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
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and
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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
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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,
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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
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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
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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
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The restriction of a representation of
to the subgroup
can be found by the ramification rule
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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) |
[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) |
[4] | C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) |
[5] | G.D. James, "The representation theory of the symmetric groups" , Springer (1978) |
Comments
Let be the free Abelian group generated by the complex irreducible representations of the symmetric group on
letters,
. Now consider the direct sum
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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
:
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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
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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 ,
,
,
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A co-algebra structure is given by
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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],
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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:
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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 |
[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 |
[a4] | D. Knutson, "![]() |
[a5] | A.V. Zelevinsky, "Representations of finite classical groups" , Springer (1981) |
[a6] | D.C. Ravenel, "The Hopf ring for complex cobordism" J. Pure Appl. Algebra , 9 (1977) pp. 241–280 |
[a7] | S. Roman, "The umbral calculus" , Acad. Press (1984) |
[a8] | G. James, A. Kerber, "The representation theory of the symmetric group" , Addison-Wesley (1981) |
[a9] | G. de B. Robinson, "Representation theory of the symmetric group" , Univ. Toronto Press (1961) |
[a10] | J.A. Green, "Polynomial representations of ![]() |
Representation of the symmetric groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_the_symmetric_groups&oldid=15150