Abstract
Starting from each finite union of orbits (called G- cluster) of an R-irreducible orthogonal representation of a finite group G, we define a representation of G in a higher-dimensional space (called permutation representation), and we prove that it can be decomposed into an orthogonal sum of two representations such that one of them is equivalent to the initial representation. This decomposition allows us to use the strip projection method and to obtain some patterns useful in quasicrystal physics. We show that certain self-similarities of such a pattern can be obtained by using the decomposition into R-irreducible components of the corresponding permutation representation, and we present two examples.
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Cotfas, N. Permutation Representations Defined by G-Clusters with Application to Quasicrystals. Letters in Mathematical Physics 47, 111–123 (1999). https://doi.org/10.1023/A:1007557230826
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DOI: https://doi.org/10.1023/A:1007557230826