# 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. [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

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=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