Non-crystallographic reduction of generalized Calogero–Moser models

Fring, Andreas; Korff, Christian

doi:10.1088/0305-4470/39/5/007

Cited by 7 publications

(9 citation statements)

References 45 publications

(112 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, the organisation of different fullerene shells of carbon onions has been modelled with previous approaches ( [19], [42]), and we expect that our new approach should be relevant in this context as well. Moreover, a mathematical formulation of systems with non-crystallographic symmetries is a challenge in wider areas of physics, such as integrable systems, where models in terms of non-crystallographic root systems have been introduced ([43], [44]); we expect that the use of projections of the higher dimensional symmetry groups, that contain non-crystallographic symmetries as crystallographic embeddings, could provide a new perspective also in this context.…”

Section: Resultsmentioning

confidence: 99%

Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

2015

View full text Add to dashboard Cite

The architecture of infinite structures with non-crystallographic symmetries can be modelled via aperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. This paper presents a group theoretical method for the construction of finite nested point sets with non-crystallographic symmetry. Akin to the construction of quasicrystals, a non-crystallographic group G is embedded into the point group P of a higher-dimensional lattice and the chains of all G-containing subgroups are constructed. The orbits of lattice points under such subgroups are determined, and it is shown that their projection into a lower-dimensional G-invariant subspace consists of nested point sets with G-symmetry at each radial level. The number of different radial levels is bounded by the index of G in the subgroup of P. In the case of icosahedral symmetry, all subgroup chains are determined explicitly and it is illustrated that these point sets in projection provide blueprints that approximate the organization of simple viral capsids, encoding information on the structural organization of capsid proteins and the genomic material collectively, based on two case studies. Contrary to the affine extensions previously introduced, these orbits endow virus architecture with an underlying finite group structure, which lends itself better to the modelling of dynamic properties than its infinite-dimensional counterpart.

show abstract

Section: Resultsmentioning

confidence: 99%

Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

2015

View full text Add to dashboard Cite

show abstract

“…Finally let us note a few more open questions. Do there exist Sutherland models for the exceptional complex reflection groups, perhaps via reduction as in [36]? The Sutherland model is the trigonometric member of the Calogero-Moser family: can one generalize to elliptic potentials?…”

Section: Discussionmentioning

confidence: 99%

Sutherland models for complex reflection groups

Crampé¹,

Young²

2008

Nuclear Physics B

View full text Add to dashboard Cite

There are known to be integrable Sutherland models associated to every real root system -or, which is almost equivalent, to every real reflection group. Real reflection groups are special cases of complex reflection groups. In this paper we associate certain integrable Sutherland models to the classical family of complex reflection groups. Internal degrees of freedom are introduced, defining dynamical spin chains, and the freezing limit taken to obtain static chains of Haldane-Shastry type. By considering the relation of these models to the usual BC N case, we are led to systems with both real and complex reflection groups as symmetries. We demonstrate their integrability by means of new Dunkl operators, associated to wreath products of dihedral groups.

show abstract

“…For instance in our forthcoming publication [51] we will apply this method to the generalized Calogero-Moser models.…”

Section: Discussionmentioning

confidence: 99%

“…Our detailed analysis of the embedding of non-crystallographic into crystallographic Coxeter groups allows one to apply the aforementioned reduction method to a wide range of application in physics, chemistry and biology, where Coxeter groups play a role. For instance in our forthcoming publication [51] we will apply this method to the generalized Calogero-Moser models.…”

Section: Discussionmentioning

confidence: 99%

Affine Toda field theories related to Coxeter groups of noncrystallographic type

Fring¹,

Korff²

2005

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

Abstract:We propose affine Toda field theories related to the non-crystallographic Coxeter groups H 2 , H 3 and H 4 . The classical mass spectrum, the classical three-point couplings and the one-loop corrections to the mass renormalisation are determined. The construction is carried out by means of a reduction procedure from crystallographic to non-crystallographic Coxeter groups. The embedding structure explains for various affine Toda field theories that their particles can be organised in pairs, such that their relative masses differ by the golden ratio.

show abstract

Non-crystallographic reduction of generalized Calogero–Moser models

Cited by 7 publications

References 45 publications

Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

Sutherland models for complex reflection groups

Affine Toda field theories related to Coxeter groups of noncrystallographic type

Contact Info

Product

Resources

About