Gil Alon

The probability of long cycles in interchange processes

We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles.

Cohomology of discrete groups in harmonic cochains on buildings

Modules of harmonic cochains on the Bruhat-Tits building of the projective general linear group over ap-adic field were defined by one of the authors, and were shown to represent the cohomology of Drinfel’d’sp-adic symmetric domain. Here we define certain non-trivial natural extensions of these modules and study their properties. In particular, for a quotient of Drinfel’d’s space by a discrete cocompact group, we are able to define maps between consecutive graded pieces of its de Rham cohomology, which we show to be (essentially) isomorphisms. We believe that these maps are graded versions of the Hyodo-Kato monodromy operatorN.

On the cohomology of Drinfel’d’sp-adic symmetric domain

There are, by now, three approaches to the de-Rham cohomology of Drinfel’d’sp-adic symmetric domain: the original work of Schneider and Stuhler, and more recent work of Iovita and Spiess, and of de Shalit. In the first part of this paper we compare all three approaches and clarify a few points which remained obscure. In the second half we give a short proof of a conjecture of Schneider and Stuhler, previously proven by Iovita and Spiess, on a Hodge-like decomposition of the cohomology ofp-adically uniformized varieties.

Improved algorithms for symmetry analysis: structure preserving permutations

We propose an improved algorithm for calculating Avnir’s continuous symmetry and chirality measures of molecules. These measures evaluate the deviation of a given structure from symmetry by calculating the distance between the structure and its nearest symmetric counterpart. Our new algorithm utilizes structural properties of the given molecule to increase the accuracy of the calculation and dramatically reduce the running time by up to tens orders of magnitude. Consequently, a wide variety of molecules of medium size with ca. 100 atoms and even more can be analyzed within seconds. Numerical evidence of the algorithm’s efficiency is presented for several families of molecules such as helicenes, porphyrins, dendrimers building blocks, fullerene and more. The ease and efficiency of the calculation make the continuous symmetry and chirality measures promising descriptors for integration in quantitative structure–activity relationship tools, as well as chemical databases and molecular visualization software.

Semicharacters of Groups

We define the notion of a semicharacter of a group G: A function from the group to ℂ*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is divisible by the order of the group. We prove that the conjecture holds for some important families of groups, including the Symmetric groups and the groups GL(2, q).