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- Representation_theory_of_the_symmetric_group abstract "In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles.The symmetric group Sn has order n!. Its conjugacy classes are labeled by partitions of n. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of n. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of n or equivalently Young diagrams of size n.Each such irreducible representation can in fact be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram.To each irreducible representation ρ we can associate an irreducible character, χρ.To compute χρ(π) where π is a permutation, one can use the combinatorial Murnaghan–Nakayama rule. Note that χρ is constant on conjugacy classes,that is, χρ(π) = χρ(σ−1πσ) for all permutations σ.Over other fields the situation can become much more complicated. If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KSn is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary).However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of modules rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called Specht modules, and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimensions are not known in general.The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory.".
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- Representation_theory_of_the_symmetric_group wikiPageWikiLink Abelianization.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Alternating_group.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Alternating_polynomial.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Alternating_polynomials.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Category:Permutations.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Category:Representation_theory_of_finite_groups.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Characteristic_(algebra).
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Commutator_subgroup.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Complex_number.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Conjugacy_class.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Cyclic_group.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Dimension_(vector_space).
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Discrete_Fourier_transform.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Dover_Publications.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Field_(mathematics).
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Group_algebra.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Icosahedral_symmetry.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Identical_particles.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Integer_partition.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Irreducible_representation.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Jucys–Murphy_element.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Maschkes_theorem.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Mathematics.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Module_(mathematics).
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Murnaghan–Nakayama_rule.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Parity_of_a_permutation.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Partition_(number_theory).
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Quantum_mechanics.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Representation_theory_of_finite_groups.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Robinson–Schensted_correspondence.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Schur_functor.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Schur–Weyl_duality.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Sign_of_a_permutation.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Specht_module.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Specht_modules.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Symmetric_function.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Symmetric_group.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Symmetric_polynomial.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Symmetric_polynomials.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Young_diagram.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Young_symmetrizer.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink Young_tableau.
- Representation_theory_of_the_symmetric_group wikiPageWikiLink File:Compound_of_five_tetrahedra.png.
- Representation_theory_of_the_symmetric_group wikiPageWikiLinkText "Representation theory of the symmetric group".
- Representation_theory_of_the_symmetric_group wikiPageWikiLinkText "Representation theory of the symmetric group#permutation representation".
- Representation_theory_of_the_symmetric_group wikiPageWikiLinkText "irreducible character".
- Representation_theory_of_the_symmetric_group wikiPageWikiLinkText "irreducible representations".
- Representation_theory_of_the_symmetric_group wikiPageWikiLinkText "representation theory of the symmetric group".
- Representation_theory_of_the_symmetric_group wikiPageWikiLinkText "representation".
- Representation_theory_of_the_symmetric_group wikiPageWikiLinkText "the action of the symmetric group".
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- Representation_theory_of_the_symmetric_group subject Category:Permutations.
- Representation_theory_of_the_symmetric_group subject Category:Representation_theory_of_finite_groups.
- Representation_theory_of_the_symmetric_group hypernym Case.
- Representation_theory_of_the_symmetric_group type Group.
- Representation_theory_of_the_symmetric_group type Person.
- Representation_theory_of_the_symmetric_group type Combinatoric.
- Representation_theory_of_the_symmetric_group type Function.
- Representation_theory_of_the_symmetric_group type Group.
- Representation_theory_of_the_symmetric_group comment "In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles.The symmetric group Sn has order n!. Its conjugacy classes are labeled by partitions of n.".
- Representation_theory_of_the_symmetric_group label "Representation theory of the symmetric group".
- Representation_theory_of_the_symmetric_group sameAs Représentations_du_groupe_symétrique.
- Representation_theory_of_the_symmetric_group sameAs m.03kpp2.
- Representation_theory_of_the_symmetric_group sameAs Q7314231.
- Representation_theory_of_the_symmetric_group sameAs Q7314231.
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