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.

Abstract. It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the Macdonald conjectures. It further initiated the study of q-analogues, which in turn enriched the Macdonald conjectures. We review these developments and proceed to exhibit the sustained prominence of the Selberg integral as evidenced by its central role in random matrix theory, Calogero–Sutherland quantum many-body systems, Knizhnik–Zamolodchikov equations, and multivariable orthogonal polynomial

The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [8] based on the lattice path method. The second aim is to study the applications of the minor summation formula to obtain several identities such as a variant of the Sundquist formula established in [31] about some extension of Pfaffian’s version of the Cauchy determinant, a simple proof of Kawanaka’s formula concerning a q-series identity involving Schur functions [15] (also, of the identity in [16] which is regarded as a determinant version of the previous one from our point of view). We also establish a certain identity similar to the Kawanaka formula.

The decomposition of the Laughlin wave function in the Slater orthogonal basis appears in the discussion on the second-quantized form of the Laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the Vandermonde determinants in the Schur basis. Such a computation is notoriously difficult and the coefficients of the expansion have not yet been interpreted. In our paper, we give an expression of these coefficients in terms of hyperdeterminants of sparse tensors. We use this result to construct an algorithm allowing to compute one coefficient of the development without computing the others. Thanks to a program in C, we performed the calculation for the square of the Vandermonde up to an alphabet of eleven lettres. 1

The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [8] based on the lattice path method. The second aim is to study the applications of the minor summation formula to obtain several identities such as a variant of the Sundquist formula established in [31] about some extension of Pfaffian’s version of the Cauchy determinant, a simple proof of Kawanaka’s formula concerning a q-series identity involving Schur functions [15] (also, of the identity in [16] which is regarded as a determinant version of the previous one from our point of view). We also establish a certain identity similar to the Kawanaka formula.