Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. 1.

Dedicated to the memory of Gian-Carlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the η-function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing

The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.

Research supported by the Australian Research Council.

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. 1

Introduction An alternating-sign matrix of order n is an n-by-n array of 0's, +1's and 1's with the property that in each row and each column, the non-zero entries alternate in sign, beginning and ending with a +1. For example, Figure 1 shows an alternating-sign matrix (ASM for short) of order 4. 0 B B @ 0 +1 0 0 +1 1 +1 0 0 0 0 +1 0 +1 0 0 1 C C A Figure 1: An alternating-sign matrix of order 4. Figure 2 exhibits all seven of the ASMs of order 3. 0 @ 0 0 +1 0 +1 0 +1 0 0 1 A 0 @ 0 0 +1 +1 0 0 0 +1 0 1 A 0 @ 0 +1 0 0 0 +1 +1 0 0 1 A 0 @ 0 +1 0 +1 1 +1 0 +1 0 1 A 0 @

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. 1 1

We prove a conjecture of Cohn and Propp, which refines a conjecture of Bosley and Fidkowski about the symmetry of the set of alternating sign matrices (ASMs). We examine data arising from the representation of an ASM as a collection of paths connecting 2n vertices and show it to be invariant under the dihedral group D2n rearranging those vertices, which is much bigger than the group of symmetries of the square. We also generalize conjectures of Propp and Wilson relating some of this data for different values of n. 1

We prove that the sum of entries of the suitably normalized groundstate vector of the O(1) loop model with periodic boundary conditions on a periodic strip of size 2n is equal to the total number of n×n alternating sign matrices. This is done by identifying the state sum of a multi-parameter inhomogeneous version of the O(1) model with the partition function of the inhomogeneous six-vertex model on a n × n square grid with domain wall boundary conditions. 1.

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