Buchbesprechungen

Ohne Zusammenfassung

---

Book reviews

     

The Interaction of a Gap with a Free Boundary in a Two Dimensional Dimer System

Let ℓ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let H{\mathcal{H}} be the half plane to the left of ℓ. We consider lozenge tilings of H{\mathcal{H}} that have a triangular gap of side-length two and in which ℓ is a free boundary — i.e., tiles are allowed to protrude out half-way across ℓ. We prove that the correlation function of this gap near the free boundary has asymptotics \frac14pr{\frac{1}{4\pi r}}, r → ∞, where r is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.     

A new matrix inverse

We compute the inverse of a specific infinite-dimensional matrix, thus unifying a number of previous matrix inversions. Our inversion theorem is applied to derive a number of summation formulas of hypergeometric type.

     

Buchbesprechungen

Ohne Zusammenfassung     

On bideterminantal formulas for characters of classical groups

New bideterminantal formulas for the irreducible symplectic and orthogonal characters are given that generalize the classical bideterminantal formulas. These formulas are analogous to Regev’s (Israel J. Math. 80 (1992), 155–160) bideterminantal formulas for Schur functions, the irreducible general linear characters. Also, new bideterminantal formulas for Proctor’s intermediate symplectic characters are derived.     

2-Adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman-Friedland-Loewy on the parity of the degrees of projective varieties of n×n complex symmetric matrices of rank at most k. We also characterize the parity of the degrees of projective varieties of n×n complex skew symmetric matrices of rank at most 2p. We give recursive relations which determine the parity of the degrees of projective varieties of m×n complex matrices of rank at most k. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of n×n skew symmetric real matrices and of m×n real matrices containing a nonzero matrix of rank at most k. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.     

Buchbesprechungen

Ohne Zusammenfassung

---

Book reviews

     

An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series

We give a new proof of a theorem of Zudilin that equates a very-well-poised hypergeometric series and a particular multiple integral. This integral generalizes integrals of Vasilenko and Vasilyev which were proposed as tools in the study of the arithmetic behaviour of values of the Riemann zeta function at integers. Our proof is based on limiting cases of a basic hypergeometric identity of Andrews. Dedicated to Richard Askey on the occasion of his 70th birthday. Research partially supported by the programme “Improving the Human Research Potential” of the European Commission, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”. 2000 Mathematics Subject Classification Primary—33C20; Secondary—11J72     

Buchbesprechungen

Ohne Zusammenfassung

---

Book reviews