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In mathematics, a '''Specht module''' is one of the representations of [[Symmetric group|symmetric group]]s studied by {{harvs|txt|first=Wilhelm|last=Specht|year=1935}}.
They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of ''n'' form a complete set of [[Irreducible representation|irreducible representation]]s of the symmetric group on ''n'' points.

==Definition==
Fix a [[Partition (number theory)|partition]] &lambda; of ''n'' and a commutative ring ''k''.  The partition determines a Young diagram with ''n'' boxes.   A [[Young tableau]]  of shape &lambda; is a way of labelling the boxes of this Young diagram by distinct numbers <math>1, \dots, n</math>.

A '''tabloid''' is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row.  For each Young tableau ''T'' of shape &lambda; let <math>\{T\}</math> be the corresponding tabloid.  The symmetric group on ''n'' points acts on the set of Young tableaux of shape &lambda;. Consequently, it acts on tabloids, and on the free ''k''-module ''V'' with the tabloids as basis. 

Given a Young tableau ''T'' of shape &lambda;, let
:<math>E_T=\sum_{\sigma\in Q_T}\epsilon(\sigma)\{\sigma(T)\} \in V</math>
where ''Q''<sub>''T''</sub> is the subgroup of permutations, preserving (as sets) all columns of ''T''  and <math>\epsilon(\sigma)</math> is the sign of the permutation &sigma;. The Specht module of the partition &lambda; is the module generated by the elements  ''E''<sub>''T''</sub> as ''T'' runs through all tableaux of shape &lambda;.

The Specht module  has a basis of elements ''E''<sub>''T''</sub> for ''T'' a  standard Young tableau.

A gentle introduction to the construction of the Specht module may be found in Section 1 of "Specht Polytopes and Specht Matroids".<ref>{{Citation |arxiv = 1701.05277|last1 = Wiltshire-Gordon|first1 = John D.|last2 = Woo|first2 = Alexander|last3 = Zajaczkowska|first3 = Magdalena|date = 2017 | contribution=Specht Polytopes and Specht Matroids | title=Combinatorial Algebraic Geometry | series=Fields Institute Communications | volume=80 | pages=201–228 | doi=10.1007/978-1-4939-7486-3_10 | doi-access=free}}</ref>

==Structure==

The dimension of the Specht module <math>V_\lambda</math> is the number of [[Young tableau#Dimension of a representation|standard Young tableaux]] of shape <math>\lambda</math>. It is given by the [[Hook length formula|hook length formula]].

Over fields of characteristic 0 the Specht modules are irreducible, and form a complete set of irreducible representations of the symmetric group.

A partition is called ''p''-regular (for a prime number ''p'') if it does not have ''p'' parts of the same (positive) size. 
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.

==See also==
* [[Garnir relations]], a more detailed description of the structure of Specht modules.

==References==
{{Reflist}}
*{{SpringerEOM|title=Specht module|first=Henning Haahr |last=Andersen}}
*{{Citation | last1=James | first1=G. D. | chapter=Chapter 4: Specht modules|title=The representation theory of the symmetric groups | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-08948-3 |mr=513828 | year=1978 | volume=682|doi=10.1007/BFb0067712 | pages=13}}
*{{Citation | last1=James | first1=Gordon | last2=Kerber | first2=Adalbert | title=The representation theory of the symmetric group | url=https://archive.org/details/representationth0000jame | publisher=Addison-Wesley Publishing Co., Reading, Mass. | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-201-13515-2 | mr=644144 | year=1981 | volume=16 | url-access=registration }}
*{{Citation |  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}}

[[Category:Representation theory of finite groups]]

{{Sourceattribution|Specht module}}