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| In mathematics, a '''Specht module''' is one of the representations of [[symmetric group]]s studied by {{harvs|txt|authorlink=Wilhelm Specht|first=Wilhelm|last=Specht|year=1935}}.
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| They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of ''n'' form a complete set of [[irreducible representation]]s of the symmetric group on ''n'' points.
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| ==Definition==
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| Fix a partition λ of ''n''.
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| A '''tabloid''' is an equivalence class of labelings of the [[Young diagram]] of shape λ, where two labelings are equivalent if one is obtained from the other by permuting the entries of each row. The symmetric group on ''n'' points acts on the set of tabloids, and therefore on the module ''V'' with the tabloids as basis. For each [[Young tableau]] ''T'' of shape λ, form the element
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| :<math>E_T=\sum_{\sigma\in Q_T}\epsilon(\sigma)\sigma(T)</math>
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| where ''Q''<sub>''T''</sub> is the subgroup fixing all columns of ''T'', and ε is the sign of a permutation. | |
| The elements ''E''<sub>''T''</sub> can be considered as elements of the module ''V'', by mapping each tableau to the tabloid it generates. The Specht module of the partition λ is the module generated by the elements ''E''<sub>''T''</sub> as ''T'' runs through all tableaux of shape λ.
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| The Specht module has a basis of elements ''E''<sub>''T''</sub> for ''T'' a [[standard Young tableau]].
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| ==Structure==
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| Over fields of characteristic 0 the Specht modules are irreducible, and form a complete set of irreducible representations of the symmetric group.
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| A partition is called ''p''-regular if it does not have ''p'' parts of the same (positive) size.
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| Over fields of characteristic ''p''>0 the Specht modules can be reducible. For ''p''-regular partitions they have a unique irreducible quotient, and these irreducible quotients form a complete set of irreducible representations.
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| ==References==
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| *{{eom|id=s/s120200|first=Henning Haahr |last=Andersen}}
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| *{{Citation | last1=James | first1=G. D. | chapter=Chapter 4: Specht modules|title=The representation theory of the symmetric groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-08948-3 | id={{MathSciNet | id = 513828}} | year=1978 | volume=682|doi=10.1007/BFb0067712 | pages=13}}
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| *{{Citation | last1=James | first1=Gordon | last2=Kerber | first2=Adalbert | title=The representation theory of the symmetric group | url=http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521104128 | publisher=Addison-Wesley Publishing Co., Reading, Mass. | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-201-13515-2 | id={{MathSciNet | id = 644144}} | year=1981 | volume=16}}
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| *{{Citation | authorlink=Wilhelm Specht | last1=Specht | first1=W. | title=Die irreduziblen Darstellungen der symmetrischen Gruppe | doi=10.1007/BF01201387 | year=1935 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=39 | issue=1 | pages=696–711}}
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| [[Category:Representation theory of finite groups]]
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