Representation of the symmetric groups

A linear representation of the group over a field . If , then all finite-dimensional representations of the symmetric groups are completely reducible (cf. Reducible representation) and defined over (in other words, irreducible finite-dimensional representations over are absolutely irreducible).

The irreducible finite-dimensional representations of over are classified as follows. Let be a Young diagram corresponding to a partition of the number , let (respectively, ) be the subgroup of consisting of all permutations mapping each of the numbers into a number in the same row (respectively, column) of . Then

and

where is the partition of dual to . There exists a unique irreducible representation of (depending on only) with the following properties: 1) in the space there is a non-zero vector such that for any ; and 2) in there exists a non-zero vector such that for any , where is the parity (sign) of . Representations corresponding to distinct partitions are not equivalent, and they exhaust all irreducible representations of over .

The vectors and are defined uniquely up to multiplication by a scalar. For all diagrams corresponding to a partition these vectors are normalized such that and for any . Here denotes the diagram obtained from by applying to all numbers the permutation . The vectors (respectively, ) corresponding to standard diagrams form a basis for . In this basis the operators of the representation have the form of integral matrices. The dimension of is

where , , and the product in the denominator of the last expression is taken over all cells of the Young tableau ; denotes the length of the corresponding hook.

To the partition corresponds the trivial one-dimensional representation of , while to the partition corresponds the non-trivial one-dimensional representation (the parity or sign representation). To the partition dual to corresponds the representation . The space can be identified (in a canonical way, up to a homothety) with , so that for any . Moreover, one may take , where is the diagram obtained from 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 is the Young diagram corresponding to a partition , then the representation is equivalent to the representation of in the left ideal of the group algebra generated by the Young symmetrizer . An a posteriori description of is the following: for , and is the operator, of rank 1, acting by the formula for any . Here denotes the invariant scalar product in , normalized in a suitable manner. Moreover,

The Frobenius formula gives a generating function for the characters of . However, recurrence relations are more convenient to use when computing individual values of characters. The Murnagan–Nakayama rule is most effective: Let be the value of a character of on the class of conjugate elements of defined by a partition of , and suppose that contains a number . Denote by the partition of obtained from by deleting . Then

where the sum is over all partitions of obtained by deleting a skew hook of length from the Young tableau , and where 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 , i.e. the matrix . Let be the representation of induced by the trivial one-dimensional representation of the subgroup , where is the Young diagram corresponding to the partition . Let and . If one assumes that the rows and columns of are positioned in order of lexicographically decreasing indices (partitions), then is a lower-triangular matrix with 1's on the diagonal. The value of a character of on a class is equal to

where is the order of the centralizer of the permutations (a representative) from . The matrix is upper triangular, and one has , where , from which can be uniquely found. Then the matrix is determined by

The restriction of a representation of to the subgroup can be found by the ramification rule

where the summation extends over all for which (including ). The restriction of to the subgroup is absolutely irreducible for and splits for over a quadratic extension of into a sum of two non-equivalent absolutely-irreducible representations of equal dimension. The representations of thus obtained exhaust all its irreducible representations over .

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 be the free Abelian group generated by the complex irreducible representations of the symmetric group on letters, . Now consider the direct sum

It is possible to define a Hopf algebra structure on , as follows. First the multiplication. Let and be, respectively, representations of and . Taking the tensor product defines a representation of . Consider as a subgroup of in the natural way. The product of and in is now defined by taking the induced representation to :

For the comultiplication restriction is used. Let be a representation of . For every , , consider the restriction of to to obtain an element of . The comultiplication of is now defined by

There is a unit mapping , defined by identifying and , and an augmentation , defined by identity on and if . It is a theorem that define a graded bi-algebra structure on . There is also an antipode, making a graded Hopf algebra.

This Hopf algebra can be explicitly described as follows. Consider the commutative ring of polynomials in infinitely many variables , , ,

A co-algebra structure is given by

and a co-unit by , for . There is also an antipode, making also a graded Hopf algebra. Perhaps the basic result in the representation theory of the symmetric groups is that and are isomorphic as Hopf algebras. The isomorphism is very nearly unique because, [a1],

The individual components of are also rings in themselves under the product of representations , . This defines a second multiplication on , which is distributive over the first, and becomes a ring object in the category of co-algebras over . Such objects have been called Hopf algebras, [a6], and quite a few of them occur naturally in algebraic topology. The ring occurs in algebraic topology as , the cohomology of the classifying space of complex -theory, and there is a "natural direct isomorphism" , [a3]. (This explains the notation used above for : the "ci" stand for Chern classes, cf. Chern class.)

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

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

As a coring object in the category of algebras , being the representing object 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 with in this manifestation.

The ring also carries the structure of a -ring and it is in fact the universal -ring on one generator, , [a4], and this gives a natural isomorphism , cf. -ring for some more details.

Finally there is a canonical notion of positivity on : 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, is the unique PSH-algebra on one generator and all other are tensor products of graded shifted copies of . This can be applied to other series of classical groups than the , [a5].

In combinatorics the algebra 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