proof of the ASM theorem avoiding the six-vertex model

We introduce a new family 𝒜 n,k of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for k=1, this family is equal to a multivariate generating function involving n+3 variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. The proof is based on a new antisymmetrizer-to-determinant formula for which we also provide a combinatorial proof, and, although this proof is complicated, it is an important step forward as it is very hard to find combinatorial proofs in this field. In the second part, we relate three specialisations of 𝒜 n,k to weighted enumerations of certain well-known classes of column strict shifted plane partitions that generalise descending plane partitions.

We demonstrate a natural bijection between a subclass of alternating sign matrices (ASMs) defined by a condition on the corresponding monotone triangle which we call the gapless condition and a subclass of totally symmetric self-complementary plane partitions defined by a similar condition on the corresponding fundamental domains or Magog triangles. We prove that, when restricted to permutations, this class of ASMs reduces to 312-avoiding permutations. This leads us to generalize pattern avoidance on permutations to a family of words associated to ASMs, which we call Gog words. We translate the gapless condition on monotone triangles into a pattern avoidance-like condition on Gog words associated. We estimate the number of gapless monotone triangles using a bijection with $p$-branchings.

The six-vertex model with domain wall boundary conditions, or squareice, is considered for particular values of its parameters, corresponding to1-,2-,and 3-enumerations of alternating sign matrices (ASMs). Using Hankel determinant representationsfor the partition function and the boundary correlator of homogeneous square ice, it is shownhow the ordinary and refined enumerations can be derived in a very simple and straightforwardway. The derivation is based on the standard relationship between Hankel determinantsand orthogonal polynomials. For the particular sets of parameters corresponding to1-,2-,and 3-enumerations of ASMs, the Hankel determinant can be naturally related to ContinuousHahn, Meixner–Pollaczek, and Continuous Dual Hahn polynomials, respectively. Thisobservation allows for a unified and simplified treatment of ASM enumerations.In particular, along the lines of the proposed approach, we provide a completesolution to the long-standing problem of the refined 3-enumeration of ASMs.

We survey the connections between the six-vertex (square ice) model of 2D statistical mechanics and random matrix theory. We highlight the same universal probability distributions appearing on both sides, and also indicate related open questions and conjectures. We present full proofs of two asymptotic theorems for the six-vertex model: in the first one the Gaussian Unitary Ensemble (GUE) and GUE–corners process appear; the second one leads to the Tracy–Widom distribution F 2 F_2 . While both results are not new, we found shorter transparent proofs for this text. On our way we introduce the key tools in the study of the six-vertex model, including the Yang–Baxter equation and the Izergin–Korepin formula.

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 function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

A new combinatorial proof of the Protasov-Voynov theorem on the structure of irreducible semigroups of nonnegative matrices is proposed. The original proof was obtained by geometric methods.

In this note we shall fill a gap in the original proof of the Main Theorem in the previous paper ^4], and sharpen the Theorem by showing that the condition (a) alone is sufficient to derive the conclusion of the theorem. This sharpened theorem enables us to solve a problem posed by K. Kodaira in [2]. The authors are grateful to Professor Mudumbai S. Narasimhan for calling our attention to the gap, to him, to Takahiro Kawai and to other friends for discussion about this. § 1. Vanishing of Cohomology In the proof of theorem 1 in Q4], we made too easy use of Serre's duality theorem. Let V be a connected paracompact complex analytic manifold of dimension n and E a holomorphic vector bundle over V. Let us denote by Cp>q=-Cp'q(V, E) the space of E -valued differential forms of type (p, q) and of class C°°, with the topology as given in Q5]. (Cp>q stands for Ap*9 in [5].) The space K%~pt*~q of £*-valued currents of type (n—p, n — q) with compact supports is isomorphic to the topological dual of Cp>9, and the transpose of the sequence

A direct bijection between descending plane partitions with no special parts and permutation matrices

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 the integer points of the $r$-th dilate of an integral convex polytope of dimension $(n{-}1)^2$ whose vertices are the standard alternating sign matrices of size $n$. It then follows that, for fixed $n$, these matrices are enumerated by an Ehrhart polynomial in $r$.

A well known theorem of Torelli states that any compact Riemann surface is determined by its principally polarized jacobian variety. Torelli's original proof (see [T]) has been revisited and adapted to any characteristic of the base field by Matsusaka (see [M]). This Author slightly modified Torelli's argument avoiding the use of some Schubert's enumerative formulas, rigourosly proved only later over by Macdonald (see [MC]), and over any algebraically closed field by Ghione (unpublished). On the other hand Matsusaka's proof uses a suitable version of Castelnuovo­Humbert's theorem and a weak form of Riemann's singularity theorem. A second proof of Torelli's theorem was given in 1952 by Andreotti (see [A1]) and adapted to any by Weil and Andreotti himself (see [A2], [W]). There is however a third proof due to Comessatti (see [C]), which seems to have been ignored in the current literature. It is, in our opinion, a very elegant proof, and we believe it useful to give an account of it in this note. Comessatti's argrnnent (see § 3) basically relies upon two results (theorems (1.8) and (2.5)), both somewhat hidden in the, rather obscure, original exposition. We have worked on an algebraically closed field of any characteristic. This required some care which, in spite of the plainess of the involved ideas, has made the exposition longer than we originally believed. Anyhow tne proof can be made strictkingly simple in characteristic zero, avoiding the use of Castelnuovo­Humbert's theorem, present in comessatti's argument and in our theorem (1.8).This simplification (see 3) has been achieved by virtue of a result (theorem (9.1)) very easy to prove, but only in characteristic zero.

We give proofs of summation theorems and continued fraction evaluations conjectured by R. W. Gosper.We also give two new elementary proofs of a theorem of Gosper whose original proof uses matrix methods.One proof uses iteration of two term recurrence relation.The latter proof is also used to give elementary proofs of three other identities due to Gosper.

We use statistical mechanics--variants of the six-vertex model in the plane studied by means of the Yang---Baxter equation--to give new deformations of Weyl's character formula for classical groups of Cartan type B, C, and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated with the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.

Weak order and descents for monotone triangles

Michael Somos conjectured a relation between Hankel determinants whose entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of $3$, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we have found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for non-integrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappearand the Poisson distribution be recovered on (non Yang--Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion we have tested the generic calculability of the model on these algebraic varieties.