More refined enumerations of alternating sign matrices

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general “d-refined” enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_n,i,j that enumerate such matrices. We give a conjectural explicit formula for A_n,i,j and formulate several other conjectures about the sufficiency of the linear equations to determine the A_n,i,j's and about an extension of the linear equations to the general d-refined enumerations.     

New enumeration formulas for alternating sign matrices and square ice partition functions

The refined enumeration of alternating sign matrices (ASMs) of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study, starting with the Refined Alternating Sign Matrix Conjecture of Mills–Robbins–Rumsey (1983) [25], its proof by Zeilberger (1996) [31], and more recent work on doubly-refined and triply-refined enumerations by several authors. In this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the “top–left–bottom” (triply-refined) and “top–left–bottom–right” (quadruply-refined) enumerations. The latter case solves the problem of computing the full boundary correlation function for ASMs. The enumeration formulas are proved by deriving new representations, which are of independent interest, for the partition function of the square ice model with domain wall boundary conditions at the “combinatorial point” η=2π/3$\eta =2\pi /3$.     

Linear relations of refined enumerations of alternating sign matrices

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 some left and right columns are fixed. The main result is a simple linear relation between the number of n×n alternating sign matrices where the top row as well as the left and the right column is fixed and the number of n×n alternating sign matrices where the two top rows and the bottom row are fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only some top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely.     

Multiply-refined enumeration of alternating sign matrices

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1’s in the first and last rows and columns of an ASM, and the numbers of generalized inversions and −1’s in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. The derivation of the relation involves combining the Desnanot–Jacobi determinant identity with the Izergin–Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions.     

q-Enumeration of alternating sign matrices with exactly one −1

We enumerate the alternating sign matrices that contain exactly one −1 according to their number of inversions (possibly taking into account the position of the unique non-zero entry in the first row). In conformity with the Mills, Robbins and Rumsey conjectures, this is the same as the enumeration, according to the number of parts, of descending plane partitions with exactly one special part. This is shown by finding a determinantal expression for the generating function of descending plane partitions, transforming it algebraically and extracting recurrences for those with one special part. Finally, we show that the generating function of alternating sign matrices that contain exactly one −1 follows the same recurrences.RésuméOn énumère les matrices à signes alternants qui ne contiennent qu'un seul −1 selon leur nombre d'inversions (en tenant compte possiblement de la position de la seule entrée non nulle de la première ligne). Conformément aux conjectures de Mills, Robbins et Rumsey, ceci revient à l’énumération, selon le nombre de parts, des partitions planes descendantes qui n'ont qu'une seule part spéciale. Pour le démontrer, on obtient d'abord la fonction génératrice des partitions planes descendantes sous forme d'un déterminant qu'on transforme algébriquement pour en extraire des récurrences qui caractérisent celles n'ayant qu'une part spéciale. Finalement, on montre que la fonction génératrice des matrices à signes alternants vérifie les mêmes récurrences.     

Sign conjugacy classes of the alternating groups

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0,1 or ?1 on C. In this paper, we classify the sign conjugacy classes of alternating groups.     

On the weighted enumeration of alternating sign matrices and descending plane partitions

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (3) (1983) 340-359] that, for any n, k, m and p, the number of n×n alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m −1?s and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of n×n matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindström-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.     

Bijective proofs of shifted tableau and alternating sign matrix identities

We give a bijective proof of an identity relating primed shifted gl(n)-standard tableaux to the product of a gl(n) character in the form of a Schur function and . This result generalises a number of well-known results due to Robbins and Rumsey, Chapman, Tokuyama, Okada and Macdonald. An analogous result is then obtained in the case of primed shifted sp(2n)-standard tableaux which are bijectively related to the product of a t-deformed sp(2n) character and . All results are also interpreted in terms of alternating sign matrix (ASM) identities, including a result regarding subsets of ASMs specified by conditions on certain restricted column sums.     

Normal coverings of finite symmetric and alternating groups

In this paper we investigate the minimum number of maximal subgroups Hi, i=1,…,k of the symmetric group Sn (or the alternating group An) such that each element in the group Sn (respectively An) lies in some conjugate of one of the Hi. We prove that this number lies between a?(n) and bn for certain constants a,b, where ?(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+?(n)/2, and we determine the exact value for Sn when n is odd and for An when n is even.     

A new proof of the refined alternating sign matrix theorem

In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of n×n alternating sign matrices with a 1 at the top of the ith 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.     

Extendible elements of the alternating groups

An element?σ?of A_n , the Alternating group of degree n, is extendible in S_n , the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

Covering the alternating groups by products of cycle classes

Given integers k,l?2, where either l is odd or k is even, we denote by n=n(k,l) the largest integer such that each element of An is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay(An,Cl), where Cl is the set of all l-cycles in An. We prove that if k?2 and l?9 is odd and divisible by 3, then . This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368-380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87-99].     

Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups

Let G be a finite alternating or symmetric group. We describe an infinite class of finite lattices, none of which is isomorphic to any interval [H,G] in the subgroup lattice of G.     

Character formulae for classical groups

We give formulae relating the value X_λ (g) of an irreducible character of a classical group G to entries of powers of the matrix g ε G. This yields a far-reaching generalization of a result of J.L. Cisneros-Molina concerning the GL ₂ case [1]. Partially supported by OTKA grants T 042769 and T 046365     

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

We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices.     

Rational irreducible characters and rational conjugacy classes in finite groups

We prove that a finite group has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.