Difference between revisions of "Representation of the symmetric groups"
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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. | 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. | ||
− | 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. | + | 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 [[Signature (permutation)|signature]] homomorphism, giving 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. |
− | The construction of a complete system of irreducible representations of a symmetric group invokes the use of the [[ | + | 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 <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, |
<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> | <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> | ||
Line 43: | Line 43: | ||
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 <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" />. | ||
− | For representations of the symmetric groups in tensors see [[ | + | 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]]]). |
Revision as of 18:39, 30 November 2016
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 signature homomorphism, giving 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, "![]() |
[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 ![]() |
Representation of the symmetric groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_the_symmetric_groups&oldid=24125