# Representation of the symmetric groups

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. ) 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. ).

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=52077
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article