Abstract
The four components of an icosahedral g orbital are represented by the irreducible spinor representation of (or, equivalently, SO(4)), where the two SO(3) groups are associated with the irreducible representations (IRs) and of the icosahedral group I. This enables the properties of the icosahedral configurations to be calculated by the familiar techniques of angular-momentum theory. The Coulomb interaction is broken into three parts , two of which ( and ) are SO(4) scalars, the third belonging to a combination of the various components and of the IR (22) that form the scalar IR A of I. The similar matrix elements of for different N are explained by introducing the concepts of quasispin and complementarity that are analogous to those used in atomic shell theory. Our angular-momentum basis is related to the icosahedral basis of Pooler with the aid of automorphisms of I that interchange and . This is formalized through the introduction of the kaleidoscope operator , and the degeneracy of the and terms for all is expressed as a result of the invariance of the Coulomb interaction to the operations of the cyclic automorphism group .
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