Symmetrization

About: Symmetrization is a research topic. Over the lifetime, 1598 publications have been published within this topic receiving 24342 citations.

Permutationally invariant potential energy surfaces in high dimensionality

Abstract: We review recent progress in developing potential energy and dipole moment surfaces for polyatomic systems with up to 10 atoms. The emphasis is on global linear least squares fitting of tens of thousands of scattered ab initio energies using a special, compact fitting basis of permutationally invariant polynomials in Morse-type variables of all the internuclear distances. The computational mathematics underlying this approach is reviewed first, followed by a review of the practical approaches used to obtain the data for the fits. A straightforward symmetrization approach is also given, mainly for pedagogical purposes. The methods are illustrated for potential energy surfaces for , (H2O)2 and CH3CHO. The relationship of this approach to other approaches is also briefly reviewed.

Partial and approximate symmetry detection for 3D geometry

Abstract: "Symmetry is a complexity-reducing concept [...]; seek it every-where." - Alan J. PerlisMany natural and man-made objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important high-level information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc.

A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras.

Abstract: In the representation theory of the group GLn(C), an important tool are the Young tableaux. The irreducible representations are in one-to-one correspondence with the shapes of these tableaux. Let T be the subgroup of diagonal matrices in GLn(C). Then there is a canonical way to assign a weight of T to any Young tableau such that the sum over the weights of all tableaux of a fixed shape is the character CharV of the corresponding GLn(C)-module V . Note that this gives not only a way to compute the character, it gives also a possibility to describe the multiplicity of a weight in the representation: It is the number of different tableaux of the same weight. Eventually, the Littlewood-Richardson rule describes the decomposition of tensor products of GLn(C)modules purely in terms of the combinatoric of these Young tableaux.

Shifted Schur Functions

Abstract: The classical algebra $\Lambda$ of symmetric functions has a remarkable deformation $\Lambda^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions $s^*_\mu$, where $\mu$ ranges over the set of all partitions. The main significance of the shifted Schur functions is that they determine a natural basis in $Z(\frak{gl}(n))$, the center of the universal enveloping algebra $U(\frak{gl}(n))$, $n=1,2,\ldots$. The functions $s^*_\mu$ are closely related to the factorial Schur functions introduced by Biedenharn and Louck and further studied by Macdonald and other authors. A part of our results about the functions $s^*_\mu$ has natural classical analogues (combinatorial presentation, generating series, Jacobi--Trudi identity, Pieri formula). Other results are of different nature (connection with the binomial formula for characters of $GL(n)$, an explicit expression for the dimension of skew shapes $\lambda/\mu$, Capelli--type identities, a characterization of the functions $s^*_\mu$ by their vanishing properties, `coherence property', special symmetrization map $S(\frak{gl}(n))\to U(\frak{gl}(n))$. The main application that we have in mind is the asymptotic character theory for the unitary groups $U(n)$ and symmetric groups $S(n)$ as $n\to\infty$. The results of this paper were used in \cite{Ok1--3}.