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A [[Linear representation|linear representation]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814701.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814702.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814703.png" />, then all finite-dimensional representations of the symmetric groups are completely reducible (cf. [[Reducible representation|Reducible representation]]) and defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814704.png" /> (in other words, irreducible finite-dimensional representations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814705.png" /> are absolutely irreducible).
+
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The irreducible finite-dimensional representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814706.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814707.png" /> are classified as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814708.png" /> be a [[Young diagram|Young diagram]] corresponding to a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r0814709.png" /> of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147010.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147011.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147012.png" />) be the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147013.png" /> consisting of all permutations mapping each of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147014.png" /> into a number in the same row (respectively, column) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147015.png" />. Then
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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/r/r081/r081470/r08147016.png" /></td> </tr></table>
+
A [[Linear representation|linear representation]] of the group  $  S _ {m} $
 +
over a field  $  K $.
 +
If  $  \mathop{\rm char}  K = 0 $,
 +
then all finite-dimensional representations of the symmetric groups are completely reducible (cf. [[Reducible representation|Reducible representation]]) and defined over  $  \mathbf Q $ (in other words, irreducible finite-dimensional representations over  $  \mathbf Q $
 +
are absolutely irreducible).
 +
 
 +
The irreducible finite-dimensional representations of  $  S _ {m} $
 +
over  $  \mathbf Q $
 +
are classified as follows. Let  $  d $
 +
be a [[Young diagram|Young diagram]] corresponding to a partition  $  \lambda = ( \lambda _ {1} \dots \lambda _ {r} ) $
 +
of the number  $  m $,
 +
let  $  R _ {d} $ (respectively,  $  C _ {d} $)
 +
be the subgroup of  $  S _ {m} $
 +
consisting of all permutations mapping each of the numbers  $  1 \dots m $
 +
into a number in the same row (respectively, column) of  $  d $.  
 +
Then
 +
 
 +
$$
 +
R _ {d}  \simeq \
 +
S _ {\lambda _ {1}  }
 +
\times \dots \times
 +
S _ {\lambda _ {r}  }
 +
$$
  
 
and
 
and
  
<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/r/r081/r081470/r08147017.png" /></td> </tr></table>
+
$$
 +
C _ {d}  \simeq \
 +
S _ {\lambda _ {1}  ^  \prime  }
 +
\times \dots \times
 +
S _ {\lambda _ {s}  ^  \prime  } ,
 +
$$
 +
 
 +
where  $  \lambda  ^  \prime  = ( \lambda _ {1}  ^  \prime  \dots \lambda _ {s}  ^  \prime  ) $
 +
is the partition of  $  m $
 +
dual to  $  \lambda $.
 +
There exists a unique irreducible representation  $  T _  \lambda  :  S _ {m} \rightarrow  \mathop{\rm GL} ( U _  \lambda  ) $
 +
of  $  S _ {m} $ (depending on  $  \lambda $
 +
only) with the following properties: 1) in the space  $  U _  \lambda  $
 +
there is a non-zero vector  $  u _ {d} $
 +
such that  $  T _  \lambda  ( g) u _ {d} = u _ {d} $
 +
for any  $  g \in R _ {d} $;  
 +
and 2) in  $  U _  \lambda  $
 +
there exists a non-zero vector  $  u _ {d}  ^  \prime  $
 +
such that  $  T _  \lambda  ( g) u _ {d}  ^  \prime  = \epsilon ( g) u _ {d}  ^  \prime  $
 +
for any  $  g \in C _ {d} $,
 +
where  $  \epsilon ( g) = \pm  1 $
 +
is the parity (sign) of  $  g $.
 +
Representations corresponding to distinct partitions are not equivalent, and they exhaust all irreducible representations of  $  S _ {m} $
 +
over  $  Q $.
 +
 
 +
The vectors  $  u _ {d} $
 +
and  $  u _ {d}  ^  \prime  $
 +
are defined uniquely up to multiplication by a scalar. For all diagrams corresponding to a partition  $  \lambda $
 +
these vectors are normalized such that  $  gu _ {d} = u _ {gd} $
 +
and  $  gu _ {d}  ^  \prime  = u _ {gd}  ^  \prime  $
 +
for any  $  g \in S _ {m} $.
 +
Here  $  gd $
 +
denotes the diagram obtained from  $  d $
 +
by applying to all numbers the permutation  $  g $.  
 +
The vectors  $  u _ {d} $ (respectively,  $  u _ {d}  ^  \prime  $)
 +
corresponding to standard diagrams  $  d $
 +
form a basis for  $  U _  \lambda  $.  
 +
In this basis the operators of the representation  $  T _  \lambda  $
 +
have the form of integral matrices. The dimension of  $  T _  \lambda  $
 +
is
 +
 
 +
$$
 +
\mathop{\rm dim}  T _  \lambda  = \
 +
 
 +
\frac{m! \prod _ {i < j }
 +
( l _ {i} - l _ {j} ) }{\prod _ {i} l _ {i} ! }
 +
  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147018.png" /> is the partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147019.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147020.png" />. There exists a unique irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147022.png" /> (depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147023.png" /> only) with the following properties: 1) in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147024.png" /> there is a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147026.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147027.png" />; and 2) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147028.png" /> there exists a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147030.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147032.png" /> is the parity (sign) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147033.png" />. Representations corresponding to distinct partitions are not equivalent, and they exhaust all irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147034.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147035.png" />.
+
\frac{m! }{\prod _ {( i, j) }  \lambda _ {ij} }
 +
,
 +
$$
  
The vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147037.png" /> are defined uniquely up to multiplication by a scalar. For all diagrams corresponding to a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147038.png" /> these vectors are normalized such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147040.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147041.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147042.png" /> denotes the diagram obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147043.png" /> by applying to all numbers the permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147044.png" />. The vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147045.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147046.png" />) corresponding to standard diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147047.png" /> form a basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147048.png" />. In this basis the operators of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147049.png" /> have the form of integral matrices. The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147050.png" /> is
+
where  $  l _ {i} = \lambda _ {i} + r - i $,
 +
$  i = 1 \dots r $,
 +
and the product in the denominator of the last expression is taken over all cells  $  c _ {ij} $
 +
of the Young tableau  $  t _  \lambda  $;
 +
$  \lambda _ {ij} $
 +
denotes the length of the corresponding hook.
  
<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/r/r081/r081470/r08147051.png" /></td> </tr></table>
+
To the partition  $  ( m) $
 +
corresponds the trivial one-dimensional representation of  $  S _ {m} $,
 +
while to the partition  $  ( 1, \dots, 1) $
 +
corresponds the non-trivial one-dimensional representation  $  \epsilon $ (the [[Signature (permutation)|signature]] homomorphism, giving the parity or '''sign representation'''). To the partition  $  \lambda  ^  \prime  $
 +
dual to  $  \lambda $
 +
corresponds the representation  $  \epsilon T _  \lambda  $.
 +
The space  $  U _ {\lambda  ^  \prime  } $
 +
can be identified (in a canonical way, up to a [[Homothety|homothety]]) with  $  U _  \lambda  $,
 +
so that  $  T _ {\lambda  ^  \prime  } ( g) = \epsilon ( g) T _  \lambda  ( g) $
 +
for any  $  g \in S _ {m} $.
 +
Moreover, one may take  $  u _ {d}  ^  \prime  = u _ {d  ^  \prime  } $,
 +
where  $  d  ^  \prime  $
 +
is the diagram obtained from  $  d $
 +
by transposition.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147053.png" />, and the product in the denominator of the last expression is taken over all cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147054.png" /> of the Young tableau <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147055.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147056.png" /> denotes the length of the corresponding hook.
+
The construction of a complete system of irreducible representations of a symmetric group invokes the use of the [[Young symmetrizer]], and allows one to obtain the decomposition of the [[regular representation]]. If  $  d $
 +
is the Young diagram corresponding to a partition  $  \lambda $,  
 +
then the representation  $  T _  \lambda  $
 +
is equivalent to the representation of  $  S _ {m} $
 +
in the left ideal of the group algebra  $  \mathbf Q S _ {m} $
 +
generated by the Young symmetrizer  $  e _ {d} $.  
 +
An a posteriori description of $  e _ {d} $
 +
is the following: $  T _  \mu  ( e _ {d} ) = 0 $
 +
for  $  \lambda \neq \mu $,
 +
and  $  T _  \lambda  ( e _ {d} ) $
 +
is the operator, of rank 1, acting by the formula  $  T _  \lambda  ( e _ {d} ) u = ( u _ {d} , u ) u _ {d}  ^  \prime  $
 +
for any  $  u \in U _  \lambda  $.  
 +
Here  $  (  , ) $
 +
denotes the invariant scalar product in  $  U _  \lambda  $,
 +
normalized in a suitable manner. Moreover,
  
To the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147057.png" /> corresponds the trivial one-dimensional representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147058.png" />, while to the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147059.png" /> corresponds the non-trivial one-dimensional representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147060.png" /> (the parity or sign representation). To the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147061.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147062.png" /> corresponds the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147063.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147064.png" /> can be identified (in a canonical way, up to a [[Homothety|homothety]]) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147065.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147066.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147067.png" />. Moreover, one may take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147068.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147069.png" /> is the diagram obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147070.png" /> by transposition.
+
$$
 +
( u _ {d} , u _ {d}  ^  \prime  ) = \
  
The construction of a complete system of irreducible representations of a symmetric group invokes the use of the [[Young symmetrizer|Young symmetrizer]], and allows one to obtain the decomposition of the [[Regular representation|regular representation]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147071.png" /> is the Young diagram corresponding to a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147072.png" />, then the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147073.png" /> is equivalent to the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147074.png" /> in the left ideal of the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147075.png" /> generated by the Young symmetrizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147076.png" />. An a posteriori description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147077.png" /> is the following: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147078.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147079.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147080.png" /> is the operator, of rank 1, acting by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147081.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147082.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147083.png" /> denotes the invariant scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147084.png" />, normalized in a suitable manner. Moreover,
+
\frac{m! }{ \mathop{\rm dim}  U _  \lambda  }
 +
.
 +
$$
  
<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/r/r081/r081470/r08147085.png" /></td> </tr></table>
+
The [[Frobenius formula|Frobenius formula]] gives a generating function for the characters of  $  T _  \lambda  $.
 +
However, recurrence relations are more convenient to use when computing individual values of characters. The Murnagan–Nakayama rule is most effective: Let  $  a _ {\lambda \mu }  = a _ {\lambda \mu }  ^ {(m)} $
 +
be the value of a character of  $  T _  \lambda  $
 +
on the class  $  [ \mu ] $
 +
of conjugate elements of  $  S _ {m} $
 +
defined by a partition  $  \mu $
 +
of  $  m $,
 +
and suppose that  $  \mu $
 +
contains a number  $  p $.
 +
Denote by  $  \overline \mu $
 +
the partition of  $  m - p $
 +
obtained from  $  \mu $
 +
by deleting  $  p $.  
 +
Then
  
The [[Frobenius formula|Frobenius formula]] gives a generating function for the characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147086.png" />. However, recurrence relations are more convenient to use when computing individual values of characters. The Murnagan–Nakayama rule is most effective: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147087.png" /> be the value of a character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147088.png" /> on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147089.png" /> of conjugate elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147090.png" /> defined by a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147091.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147092.png" />, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147093.png" /> contains a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147094.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147095.png" /> the partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147096.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147097.png" /> by deleting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r08147098.png" />. Then
+
$$
 +
a _ {\lambda \mu }  ^ {(m)}  = \
 +
\sum _ {\overline \lambda }
 +
(- 1) ^ {i ( \overline \lambda ) }
 +
a _ {\overline \lambda \overline \mu }  ^ {( m - p) } ,
 +
$$
  
<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/r/r081/r081470/r08147099.png" /></td> </tr></table>
+
where the sum is over all partitions  $  \overline \lambda $
 +
of  $  m - p $
 +
obtained by deleting a skew hook of length  $  p $
 +
from the [[Young tableau|Young tableau]]  $  t _  \lambda  $,
 +
and where  $  i ( \overline \lambda ) $
 +
denotes the height of the skew hook taken out.
  
where the sum is over all partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470100.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470101.png" /> obtained by deleting a skew hook of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470102.png" /> from the [[Young tableau|Young tableau]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470103.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470104.png" /> denotes the height of the skew hook taken out.
+
There is also a method (cf. [[#References|[5]]]) by which one can find the entire table of characters of  $  S _ {m} $,
 +
i.e. the matrix  $  A = \| a _ {\lambda \mu }  \| $.  
 +
Let  $  M _  \lambda  $
 +
be the representation of  $  S _ {m} $
 +
induced by the trivial one-dimensional representation of the subgroup  $  R _  \lambda  = R _ {d} $,  
 +
where $  d $
 +
is the Young diagram corresponding to the partition  $  \lambda $.
 +
Let  $  M _  \lambda  = \sum _  \mu  m _ {\lambda \mu }  T _  \mu  $
 +
and  $  M = \| m _ {\lambda \mu }  \| $.  
 +
If one assumes that the rows and columns of  $  M $
 +
are positioned in order of lexicographically decreasing indices (partitions), then  $  M $
 +
is a lower-triangular matrix with 1's on the diagonal. The value of a character of  $  M _  \lambda  $
 +
on a class  $  [ \mu ] $
 +
is equal to
  
There is also a method (cf. [[#References|[5]]]) by which one can find the entire table of characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470105.png" />, i.e. the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470106.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470107.png" /> be the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470108.png" /> induced by the trivial one-dimensional representation of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470110.png" /> is the Young diagram corresponding to the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470111.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470113.png" />. If one assumes that the rows and columns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470114.png" /> are positioned in order of lexicographically decreasing indices (partitions), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470115.png" /> is a lower-triangular matrix with 1's on the diagonal. The value of a character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470116.png" /> on a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470117.png" /> is equal to
+
$$
 +
b _ {\lambda \mu }  = \
  
<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/r/r081/r081470/r081470118.png" /></td> </tr></table>
+
\frac{c _  \mu  | R _  \lambda  \cap [ \mu ] | }{| R _  \lambda  | }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470119.png" /> is the order of the centralizer of the permutations (a representative) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470120.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470121.png" /> is upper triangular, and one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470122.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470123.png" />, from which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470124.png" /> can be uniquely found. Then the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470125.png" /> is determined by
+
where $  c _  \mu  $
 +
is the order of the centralizer of the permutations (a representative) from $  [ \mu ] $.  
 +
The matrix $  B = \| b _ {\lambda \mu }  \| $
 +
is upper triangular, and one has $  MM  ^ {T} = BC  ^ {-1} B  ^ {T} $,  
 +
where $  C = \mathop{\rm diag} ( c _  \mu  ) $,  
 +
from which $  M $
 +
can be uniquely found. Then the matrix $  A $
 +
is determined by
  
<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/r/r081/r081470/r081470126.png" /></td> </tr></table>
+
$$
 +
= M  ^ {-1} B.
 +
$$
  
The restriction of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470127.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470128.png" /> to the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470129.png" /> can be found by the ramification rule
+
The restriction of a representation $  T _  \lambda  $
 +
of $  S _ {m} $
 +
to the subgroup $  S _ {m - 1 }  $
 +
can be found by the ramification rule
  
<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/r/r081/r081470/r081470130.png" /></td> </tr></table>
+
$$
 +
T _  \lambda  \mid  _ {S _ {m - 1 }  }  = \
 +
\sum _ { i } T _ {( \lambda _ {1}  \dots \lambda _ {i} - 1 \dots \lambda _ {r} ) } ,
 +
$$
  
where the summation extends over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470131.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470132.png" /> (including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470133.png" />). The restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470134.png" /> to the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470135.png" /> is absolutely irreducible for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470136.png" /> and splits for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470137.png" /> over a quadratic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470138.png" /> into a sum of two non-equivalent absolutely-irreducible representations of equal dimension. The representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470139.png" /> thus obtained exhaust all its irreducible representations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470140.png" />.
+
where the summation extends over all $  i $
 +
for which $  \lambda _ {i} > \lambda _ {i + 1 }  $ (including r $).  
 +
The restriction of $  T _  \lambda  $
 +
to the subgroup $  A _ {m} $
 +
is absolutely irreducible for $  \lambda \neq \lambda  ^  \prime  $
 +
and splits for $  \lambda = \lambda  ^  \prime  $
 +
over a quadratic extension of $  \mathbf Q $
 +
into a sum of two non-equivalent absolutely-irreducible representations of equal dimension. The representations of $  A _ {m} $
 +
thus obtained exhaust all its irreducible representations over $  \mathbf C $.
  
For representations of the symmetric groups in tensors see [[Representation of the classical groups|Representation of the classical groups]].
+
For representations of the symmetric groups in tensors see [[Representation of the classical groups]].
  
 
The theory of modular representations of the symmetric groups has also been developed (see, e.g. [[#References|[5]]]).
 
The theory of modular representations of the symmetric groups has also been developed (see, e.g. [[#References|[5]]]).
Line 49: Line 224:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F.D. Murnagan, "The theory of group representations" , Johns Hopkins Univ. Press (1938)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M. Hamermesh, "Group theory and its application to physical problems" , Addison-Wesley (1962) {{MR|0136667}} {{ZBL|0100.36704}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) {{MR|0144979}} {{ZBL|0131.25601}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> G.D. James, "The representation theory of the symmetric groups" , Springer (1978) {{MR|0513828}} {{ZBL|0393.20009}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F.D. Murnagan, "The theory of group representations" , Johns Hopkins Univ. Press (1938)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M. Hamermesh, "Group theory and its application to physical problems" , Addison-Wesley (1962) {{MR|0136667}} {{ZBL|0100.36704}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) {{MR|0144979}} {{ZBL|0131.25601}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> G.D. James, "The representation theory of the symmetric groups" , Springer (1978) {{MR|0513828}} {{ZBL|0393.20009}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470141.png" /> be the free Abelian group generated by the complex irreducible representations of the symmetric group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470142.png" /> letters, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470143.png" />. Now consider the direct sum
+
Let $  R ( S _ {m} ) $
 +
be the free Abelian group generated by the complex irreducible representations of the symmetric group on $  m $
 +
letters, $  S _ {m} $.  
 +
Now consider the direct sum
  
<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/r/r081/r081470/r081470144.png" /></td> </tr></table>
+
$$
 +
= \oplus _ { m= 0} ^  \infty  R( S _ {m} ) ,\ \
 +
R ( S _ {0} )  = \mathbf Z .
 +
$$
  
It is possible to define a [[Hopf algebra|Hopf algebra]] structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470145.png" />, as follows. First the multiplication. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470146.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470147.png" /> be, respectively, representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470149.png" />. Taking the [[Tensor product|tensor product]] defines a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470150.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470151.png" />. Consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470152.png" /> as a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470153.png" /> in the natural way. The product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470155.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470156.png" /> is now defined by taking the [[Induced representation|induced representation]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470157.png" />:
+
It is possible to define a [[Hopf algebra|Hopf algebra]] structure on $  R $,  
 +
as follows. First the multiplication. Let $  \rho $
 +
and $  \tau $
 +
be, respectively, representations of $  S _ {n} $
 +
and $  S _ {m} $.  
 +
Taking the [[Tensor product|tensor product]] defines a representation $  ( g, h) \mapsto \rho ( g) \otimes \sigma ( h) $
 +
of $  S _ {n} \times S _ {m} $.  
 +
Consider $  S _ {n} \times S _ {m} $
 +
as a subgroup of $  S _ {n+m} $
 +
in the natural way. The product of $  \rho $
 +
and $  \tau $
 +
in $  R $
 +
is now defined by taking the [[Induced representation|induced representation]] to $  S _ {n+m} $:
  
<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/r/r081/r081470/r081470158.png" /></td> </tr></table>
+
$$
 +
\rho \sigma  =   \mathrm{Ind} _ {S _ {n}  \times S _ {m} } ^ {S _ {n+m} } ( \rho \otimes \sigma ) .
 +
$$
  
For the comultiplication restriction is used. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470159.png" /> be a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470160.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470162.png" />, consider the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470163.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470164.png" /> to obtain an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470165.png" />. The comultiplication of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470166.png" /> is now defined by
+
For the comultiplication restriction is used. Let $  \rho $
 +
be a representation of $  S _ {n} $.  
 +
For every $  p, q \in \{ 0, 1 , \dots \} $,
 +
$  p+ q = n $,  
 +
consider the restriction of $  \rho $
 +
to $  S _ {p} \times S _ {q} $
 +
to obtain an element of $  R ( S _ {p} \times S _ {q} ) = R ( S _ {p} ) \otimes R( S _ {q} ) $.  
 +
The comultiplication of $  R $
 +
is now defined by
  
<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/r/r081/r081470/r081470167.png" /></td> </tr></table>
+
$$
 +
\mu  = \sum _ { p+ q= n}  \mathrm{Res} _ {S _ {p}  \times S _ {q} } ^ {S _ {n} } ( \rho ) .
 +
$$
  
There is a unit mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470168.png" />, defined by identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470170.png" />, and an augmentation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470171.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470172.png" /> identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470173.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470174.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470175.png" />. It is a theorem that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470176.png" /> define a graded bi-algebra structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470177.png" />. There is also an antipode, making <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470178.png" /> a graded Hopf algebra.
+
There is a unit mapping $  e : \mathbf Z \rightarrow R $,  
 +
defined by identifying $  \mathbf Z $
 +
and $  R( S _ {0} ) $,  
 +
and an augmentation $  \epsilon : R \rightarrow \mathbf Z $,  
 +
defined by $  \epsilon = $
 +
identity on $  R( S _ {0} ) = \mathbf Z $
 +
and $  \epsilon ( R( S _ {m} )) = 0 $
 +
if  $  m > 0 $.  
 +
It is a theorem that $  ( m, \mu , e , \epsilon ) $
 +
define a graded bi-algebra structure on $  R $.  
 +
There is also an antipode, making $  R $
 +
a graded Hopf algebra.
  
This Hopf algebra can be explicitly described as follows. Consider the commutative [[Ring of polynomials|ring of polynomials]] in infinitely many variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470179.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470181.png" />,
+
This Hopf algebra can be explicitly described as follows. Consider the commutative [[Ring of polynomials|ring of polynomials]] in infinitely many variables $  c _ {i} $,
 +
$  i = 1, 2 , \dots $,  
 +
$  c _ {0} = 1 $,
  
<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/r/r081/r081470/r081470182.png" /></td> </tr></table>
+
$$
 +
= \mathbf Z [ c _ {1} , c _ {2} , \dots ] .
 +
$$
  
 
A [[Co-algebra|co-algebra]] structure is given by
 
A [[Co-algebra|co-algebra]] structure is given by
  
<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/r/r081/r081470/r081470183.png" /></td> </tr></table>
+
$$
 +
c _ {n}  \mapsto  \sum _ { p+ q= n} c _ {p} \otimes c _ {q} ,
 +
$$
  
and a co-unit by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470185.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470186.png" />. There is also an antipode, making <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470187.png" /> also a graded Hopf algebra. Perhaps the basic result in the representation theory of the symmetric groups is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470188.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470189.png" /> are isomorphic as Hopf algebras. The isomorphism is very nearly unique because, [[#References|[a1]]],
+
and a co-unit by $  \epsilon ( c _ {0} ) = 1 $,  
 +
$  \epsilon ( c _ {n} ) = 0 $
 +
for $  n \geq  1 $.  
 +
There is also an antipode, making $  U $
 +
also a graded Hopf algebra. Perhaps the basic result in the representation theory of the symmetric groups is that $  R $
 +
and $  U $
 +
are isomorphic as Hopf algebras. The isomorphism is very nearly unique because, [[#References|[a1]]],
  
<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/r/r081/r081470/r081470190.png" /></td> </tr></table>
+
$$
 +
\mathrm{Aut} _ { {\rm Hopf}  } ( U)  = \
 +
\mathbf Z /( 2) \times \mathbf Z / ( 2) .
 +
$$
  
The individual components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470191.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470192.png" /> are also rings in themselves under the product of representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470193.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470194.png" />. This defines a second multiplication on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470195.png" />, which is distributive over the first, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470196.png" /> becomes a ring object in the category of co-algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470197.png" />. Such objects have been called Hopf algebras, [[#References|[a6]]], and quite a few of them occur naturally in algebraic topology. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470198.png" /> occurs in algebraic topology as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470199.png" />, the cohomology of the [[Classifying space|classifying space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470200.png" /> of complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470201.png" />-theory, and there is a "natural direct isomorphism" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470202.png" />, [[#References|[a3]]]. (This explains the notation used above for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470203.png" />: the "ci" stand for Chern classes, cf. [[Chern class|Chern class]].)
+
The individual components $  R( S _ {m} ) $
 +
of $  R $
 +
are also rings in themselves under the product of representations $  \rho , \sigma \mapsto \rho \times \sigma $,  
 +
$  ( \rho \times \sigma ) ( g) = \rho ( g) \otimes \sigma ( g) $.  
 +
This defines a second multiplication on $  R $,  
 +
which is distributive over the first, and $  R $
 +
becomes a ring object in the category of co-algebras over $  \mathbf Z $.  
 +
Such objects have been called Hopf algebras, [[#References|[a6]]], and quite a few of them occur naturally in algebraic topology. The ring $  U \simeq R $
 +
occurs in algebraic topology as $  H  ^  \star  ( \mathbf{BU} ) $,  
 +
the cohomology of the [[Classifying space|classifying space]] $  \mathbf{BU} $
 +
of complex $  K $-theory, and there is a "natural direct isomorphism" $  R \simeq H  ^  \star  ( \mathbf{BU} ) $,  
 +
[[#References|[a3]]]. (This explains the notation used above for $  U $:  
 +
the "ci" stand for Chern classes, cf. [[Chern class|Chern class]].)
  
There is also an inner product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470204.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470205.png" /> counts the number of irreducible representations that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470207.png" /> have in common, and with respect to this inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470208.png" /> is (graded) self-dual. In particular, the multiplication and comultiplication are adjoint to one another:
+
There is also an inner product on $  R = U $:  
 +
$  \langle  \rho , \sigma \rangle $
 +
counts the number of irreducible representations that $  \rho $
 +
and $  \sigma $
 +
have in common, and with respect to this inner product $  R $
 +
is (graded) self-dual. In particular, the multiplication and comultiplication are adjoint to one another:
  
<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/r/r081/r081470/r081470209.png" /></td> </tr></table>
+
$$
 +
\langle  \rho , \sigma \tau \rangle  = \
 +
\langle  \mu ( \rho ) , \sigma \otimes \tau \rangle ,
 +
$$
  
 
which is the same as Frobenius reciprocity, cf. [[Induced representation|Induced representation]], in this case.
 
which is the same as Frobenius reciprocity, cf. [[Induced representation|Induced representation]], in this case.
  
As a coring object in the category of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470210.png" />, being the representing object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470211.png" /> of the functor of Witt vectors, [[#References|[a2]]], plays an important role in formal group theory. But, so far, no direct natural isomorphism has been found linking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470212.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470213.png" /> in this manifestation.
+
As a coring object in the category of algebras $  U $,  
 +
being the representing object $  R( W) $
 +
of the functor of Witt vectors, [[#References|[a2]]], plays an important role in formal group theory. But, so far, no direct natural isomorphism has been found linking $  R $
 +
with $  U = R( W) $
 +
in this manifestation.
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470214.png" /> also carries the structure of a [[Lambda-ring|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470215.png" />-ring]] and it is in fact the universal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470217.png" />-ring on one generator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470218.png" />, [[#References|[a4]]], and this gives a natural isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470219.png" />, cf. [[Lambda-ring|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470220.png" />-ring]] for some more details.
+
The ring $  U $
 +
also carries the structure of a [[Lambda-ring| $  \lambda $-ring]] and it is in fact the universal $  \lambda $-ring on one generator, $  U( \Lambda ) $,  
 +
[[#References|[a4]]], and this gives a natural isomorphism $  U( \Lambda ) \simeq R( W) $,  
 +
cf. [[Lambda-ring| $  \lambda $-ring]] for some more details.
  
Finally there is a canonical notion of positivity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470221.png" />: the actual (i.e. not virtual) representations are positive and the multiplication and comultiplication preserve positivity. This has led to the notion of a PSH-algebra, which stands for positive self-adjoint Hopf algebra, [[#References|[a5]]]. Essentially, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470222.png" /> is the unique PSH-algebra on one generator and all other are tensor products of graded shifted copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470223.png" />. This can be applied to other series of classical groups than the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470224.png" />, [[#References|[a5]]].
+
Finally there is a canonical notion of positivity on $  \oplus R( S _ {n} ) $:  
 +
the actual (i.e. not virtual) representations are positive and the multiplication and comultiplication preserve positivity. This has led to the notion of a PSH-algebra, which stands for positive self-adjoint Hopf algebra, [[#References|[a5]]]. Essentially, $  U $
 +
is the unique PSH-algebra on one generator and all other are tensor products of graded shifted copies of $  U $.  
 +
This can be applied to other series of classical groups than the $  S _ {m} $,  
 +
[[#References|[a5]]].
  
In combinatorics the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081470/r081470225.png" /> also has a long history. In modern terminology it is at the basis of the so-called umbral calculus, [[#References|[a7]]].
+
In combinatorics the algebra $  U $
 +
also has a long history. In modern terminology it is at the basis of the so-called umbral calculus, [[#References|[a7]]].
  
 
A modern reference to the representation theory of the symmetric groups, both ordinary and modular, is [[#References|[a8]]].
 
A modern reference to the representation theory of the symmetric groups, both ordinary and modular, is [[#References|[a8]]].

Latest revision as of 13:04, 18 February 2022


A linear representation of the group $ S _ {m} $ over a field $ K $. If $ \mathop{\rm char} K = 0 $, then all finite-dimensional representations of the symmetric groups are completely reducible (cf. Reducible representation) and defined over $ \mathbf Q $ (in other words, irreducible finite-dimensional representations over $ \mathbf Q $ are absolutely irreducible).

The irreducible finite-dimensional representations of $ S _ {m} $ over $ \mathbf Q $ are classified as follows. Let $ d $ be a Young diagram corresponding to a partition $ \lambda = ( \lambda _ {1} \dots \lambda _ {r} ) $ of the number $ m $, let $ R _ {d} $ (respectively, $ C _ {d} $) be the subgroup of $ S _ {m} $ consisting of all permutations mapping each of the numbers $ 1 \dots m $ into a number in the same row (respectively, column) of $ d $. Then

$$ R _ {d} \simeq \ S _ {\lambda _ {1} } \times \dots \times S _ {\lambda _ {r} } $$

and

$$ C _ {d} \simeq \ S _ {\lambda _ {1} ^ \prime } \times \dots \times S _ {\lambda _ {s} ^ \prime } , $$

where $ \lambda ^ \prime = ( \lambda _ {1} ^ \prime \dots \lambda _ {s} ^ \prime ) $ is the partition of $ m $ dual to $ \lambda $. There exists a unique irreducible representation $ T _ \lambda : S _ {m} \rightarrow \mathop{\rm GL} ( U _ \lambda ) $ of $ S _ {m} $ (depending on $ \lambda $ only) with the following properties: 1) in the space $ U _ \lambda $ there is a non-zero vector $ u _ {d} $ such that $ T _ \lambda ( g) u _ {d} = u _ {d} $ for any $ g \in R _ {d} $; and 2) in $ U _ \lambda $ there exists a non-zero vector $ u _ {d} ^ \prime $ such that $ T _ \lambda ( g) u _ {d} ^ \prime = \epsilon ( g) u _ {d} ^ \prime $ for any $ g \in C _ {d} $, where $ \epsilon ( g) = \pm 1 $ is the parity (sign) of $ g $. Representations corresponding to distinct partitions are not equivalent, and they exhaust all irreducible representations of $ S _ {m} $ over $ Q $.

The vectors $ u _ {d} $ and $ u _ {d} ^ \prime $ are defined uniquely up to multiplication by a scalar. For all diagrams corresponding to a partition $ \lambda $ these vectors are normalized such that $ gu _ {d} = u _ {gd} $ and $ gu _ {d} ^ \prime = u _ {gd} ^ \prime $ for any $ g \in S _ {m} $. Here $ gd $ denotes the diagram obtained from $ d $ by applying to all numbers the permutation $ g $. The vectors $ u _ {d} $ (respectively, $ u _ {d} ^ \prime $) corresponding to standard diagrams $ d $ form a basis for $ U _ \lambda $. In this basis the operators of the representation $ T _ \lambda $ have the form of integral matrices. The dimension of $ T _ \lambda $ is

$$ \mathop{\rm dim} T _ \lambda = \ \frac{m! \prod _ {i < j } ( l _ {i} - l _ {j} ) }{\prod _ {i} l _ {i} ! } = \ \frac{m! }{\prod _ {( i, j) } \lambda _ {ij} } , $$

where $ l _ {i} = \lambda _ {i} + r - i $, $ i = 1 \dots r $, and the product in the denominator of the last expression is taken over all cells $ c _ {ij} $ of the Young tableau $ t _ \lambda $; $ \lambda _ {ij} $ denotes the length of the corresponding hook.

To the partition $ ( m) $ corresponds the trivial one-dimensional representation of $ S _ {m} $, while to the partition $ ( 1, \dots, 1) $ corresponds the non-trivial one-dimensional representation $ \epsilon $ (the signature homomorphism, giving the parity or sign representation). To the partition $ \lambda ^ \prime $ dual to $ \lambda $ corresponds the representation $ \epsilon T _ \lambda $. The space $ U _ {\lambda ^ \prime } $ can be identified (in a canonical way, up to a homothety) with $ U _ \lambda $, so that $ T _ {\lambda ^ \prime } ( g) = \epsilon ( g) T _ \lambda ( g) $ for any $ g \in S _ {m} $. Moreover, one may take $ u _ {d} ^ \prime = u _ {d ^ \prime } $, where $ d ^ \prime $ is the diagram obtained from $ d $ by transposition.

The construction of a complete system of irreducible representations of a symmetric group invokes the use of the Young symmetrizer, and allows one to obtain the decomposition of the regular representation. If $ d $ is the Young diagram corresponding to a partition $ \lambda $, then the representation $ T _ \lambda $ is equivalent to the representation of $ S _ {m} $ in the left ideal of the group algebra $ \mathbf Q S _ {m} $ generated by the Young symmetrizer $ e _ {d} $. An a posteriori description of $ e _ {d} $ is the following: $ T _ \mu ( e _ {d} ) = 0 $ for $ \lambda \neq \mu $, and $ T _ \lambda ( e _ {d} ) $ is the operator, of rank 1, acting by the formula $ T _ \lambda ( e _ {d} ) u = ( u _ {d} , u ) u _ {d} ^ \prime $ for any $ u \in U _ \lambda $. Here $ ( , ) $ denotes the invariant scalar product in $ U _ \lambda $, normalized in a suitable manner. Moreover,

$$ ( u _ {d} , u _ {d} ^ \prime ) = \ \frac{m! }{ \mathop{\rm dim} U _ \lambda } . $$

The Frobenius formula gives a generating function for the characters of $ T _ \lambda $. However, recurrence relations are more convenient to use when computing individual values of characters. The Murnagan–Nakayama rule is most effective: Let $ a _ {\lambda \mu } = a _ {\lambda \mu } ^ {(m)} $ be the value of a character of $ T _ \lambda $ on the class $ [ \mu ] $ of conjugate elements of $ S _ {m} $ defined by a partition $ \mu $ of $ m $, and suppose that $ \mu $ contains a number $ p $. Denote by $ \overline \mu $ the partition of $ m - p $ obtained from $ \mu $ by deleting $ p $. Then

$$ a _ {\lambda \mu } ^ {(m)} = \ \sum _ {\overline \lambda } (- 1) ^ {i ( \overline \lambda ) } a _ {\overline \lambda \overline \mu } ^ {( m - p) } , $$

where the sum is over all partitions $ \overline \lambda $ of $ m - p $ obtained by deleting a skew hook of length $ p $ from the Young tableau $ t _ \lambda $, and where $ i ( \overline \lambda ) $ denotes the height of the skew hook taken out.

There is also a method (cf. [5]) by which one can find the entire table of characters of $ S _ {m} $, i.e. the matrix $ A = \| a _ {\lambda \mu } \| $. Let $ M _ \lambda $ be the representation of $ S _ {m} $ induced by the trivial one-dimensional representation of the subgroup $ R _ \lambda = R _ {d} $, where $ d $ is the Young diagram corresponding to the partition $ \lambda $. Let $ M _ \lambda = \sum _ \mu m _ {\lambda \mu } T _ \mu $ and $ M = \| m _ {\lambda \mu } \| $. If one assumes that the rows and columns of $ M $ are positioned in order of lexicographically decreasing indices (partitions), then $ M $ is a lower-triangular matrix with 1's on the diagonal. The value of a character of $ M _ \lambda $ on a class $ [ \mu ] $ is equal to

$$ b _ {\lambda \mu } = \ \frac{c _ \mu | R _ \lambda \cap [ \mu ] | }{| R _ \lambda | } , $$

where $ c _ \mu $ is the order of the centralizer of the permutations (a representative) from $ [ \mu ] $. The matrix $ B = \| b _ {\lambda \mu } \| $ is upper triangular, and one has $ MM ^ {T} = BC ^ {-1} B ^ {T} $, where $ C = \mathop{\rm diag} ( c _ \mu ) $, from which $ M $ can be uniquely found. Then the matrix $ A $ is determined by

$$ A = M ^ {-1} B. $$

The restriction of a representation $ T _ \lambda $ of $ S _ {m} $ to the subgroup $ S _ {m - 1 } $ can be found by the ramification rule

$$ T _ \lambda \mid _ {S _ {m - 1 } } = \ \sum _ { i } T _ {( \lambda _ {1} \dots \lambda _ {i} - 1 \dots \lambda _ {r} ) } , $$

where the summation extends over all $ i $ for which $ \lambda _ {i} > \lambda _ {i + 1 } $ (including $ r $). The restriction of $ T _ \lambda $ to the subgroup $ A _ {m} $ is absolutely irreducible for $ \lambda \neq \lambda ^ \prime $ and splits for $ \lambda = \lambda ^ \prime $ over a quadratic extension of $ \mathbf Q $ into a sum of two non-equivalent absolutely-irreducible representations of equal dimension. The representations of $ A _ {m} $ thus obtained exhaust all its irreducible representations over $ \mathbf C $.

For representations of the symmetric groups in tensors see Representation of the classical groups.

The theory of modular representations of the symmetric groups has also been developed (see, e.g. [5]).

References

[1] H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502
[2] F.D. Murnagan, "The theory of group representations" , Johns Hopkins Univ. Press (1938)
[3] M. Hamermesh, "Group theory and its application to physical problems" , Addison-Wesley (1962) MR0136667 Zbl 0100.36704
[4] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) MR0144979 Zbl 0131.25601
[5] G.D. James, "The representation theory of the symmetric groups" , Springer (1978) MR0513828 Zbl 0393.20009

Comments

Let $ R ( S _ {m} ) $ be the free Abelian group generated by the complex irreducible representations of the symmetric group on $ m $ letters, $ S _ {m} $. Now consider the direct sum

$$ R = \oplus _ { m= 0} ^ \infty R( S _ {m} ) ,\ \ R ( S _ {0} ) = \mathbf Z . $$

It is possible to define a Hopf algebra structure on $ R $, as follows. First the multiplication. Let $ \rho $ and $ \tau $ be, respectively, representations of $ S _ {n} $ and $ S _ {m} $. Taking the tensor product defines a representation $ ( g, h) \mapsto \rho ( g) \otimes \sigma ( h) $ of $ S _ {n} \times S _ {m} $. Consider $ S _ {n} \times S _ {m} $ as a subgroup of $ S _ {n+m} $ in the natural way. The product of $ \rho $ and $ \tau $ in $ R $ is now defined by taking the induced representation to $ S _ {n+m} $:

$$ \rho \sigma = \mathrm{Ind} _ {S _ {n} \times S _ {m} } ^ {S _ {n+m} } ( \rho \otimes \sigma ) . $$

For the comultiplication restriction is used. Let $ \rho $ be a representation of $ S _ {n} $. For every $ p, q \in \{ 0, 1 , \dots \} $, $ p+ q = n $, consider the restriction of $ \rho $ to $ S _ {p} \times S _ {q} $ to obtain an element of $ R ( S _ {p} \times S _ {q} ) = R ( S _ {p} ) \otimes R( S _ {q} ) $. The comultiplication of $ R $ is now defined by

$$ \mu = \sum _ { p+ q= n} \mathrm{Res} _ {S _ {p} \times S _ {q} } ^ {S _ {n} } ( \rho ) . $$

There is a unit mapping $ e : \mathbf Z \rightarrow R $, defined by identifying $ \mathbf Z $ and $ R( S _ {0} ) $, and an augmentation $ \epsilon : R \rightarrow \mathbf Z $, defined by $ \epsilon = $ identity on $ R( S _ {0} ) = \mathbf Z $ and $ \epsilon ( R( S _ {m} )) = 0 $ if $ m > 0 $. It is a theorem that $ ( m, \mu , e , \epsilon ) $ define a graded bi-algebra structure on $ R $. There is also an antipode, making $ R $ a graded Hopf algebra.

This Hopf algebra can be explicitly described as follows. Consider the commutative ring of polynomials in infinitely many variables $ c _ {i} $, $ i = 1, 2 , \dots $, $ c _ {0} = 1 $,

$$ U = \mathbf Z [ c _ {1} , c _ {2} , \dots ] . $$

A co-algebra structure is given by

$$ c _ {n} \mapsto \sum _ { p+ q= n} c _ {p} \otimes c _ {q} , $$

and a co-unit by $ \epsilon ( c _ {0} ) = 1 $, $ \epsilon ( c _ {n} ) = 0 $ for $ n \geq 1 $. There is also an antipode, making $ U $ also a graded Hopf algebra. Perhaps the basic result in the representation theory of the symmetric groups is that $ R $ and $ U $ are isomorphic as Hopf algebras. The isomorphism is very nearly unique because, [a1],

$$ \mathrm{Aut} _ { {\rm Hopf} } ( U) = \ \mathbf Z /( 2) \times \mathbf Z / ( 2) . $$

The individual components $ R( S _ {m} ) $ of $ R $ are also rings in themselves under the product of representations $ \rho , \sigma \mapsto \rho \times \sigma $, $ ( \rho \times \sigma ) ( g) = \rho ( g) \otimes \sigma ( g) $. This defines a second multiplication on $ R $, which is distributive over the first, and $ R $ becomes a ring object in the category of co-algebras over $ \mathbf Z $. Such objects have been called Hopf algebras, [a6], and quite a few of them occur naturally in algebraic topology. The ring $ U \simeq R $ occurs in algebraic topology as $ H ^ \star ( \mathbf{BU} ) $, the cohomology of the classifying space $ \mathbf{BU} $ of complex $ K $-theory, and there is a "natural direct isomorphism" $ R \simeq H ^ \star ( \mathbf{BU} ) $, [a3]. (This explains the notation used above for $ U $: the "ci" stand for Chern classes, cf. Chern class.)

There is also an inner product on $ R = U $: $ \langle \rho , \sigma \rangle $ counts the number of irreducible representations that $ \rho $ and $ \sigma $ have in common, and with respect to this inner product $ R $ is (graded) self-dual. In particular, the multiplication and comultiplication are adjoint to one another:

$$ \langle \rho , \sigma \tau \rangle = \ \langle \mu ( \rho ) , \sigma \otimes \tau \rangle , $$

which is the same as Frobenius reciprocity, cf. Induced representation, in this case.

As a coring object in the category of algebras $ U $, being the representing object $ R( W) $ of the functor of Witt vectors, [a2], plays an important role in formal group theory. But, so far, no direct natural isomorphism has been found linking $ R $ with $ U = R( W) $ in this manifestation.

The ring $ U $ also carries the structure of a $ \lambda $-ring and it is in fact the universal $ \lambda $-ring on one generator, $ U( \Lambda ) $, [a4], and this gives a natural isomorphism $ U( \Lambda ) \simeq R( W) $, cf. $ \lambda $-ring for some more details.

Finally there is a canonical notion of positivity on $ \oplus R( S _ {n} ) $: the actual (i.e. not virtual) representations are positive and the multiplication and comultiplication preserve positivity. This has led to the notion of a PSH-algebra, which stands for positive self-adjoint Hopf algebra, [a5]. Essentially, $ U $ is the unique PSH-algebra on one generator and all other are tensor products of graded shifted copies of $ U $. This can be applied to other series of classical groups than the $ S _ {m} $, [a5].

In combinatorics the algebra $ U $ also has a long history. In modern terminology it is at the basis of the so-called umbral calculus, [a7].

A modern reference to the representation theory of the symmetric groups, both ordinary and modular, is [a8].

References

[a1] A. Liulevicius, "Arrows, symmetries, and representation rings" J. Pure Appl. Algebra , 19 (1980) pp. 259–273 MR0593256 Zbl 0448.55013
[a2] M. Hazewinkel, "Formal rings and applications" , Acad. Press (1978)
[a3] M.F. Atiyah, "Power operations in K-theory" Quarterly J. Math. (2) , 17 (1966) pp. 165–193 MR0202130 Zbl 0144.44901
[a4] D. Knutson, "-rings and the representation theory of the symmetric group" , Springer (1973) MR0364425 Zbl 0272.20008
[a5] A.V. Zelevinsky, "Representations of finite classical groups" , Springer (1981) MR0643482 Zbl 0465.20009
[a6] D.C. Ravenel, "The Hopf ring for complex cobordism" J. Pure Appl. Algebra , 9 (1977) pp. 241–280 MR0448337 Zbl 0373.57020
[a7] S. Roman, "The umbral calculus" , Acad. Press (1984) MR0741185 Zbl 0536.33001
[a8] G. James, A. Kerber, "The representation theory of the symmetric group" , Addison-Wesley (1981) MR0644144 Zbl 0491.20010
[a9] G. de B. Robinson, "Representation theory of the symmetric group" , Univ. Toronto Press (1961) MR1531490 MR0125885 Zbl 0102.02002
[a10] J.A. Green, "Polynomial representations of " , Lect. notes in math. , 830 , Springer (1980) MR0606556 Zbl 0451.20037
How to Cite This Entry:
Representation of the symmetric groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_the_symmetric_groups&oldid=24125
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article