The rings of n-dimensional polytopes
Identifieur interne : 000797 ( Main/Exploration ); précédent : 000796; suivant : 000798The rings of n-dimensional polytopes
Auteurs : L. Hkov [Canada] ; M. Larouche [Canada] ; J. Patera [Canada]Source :
- Journal of Physics A: Mathematical and Theoretical [ 1751-8113 ] ; 2008.
English descriptors
- KwdEn :
- Algebra, Anomaly, Anomaly number, Anomaly numbers, Cambridge university press, Cartan matrices, Check marks, Compact semisimple, Complete list, Congruence class, Congruence classes, Congruence number, Congruence numbers, Coxeter, Coxeter diagrams, Coxeter group, Coxeter group orbits, Coxeter groups, Crystallographic coxeter groups, Crystallographic groups, Dominant point, Dominant points, Dominant weight, Dominant weights, Dynkin, Dynkin diagrams, Extensive tables, Fundamental orbits, Fundamental representations, Fundamental weights, Further decomposable, General method, Geometric interpretation, Group representations, Harmonic analysis, Higher indices, Individual orbits, Irreducible, Irreducible representations, Lattice, Many places, Marcel dekker, Math, Matrix, Nite, Nite coxeter group, Nite coxeter groups, Nite order, Node, Noncrystallographic groups, Open circles, Orbit, Orbit layers, Orbit points, Orbits form, Orthogonal, Particle physics, Particular representation, Patera, Permutation, Permutation symmetry components, Phys, Physics literature, Polytope, Polytopes, Positive coordinates, Positive integers, Practical importance, Real euclidean space, Recherches math, Relative angles, Relative lengths, Representation theory, Right side, Root lattice, Root system, Same congruence class, Same time, Second column, Seed point, Seed points, Semisimple, Several variables, Simple roots, Single seed point, Stabilizer, Subgroup, Subsequent columns, Such questions, Suitable choice, Symm, Symmetrized powers, Symmetry group, Symmetry groups, Theor, Various degrees, Weight lattice, Weight system, Weight systems, Weyl, Weyl group, Weyl groups.
- Teeft :
- Algebra, Anomaly, Anomaly number, Anomaly numbers, Cambridge university press, Cartan matrices, Check marks, Compact semisimple, Complete list, Congruence class, Congruence classes, Congruence number, Congruence numbers, Coxeter, Coxeter diagrams, Coxeter group, Coxeter group orbits, Coxeter groups, Crystallographic coxeter groups, Crystallographic groups, Dominant point, Dominant points, Dominant weight, Dominant weights, Dynkin, Dynkin diagrams, Extensive tables, Fundamental orbits, Fundamental representations, Fundamental weights, Further decomposable, General method, Geometric interpretation, Group representations, Harmonic analysis, Higher indices, Individual orbits, Irreducible, Irreducible representations, Lattice, Many places, Marcel dekker, Math, Matrix, Nite, Nite coxeter group, Nite coxeter groups, Nite order, Node, Noncrystallographic groups, Open circles, Orbit, Orbit layers, Orbit points, Orbits form, Orthogonal, Particle physics, Particular representation, Patera, Permutation, Permutation symmetry components, Phys, Physics literature, Polytope, Polytopes, Positive coordinates, Positive integers, Practical importance, Real euclidean space, Recherches math, Relative angles, Relative lengths, Representation theory, Right side, Root lattice, Root system, Same congruence class, Same time, Second column, Seed point, Seed points, Semisimple, Several variables, Simple roots, Single seed point, Stabilizer, Subgroup, Subsequent columns, Such questions, Suitable choice, Symm, Symmetrized powers, Symmetry group, Symmetry groups, Theor, Various degrees, Weight lattice, Weight system, Weight systems, Weyl, Weyl group, Weyl groups.
Abstract
Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G-polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.
Url:
DOI: 10.1088/1751-8113/41/49/495202
Affiliations:
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<term>Anomaly</term>
<term>Anomaly number</term>
<term>Anomaly numbers</term>
<term>Cambridge university press</term>
<term>Cartan matrices</term>
<term>Check marks</term>
<term>Compact semisimple</term>
<term>Complete list</term>
<term>Congruence class</term>
<term>Congruence classes</term>
<term>Congruence number</term>
<term>Congruence numbers</term>
<term>Coxeter</term>
<term>Coxeter diagrams</term>
<term>Coxeter group</term>
<term>Coxeter group orbits</term>
<term>Coxeter groups</term>
<term>Crystallographic coxeter groups</term>
<term>Crystallographic groups</term>
<term>Dominant point</term>
<term>Dominant points</term>
<term>Dominant weight</term>
<term>Dominant weights</term>
<term>Dynkin</term>
<term>Dynkin diagrams</term>
<term>Extensive tables</term>
<term>Fundamental orbits</term>
<term>Fundamental representations</term>
<term>Fundamental weights</term>
<term>Further decomposable</term>
<term>General method</term>
<term>Geometric interpretation</term>
<term>Group representations</term>
<term>Harmonic analysis</term>
<term>Higher indices</term>
<term>Individual orbits</term>
<term>Irreducible</term>
<term>Irreducible representations</term>
<term>Lattice</term>
<term>Many places</term>
<term>Marcel dekker</term>
<term>Math</term>
<term>Matrix</term>
<term>Nite</term>
<term>Nite coxeter group</term>
<term>Nite coxeter groups</term>
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<term>Node</term>
<term>Noncrystallographic groups</term>
<term>Open circles</term>
<term>Orbit</term>
<term>Orbit layers</term>
<term>Orbit points</term>
<term>Orbits form</term>
<term>Orthogonal</term>
<term>Particle physics</term>
<term>Particular representation</term>
<term>Patera</term>
<term>Permutation</term>
<term>Permutation symmetry components</term>
<term>Phys</term>
<term>Physics literature</term>
<term>Polytope</term>
<term>Polytopes</term>
<term>Positive coordinates</term>
<term>Positive integers</term>
<term>Practical importance</term>
<term>Real euclidean space</term>
<term>Recherches math</term>
<term>Relative angles</term>
<term>Relative lengths</term>
<term>Representation theory</term>
<term>Right side</term>
<term>Root lattice</term>
<term>Root system</term>
<term>Same congruence class</term>
<term>Same time</term>
<term>Second column</term>
<term>Seed point</term>
<term>Seed points</term>
<term>Semisimple</term>
<term>Several variables</term>
<term>Simple roots</term>
<term>Single seed point</term>
<term>Stabilizer</term>
<term>Subgroup</term>
<term>Subsequent columns</term>
<term>Such questions</term>
<term>Suitable choice</term>
<term>Symm</term>
<term>Symmetrized powers</term>
<term>Symmetry group</term>
<term>Symmetry groups</term>
<term>Theor</term>
<term>Various degrees</term>
<term>Weight lattice</term>
<term>Weight system</term>
<term>Weight systems</term>
<term>Weyl</term>
<term>Weyl group</term>
<term>Weyl groups</term>
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<term>Anomaly</term>
<term>Anomaly number</term>
<term>Anomaly numbers</term>
<term>Cambridge university press</term>
<term>Cartan matrices</term>
<term>Check marks</term>
<term>Compact semisimple</term>
<term>Complete list</term>
<term>Congruence class</term>
<term>Congruence classes</term>
<term>Congruence number</term>
<term>Congruence numbers</term>
<term>Coxeter</term>
<term>Coxeter diagrams</term>
<term>Coxeter group</term>
<term>Coxeter group orbits</term>
<term>Coxeter groups</term>
<term>Crystallographic coxeter groups</term>
<term>Crystallographic groups</term>
<term>Dominant point</term>
<term>Dominant points</term>
<term>Dominant weight</term>
<term>Dominant weights</term>
<term>Dynkin</term>
<term>Dynkin diagrams</term>
<term>Extensive tables</term>
<term>Fundamental orbits</term>
<term>Fundamental representations</term>
<term>Fundamental weights</term>
<term>Further decomposable</term>
<term>General method</term>
<term>Geometric interpretation</term>
<term>Group representations</term>
<term>Harmonic analysis</term>
<term>Higher indices</term>
<term>Individual orbits</term>
<term>Irreducible</term>
<term>Irreducible representations</term>
<term>Lattice</term>
<term>Many places</term>
<term>Marcel dekker</term>
<term>Math</term>
<term>Matrix</term>
<term>Nite</term>
<term>Nite coxeter group</term>
<term>Nite coxeter groups</term>
<term>Nite order</term>
<term>Node</term>
<term>Noncrystallographic groups</term>
<term>Open circles</term>
<term>Orbit</term>
<term>Orbit layers</term>
<term>Orbit points</term>
<term>Orbits form</term>
<term>Orthogonal</term>
<term>Particle physics</term>
<term>Particular representation</term>
<term>Patera</term>
<term>Permutation</term>
<term>Permutation symmetry components</term>
<term>Phys</term>
<term>Physics literature</term>
<term>Polytope</term>
<term>Polytopes</term>
<term>Positive coordinates</term>
<term>Positive integers</term>
<term>Practical importance</term>
<term>Real euclidean space</term>
<term>Recherches math</term>
<term>Relative angles</term>
<term>Relative lengths</term>
<term>Representation theory</term>
<term>Right side</term>
<term>Root lattice</term>
<term>Root system</term>
<term>Same congruence class</term>
<term>Same time</term>
<term>Second column</term>
<term>Seed point</term>
<term>Seed points</term>
<term>Semisimple</term>
<term>Several variables</term>
<term>Simple roots</term>
<term>Single seed point</term>
<term>Stabilizer</term>
<term>Subgroup</term>
<term>Subsequent columns</term>
<term>Such questions</term>
<term>Suitable choice</term>
<term>Symm</term>
<term>Symmetrized powers</term>
<term>Symmetry group</term>
<term>Symmetry groups</term>
<term>Theor</term>
<term>Various degrees</term>
<term>Weight lattice</term>
<term>Weight system</term>
<term>Weight systems</term>
<term>Weyl</term>
<term>Weyl group</term>
<term>Weyl groups</term>
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<front><div type="abstract">Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G-polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.</div>
</front>
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