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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202001.png" /> and suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202002.png" /> is a (proper) partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202003.png" />. This means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202004.png" />, where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202006.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202007.png" /> is maximal with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202008.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s1202009.png" /> is a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020010.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020011.png" /> parts.
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A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020013.png" />-tableau (sometimes called a [[Young tableau|Young tableau]] associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020014.png" />) is an array consisting of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020015.png" /> listed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020016.png" /> rows with exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020017.png" /> numbers occurring in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020018.png" />th row, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020019.png" />. If, for instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020021.png" />, then the following arrays are examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020022.png" />-tableaux:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020023.png" /></td> </tr></table>
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Let $n \in \mathbf N$ and suppose $\lambda$ is a (proper) partition of $n$. This means that $\lambda = ( \lambda _ { 1 } \geq \lambda _ { 2 } \geq \ldots \geq 0 )$, where each $\lambda _ { i } \in \bf Z$ and $\sum _ { i } \lambda _ { i } = n$. If $r$ is maximal with $\lambda _ { r } > 0$, then one says that $\lambda$ is a partition of $n$ into $r$ parts.
  
One says that two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020024.png" />-tableaux are equivalent if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020026.png" /> the two sets of numbers in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020027.png" />th rows of the two arrays coincide. Clearly, the two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020028.png" />-tableaux above are not equivalent. The equivalence classes with respect to this relation are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020030.png" />-tabloids. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020031.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020032.png" />-tableau, one usually denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020033.png" />-tabloid by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020034.png" />. As examples, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020035.png" /> as above one has
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A $\lambda$-tableau (sometimes called a [[Young tableau|Young tableau]] associated with $\lambda$) is an array consisting of the numbers $1 , \dots , n$ listed in $r$ rows with exactly $\lambda _ { i }$ numbers occurring in the $i$th row, $ i  = 1 , \ldots , r$. If, for instance, $n = 9$ and $\lambda = ( 4,3,1,1 )$, then the following arrays are examples of $\lambda$-tableaux:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020036.png" /></td> </tr></table>
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\begin{equation*} \left. \begin{array} { l l l l l l l l l } { 1 } &amp; { 2 } &amp; { 3 } &amp; { 4 } &amp; { } &amp; { 9 } &amp; { 2 } &amp; { 3 } &amp; { 6 } \\ { 5 } &amp; { 6 } &amp; { 7 } &amp; {  } &amp; { \text { and } } &amp; { 7 } &amp; { 1 } &amp; { 4 } &amp; {  } \\ { 8 } &amp; {  } &amp; {  } &amp; {  } &amp; { } &amp; { 5 } &amp; {  } &amp; {  } &amp; {  } \\ { 9 } &amp; {  } &amp; {  } &amp; {  } &amp; { } &amp; { 8 } &amp; {  } &amp; {  } &amp; {  } \end{array} \right. \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020037.png" /></td> </tr></table>
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One says that two $\lambda$-tableaux are equivalent if for each $ i  = 1 , \ldots , r$ the two sets of numbers in the $i$th rows of the two arrays coincide. Clearly, the two $\lambda$-tableaux above are not equivalent. The equivalence classes with respect to this relation are called $\lambda$-tabloids. If $t$ is a $\lambda$-tableau, one usually denotes the $\lambda$-tabloid by $\{ t \}$. As examples, for $\lambda$ as above one has
  
Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020038.png" /> is a [[Field|field]]. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020039.png" /> the [[Vector space|vector space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020040.png" /> with basis equal to the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020041.png" />-tabloids. Then the symmetric group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020043.png" /> letters (cf. also [[Symmetric group|Symmetric group]]) acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020044.png" /> (or, more precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020045.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020046.png" />-module) in a natural way. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020048.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020049.png" />-tableau, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020050.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020051.png" />-tableau obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020052.png" /> by replacing each number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020053.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020054.png" />. If one uses the usual cycle presentation of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020055.png" />, then, e.g., for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020056.png" /> one has
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\begin{equation*} \left( \begin{array} { c c c c } { 1 } &amp; { 2 } &amp; { 3 } &amp; { 4 } \\ { 5 } &amp; { 6 } &amp; { 7 } &amp; {  } \\ { 8 } &amp; {  } &amp; {  } &amp; {  } \\ { 9 } &amp; {  } &amp; {  } &amp; {  } \end{array} \right) = \left( \begin{array} { c c c c } { 4 } &amp; { 2 } &amp; { 1 } &amp; { 3 } \\ { 6 } &amp; { 5 } &amp; { 7 } &amp; {  } \\ { 8 } &amp; {  } &amp; {  } &amp; {  } \\ { 9 } &amp; {  } &amp; {  } &amp; {  } \end{array} \right) \neq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020057.png" /></td> </tr></table>
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\begin{equation*} \neq \left( \begin{array} { c c c c } { 9 } &amp; { 2 } &amp; { 3 } &amp; { 6 } \\ { 7 } &amp; { 1 } &amp; { 4 } &amp; {  } \\ { 8 } &amp; {  } &amp; {  } &amp; {  } \\ { 9 } &amp; {  } &amp; {  } &amp; {  } \end{array} \right) = \left( \begin{array} { c c c c } { 2 } &amp; { 3 } &amp; { 9 } &amp; { 6 } \\ { 4 } &amp; { 1 } &amp; { 7 } &amp; {  } \\ { 8 } &amp; {  } &amp; {  } &amp; {  } \\ { 9 } &amp; {  } &amp; {  } &amp; {  } \end{array} \right). \end{equation*}
  
This action clearly induces an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020059.png" />-tabloids and this gives the desired module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020060.png" />.
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Suppose $k$ is a [[Field|field]]. Denote by $M ^ { \lambda }$ the [[Vector space|vector space]] over $k$ with basis equal to the set of $\lambda$-tabloids. Then the symmetric group $S _ { n }$ on $n$ letters (cf. also [[Symmetric group|Symmetric group]]) acts on $M ^ { \lambda }$ (or, more precisely, $M ^ { \lambda }$ is a $k S _ { n }$-module) in a natural way. Indeed, if $\sigma \in S _ { n }$ and $t$ is a $\lambda$-tableau, then $\sigma t $ is the $\lambda$-tableau obtained from $t$ by replacing each number $i$ by $\sigma i$. If one uses the usual cycle presentation of elements in $S _ { n }$, then, e.g., for $\sigma = ( 452 ) ( 89 ) ( 316 ) \in S_{9}$ one has
  
If, again, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020061.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020062.png" />-tableau, then one sets
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\begin{equation*} \sigma \left( \begin{array} { c c c c } { 9 } &amp; { 2 } &amp; { 3 } &amp; { 6 } \\ { 7 } &amp; { 1 } &amp; { 4 } &amp; {  } \\ { 5 } &amp; {  } &amp; {  } &amp; {  } \\ { 8 } &amp; {  } &amp; {  } &amp; {  } \end{array} \right) = \left( \begin{array} { c c c c } { 8 } &amp; { 4 } &amp; { 1 } &amp; { 3 } \\ { 7 } &amp; { 6 } &amp; { 5 } &amp; {  } \\ { 2 } &amp; {  } &amp; {  } &amp; {  } \\ { 9 } &amp; {  } &amp; {  } &amp; {  } \end{array} \right). \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020063.png" /></td> </tr></table>
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This action clearly induces an action of $S _ { n }$ on $\lambda$-tabloids and this gives the desired module structure on $M ^ { \lambda }$.
  
where the sum runs over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020064.png" /> that leave the set of numbers in each column in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020065.png" /> stable. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020066.png" /> is the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020067.png" />.
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If, again, $t$ is a $\lambda$-tableau, then one sets
  
The Specht module associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020068.png" /> is defined as the submodule
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\begin{equation*} e _ { t } = \sum _ { \pi } \operatorname { sgn } ( \pi ) \{ \pi t \}, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020069.png" /></td> </tr></table>
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where the sum runs over the $\pi \in S _ { n }$ that leave the set of numbers in each column in $t$ stable. Here, $\operatorname { sgn } ( \pi )$ is the sign of $\pi$.
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020070.png" />. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020071.png" /> is invariant under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020072.png" />. (In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020073.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020074.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020075.png" />-tableau <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020076.png" />.)
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The Specht module associated to $\lambda$ is defined as the submodule
  
Specht modules were introduced in 1935 by W. Specht [[#References|[a5]]]. Their importance in the representation theory for symmetric groups (cf. also [[Representation of the symmetric groups|Representation of the symmetric groups]]) comes from the fact that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020077.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020078.png" />, then each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020079.png" /> is a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020080.png" />-module. Moreover, the set
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\begin{equation*}
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  S^{\lambda} = \operatorname{span} \{ e_t : t \text{ a }\lambda\text{-tableau } \}
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\end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020081.png" /></td> </tr></table>
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of $M^{\lambda}$. Clearly, $S^{\lambda}$ is invariant under the action of $S_{n}$. (In fact, $\sigma e_{t} = e_{ \sigma  t}$ for all $\sigma \in S_{n}$ and all $\lambda$-tableau $t$.)
  
is a full set of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020082.png" />-modules.
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Specht modules were introduced in 1935 by W. Specht [[#References|[a5]]]. Their importance in the representation theory for symmetric groups (cf. also [[Representation of the symmetric groups]]) comes from the fact that when $k$ contains $\mathbf{Q}$, then each $S^{\lambda}$ is a simple $k S_{n}$-module. Moreover, the set
  
When the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020083.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020084.png" />, then the Specht modules are no longer always simple. However, they still play an important role in the classification of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020085.png" />-modules. Namely, it turns out that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020086.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020088.png" />-regular (i.e. no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020089.png" /> parts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020090.png" /> are equal), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020091.png" /> has a unique simple quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020092.png" /> and the set
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\begin{equation*}
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  \left\{S^{\lambda} : \lambda \text { a partition of } n \right\}
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\end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020093.png" /></td> </tr></table>
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is a full set of simple $k S_{n}$-modules.
  
constitutes a full set of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020094.png" />-modules. It is a major open problem (1999) to determine the dimensions of these modules.
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When the characteristic of $k$ is $p > 0$, then the Specht modules are no longer always simple. However, they still play an important role in the classification of simple $k S_{n}$-modules. Namely, it turns out that when $\lambda$ is $p$-regular (i.e. no $p$ parts of $\lambda$ are equal), then $S^{\lambda}$ has a unique simple quotient $D^{\lambda}$ and the set
  
It is possible to give a (characteristic-free) natural basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020095.png" />. This is sometimes referred to as the Specht basis. In the notation above, it is given by
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\begin{equation*}
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  \{ D^{\lambda} : \lambda \text{ a p-regular partition of } n \}
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\end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020096.png" /></td> </tr></table>
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constitutes a full set of simple $k S_{n}$-modules. It is a major open problem (1999) to determine the dimensions of these modules.
  
Here, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020097.png" />-tableau <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020098.png" /> is called standard if the numbers occurring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s120200100.png" /> are increasing along each row and down each column.
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It is possible to give a (characteristic-free) natural basis for $S^{\lambda}$. This is sometimes referred to as the Specht basis. In the notation above, it is given by
  
An immediate consequence is that the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s120200101.png" /> equals the number of standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s120200102.png" />-tableaux. For various formulas for this number (as well as many further properties of Specht modules) see [[#References|[a2]]] and [[#References|[a3]]].
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s12020096.png"/></td> </tr></table>
  
The representation theory for symmetric groups is intimately related to the corresponding theory for general linear groups (Schur duality). Under this correspondence, Specht modules play the same role for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s120200103.png" /> as do the Weyl modules for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s120200104.png" />, see e.g. [[#References|[a1]]] and [[Weyl module|Weyl module]]. For a recent result exploring this correspondence in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s120200105.png" />, see [[#References|[a4]]].
+
Here, a $\lambda$-tableau $t$ is called standard if the numbers occurring in $t$ are increasing along each row and down each column.
 +
 
 +
An immediate consequence is that the dimension of $S^{\lambda}$ equals the number of standard $\lambda$-tableaux. For various formulas for this number (as well as many further properties of Specht modules) see [[#References|[a2]]] and [[#References|[a3]]].
 +
 
 +
The representation theory for symmetric groups is intimately related to the corresponding theory for general linear groups (Schur duality). Under this correspondence, Specht modules play the same role for $S_{n}$ as do the Weyl modules for $\operatorname{GL}_{n}$, see e.g. [[#References|[a1]]] and [[Weyl module]]. For a recent result exploring this correspondence in characteristic $p > 0$, see [[#References|[a4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.A. Green,   "Polynomial representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120200/s120200106.png" />" , ''Lecture Notes Math.'' , '''830''' , Springer  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.D. James,   "The representation theory of the symmetric groups" , ''Lecture Notes Math.'' , '''682''' , Springer  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G.D. James,   A. Kerber,   "The representation theory of the symmetric group" , ''Encycl. Math. Appl.'' , '''16''' , Addison-Wesley  (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> O. Mathieu,   "On the dimension of some modular irreducible representations of the symmetric group"  ''Lett. Math. Phys.'' , '''38'''  (1996)  pp. 23–32</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Specht,   "Die irreduziblen Darstellungen der symmetrischen Gruppe"  ''Math. Z.'' , '''39'''  (1935)  pp. 696–711</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top"> J.A. Green, "Polynomial representations of $GL_n$" , ''Lecture Notes Math.'' , '''830''' , Springer  (1980)</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top"> G.D. James, "The representation theory of the symmetric groups" , ''Lecture Notes Math.'' , '''682''' , Springer  (1978)</td></tr>
 +
<tr><td valign="top">[a3]</td> <td valign="top"> G.D. James, A. Kerber, "The representation theory of the symmetric group" , ''Encycl. Math. Appl.'' , '''16''' , Addison-Wesley  (1981)</td></tr>
 +
<tr><td valign="top">[a4]</td> <td valign="top"> O. Mathieu, "On the dimension of some modular irreducible representations of the symmetric group"  ''Lett. Math. Phys.'' , '''38'''  (1996)  pp. 23–32</td></tr>
 +
<tr><td valign="top">[a5]</td> <td valign="top"> W. Specht, "Die irreduziblen Darstellungen der symmetrischen Gruppe"  ''Math. Z.'' , '''39'''  (1935)  pp. 696–711</td></tr>
 +
</table>

Latest revision as of 10:18, 11 November 2023

Let $n \in \mathbf N$ and suppose $\lambda$ is a (proper) partition of $n$. This means that $\lambda = ( \lambda _ { 1 } \geq \lambda _ { 2 } \geq \ldots \geq 0 )$, where each $\lambda _ { i } \in \bf Z$ and $\sum _ { i } \lambda _ { i } = n$. If $r$ is maximal with $\lambda _ { r } > 0$, then one says that $\lambda$ is a partition of $n$ into $r$ parts.

A $\lambda$-tableau (sometimes called a Young tableau associated with $\lambda$) is an array consisting of the numbers $1 , \dots , n$ listed in $r$ rows with exactly $\lambda _ { i }$ numbers occurring in the $i$th row, $ i = 1 , \ldots , r$. If, for instance, $n = 9$ and $\lambda = ( 4,3,1,1 )$, then the following arrays are examples of $\lambda$-tableaux:

\begin{equation*} \left. \begin{array} { l l l l l l l l l } { 1 } & { 2 } & { 3 } & { 4 } & { } & { 9 } & { 2 } & { 3 } & { 6 } \\ { 5 } & { 6 } & { 7 } & { } & { \text { and } } & { 7 } & { 1 } & { 4 } & { } \\ { 8 } & { } & { } & { } & { } & { 5 } & { } & { } & { } \\ { 9 } & { } & { } & { } & { } & { 8 } & { } & { } & { } \end{array} \right. \end{equation*}

One says that two $\lambda$-tableaux are equivalent if for each $ i = 1 , \ldots , r$ the two sets of numbers in the $i$th rows of the two arrays coincide. Clearly, the two $\lambda$-tableaux above are not equivalent. The equivalence classes with respect to this relation are called $\lambda$-tabloids. If $t$ is a $\lambda$-tableau, one usually denotes the $\lambda$-tabloid by $\{ t \}$. As examples, for $\lambda$ as above one has

\begin{equation*} \left( \begin{array} { c c c c } { 1 } & { 2 } & { 3 } & { 4 } \\ { 5 } & { 6 } & { 7 } & { } \\ { 8 } & { } & { } & { } \\ { 9 } & { } & { } & { } \end{array} \right) = \left( \begin{array} { c c c c } { 4 } & { 2 } & { 1 } & { 3 } \\ { 6 } & { 5 } & { 7 } & { } \\ { 8 } & { } & { } & { } \\ { 9 } & { } & { } & { } \end{array} \right) \neq \end{equation*}

\begin{equation*} \neq \left( \begin{array} { c c c c } { 9 } & { 2 } & { 3 } & { 6 } \\ { 7 } & { 1 } & { 4 } & { } \\ { 8 } & { } & { } & { } \\ { 9 } & { } & { } & { } \end{array} \right) = \left( \begin{array} { c c c c } { 2 } & { 3 } & { 9 } & { 6 } \\ { 4 } & { 1 } & { 7 } & { } \\ { 8 } & { } & { } & { } \\ { 9 } & { } & { } & { } \end{array} \right). \end{equation*}

Suppose $k$ is a field. Denote by $M ^ { \lambda }$ the vector space over $k$ with basis equal to the set of $\lambda$-tabloids. Then the symmetric group $S _ { n }$ on $n$ letters (cf. also Symmetric group) acts on $M ^ { \lambda }$ (or, more precisely, $M ^ { \lambda }$ is a $k S _ { n }$-module) in a natural way. Indeed, if $\sigma \in S _ { n }$ and $t$ is a $\lambda$-tableau, then $\sigma t $ is the $\lambda$-tableau obtained from $t$ by replacing each number $i$ by $\sigma i$. If one uses the usual cycle presentation of elements in $S _ { n }$, then, e.g., for $\sigma = ( 452 ) ( 89 ) ( 316 ) \in S_{9}$ one has

\begin{equation*} \sigma \left( \begin{array} { c c c c } { 9 } & { 2 } & { 3 } & { 6 } \\ { 7 } & { 1 } & { 4 } & { } \\ { 5 } & { } & { } & { } \\ { 8 } & { } & { } & { } \end{array} \right) = \left( \begin{array} { c c c c } { 8 } & { 4 } & { 1 } & { 3 } \\ { 7 } & { 6 } & { 5 } & { } \\ { 2 } & { } & { } & { } \\ { 9 } & { } & { } & { } \end{array} \right). \end{equation*}

This action clearly induces an action of $S _ { n }$ on $\lambda$-tabloids and this gives the desired module structure on $M ^ { \lambda }$.

If, again, $t$ is a $\lambda$-tableau, then one sets

\begin{equation*} e _ { t } = \sum _ { \pi } \operatorname { sgn } ( \pi ) \{ \pi t \}, \end{equation*}

where the sum runs over the $\pi \in S _ { n }$ that leave the set of numbers in each column in $t$ stable. Here, $\operatorname { sgn } ( \pi )$ is the sign of $\pi$.

The Specht module associated to $\lambda$ is defined as the submodule

\begin{equation*} S^{\lambda} = \operatorname{span} \{ e_t : t \text{ a }\lambda\text{-tableau } \} \end{equation*}

of $M^{\lambda}$. Clearly, $S^{\lambda}$ is invariant under the action of $S_{n}$. (In fact, $\sigma e_{t} = e_{ \sigma t}$ for all $\sigma \in S_{n}$ and all $\lambda$-tableau $t$.)

Specht modules were introduced in 1935 by W. Specht [a5]. Their importance in the representation theory for symmetric groups (cf. also Representation of the symmetric groups) comes from the fact that when $k$ contains $\mathbf{Q}$, then each $S^{\lambda}$ is a simple $k S_{n}$-module. Moreover, the set

\begin{equation*} \left\{S^{\lambda} : \lambda \text { a partition of } n \right\} \end{equation*}

is a full set of simple $k S_{n}$-modules.

When the characteristic of $k$ is $p > 0$, then the Specht modules are no longer always simple. However, they still play an important role in the classification of simple $k S_{n}$-modules. Namely, it turns out that when $\lambda$ is $p$-regular (i.e. no $p$ parts of $\lambda$ are equal), then $S^{\lambda}$ has a unique simple quotient $D^{\lambda}$ and the set

\begin{equation*} \{ D^{\lambda} : \lambda \text{ a p-regular partition of } n \} \end{equation*}

constitutes a full set of simple $k S_{n}$-modules. It is a major open problem (1999) to determine the dimensions of these modules.

It is possible to give a (characteristic-free) natural basis for $S^{\lambda}$. This is sometimes referred to as the Specht basis. In the notation above, it is given by

Here, a $\lambda$-tableau $t$ is called standard if the numbers occurring in $t$ are increasing along each row and down each column.

An immediate consequence is that the dimension of $S^{\lambda}$ equals the number of standard $\lambda$-tableaux. For various formulas for this number (as well as many further properties of Specht modules) see [a2] and [a3].

The representation theory for symmetric groups is intimately related to the corresponding theory for general linear groups (Schur duality). Under this correspondence, Specht modules play the same role for $S_{n}$ as do the Weyl modules for $\operatorname{GL}_{n}$, see e.g. [a1] and Weyl module. For a recent result exploring this correspondence in characteristic $p > 0$, see [a4].

References

[a1] J.A. Green, "Polynomial representations of $GL_n$" , Lecture Notes Math. , 830 , Springer (1980)
[a2] G.D. James, "The representation theory of the symmetric groups" , Lecture Notes Math. , 682 , Springer (1978)
[a3] G.D. James, A. Kerber, "The representation theory of the symmetric group" , Encycl. Math. Appl. , 16 , Addison-Wesley (1981)
[a4] O. Mathieu, "On the dimension of some modular irreducible representations of the symmetric group" Lett. Math. Phys. , 38 (1996) pp. 23–32
[a5] W. Specht, "Die irreduziblen Darstellungen der symmetrischen Gruppe" Math. Z. , 39 (1935) pp. 696–711
How to Cite This Entry:
Specht module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Specht_module&oldid=16338
This article was adapted from an original article by Henning Haahr Andersen (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article