Representation Theory of Symmetric Groups [expository notes]

23 Pages·2013·0.375 MB·English

by  João Pedro Martins dos Santos

Representation Theory of Symmetric Groups
João Pedro Martins dos Santos
2nd semester of 2012/2013
1
Contents
1 Introduction 3
2 Basic notions 4
3 Character theory 8
4 Induced representations 12
5 The group algebra 14
6 Irreducible representations of S 15
d
7 Frobenius’ formula 19
References 23
2
1 Introduction
Representation theory is very much a 20th century subject. In the 19th
century, when groups were dealt with, they were generally understood as
subsets, closed under composition and inverse, of the permutations of a
set or of the automorphisms GL(V) of a vector space V. The notion of
an abstract group was only given in the 20th century, making it possible
to make a distinction between the properties of the abstract group and the
propertiesoftheparticularrealizationasasubgroupofapermutationgroup
or of GL(V).
What would have been called in the 19th century simply ”group theory”
is now factored into two parts. First, there is the study of the structure
of abstract groups (e.g., the classification of simple groups). Second is the
companion question: given a group G, how can we describe all the ways
in which G may be embedded in (or just mapped to) linear group GL(V)?
This, of course, is the subject matter of representation theory.
Given this point of view, it makes sense when first introducing repre-
sentation theory to do so in a context where the nature of the groups G in
question is itself simple, and relatively well understood. In fact, throughout
the entire article, only representations of finite groups are mentioned.
In section 2, we start defining what a representation is, then give some
basic examples and end by establishing the Schur’s lemma and concluding
that every representation is a direct sum of irreducible ones.
In section 3, we talk about a very useful tool in representation theory:
character theory. Using character theory, we can prove that every finite
group has a finite number of irreducible representations. In fact, that num-
ber is equal to the number of conjugacy classes in the group.
Insection4, wedefinewhataninducedrepresentationis, atoolthatwill
be necessary to prove Frobenius’ formula for the characters of irreducible
representations of symmetric groups.
Insection5, wegiveabriefintroductiontotheconceptofgroupalgebra,
whichwillbeakeyconceptinstudyingtherepresentationsofthesymmetric
groups, since irreducible representations will be identified as minimal left
ideals of the group algebra.
In section 6, we finally reach our goal: determining all irreducible rep-
resentations of the symmetric groups. We even get an explicit bijection be-
tween the set of irreducible representations and the set of conjugacy classes
of the symmetric group.
Last but not least, in section 7, using symmetric polynomials and in-
duced representations, we prove Frobenius’ formula for the characters of
irreducible representations of symmetric groups, from which we can get a
simpler formula for the dimension of an irreducible representation.
3
2 Basic notions
Definition 1. A representation of a finite group G on a finite dimensional
complex vector space V is a group homomorphism ρ : G → GL(V) of G to
the group of automorphisms of V.
The map ρ gives V the structure of a G-module.
When there is little ambiguity about the map ρ, we sometimes call V
itself a representation of G and we often suppress the symbol ρ, writing g·v
or gv for ρ(g)(v). The dimension of V is also called the degree of ρ.
Definition 2. The unit or trivial representation of G is the representation
ρ : G → GL(C) such that ρ(g) = 1 for every g ∈ G.
Definition 3. If G has a subgroup H with index 2, then we can define the
alternating representation associated to the pair (G,H) as the representation
ρ : G → GL(C) such that ρ(g) = 1 if g ∈ H and ρ(g) = −1 otherwise.
The particular case of this definition that will sometimes be referred to
in this article is the case when G = S is the symmetric group and H = A
d d
is the alternating group.
Definition 4. If X is any finite set and G acts on the left on X, there is an
associated permutation representation. Let V be the vector space with basis
{e : x ∈ X} and let G act on V by
x
X X
g· a e = a e .
x x x gx
x∈X x∈X
Definition 5. The regular representation, denoted R or just R, is the
G
permutation representation corresponding to the left action of G on itself.
Definition 6. A sub-representation of a representation V is a vector sub-
space W of V which is invariant under G. In other words, W is a G-
submodule of V.
IfV andW arerepresentations,thenthedirectsumV⊕W andthetensor
product V ⊗W are also representations, the latter via g(v⊗w) = gv⊗gw.
For a representation V, the n-th tensor power V⊗n is again a repre-
sentation of G by this rule, and the exterior powers Λn(V) and symmetric
powers Symn(V) are sub-representations of it. The dual V∗ = Hom(V,C)
of V is also a representation, though not in the most obvious way: we want
the two representations of G to respect the natural pairing (denoted h , i)
between V∗ and V, so that if ρ : G → GL(V) is a representation and
ρ∗ : G → GL(V∗) is the dual, we should have
hρ∗(g)(v∗),ρ(g)(v)i = hv∗,vi
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for all g ∈ G, v ∈ V and v∗ ∈ V∗. This in turn forces us to define the dual
representation by
ρ∗(g)(v∗) = v∗◦ρ(g−1), for every g ∈ G.
Having defined the dual of a representation and the tensor product of
two representations, then Hom(V,W) is also a representation, via the iden-
tification Hom(V,W) = V∗⊗W. Unravelling this, if we view an element of
Hom(V,W) as a linear map ϕ : V → W, we have
(gϕ)(v) = gϕ(g−1v), for every g ∈ G and v ∈ V. (1)
In other words, the definition is such that the diagram
ϕ
V W
g g
V W
gϕ
commutes. Note that the dual representation is, in turn, a special case of
this: when W = C is the trivial representation, that is, gw = w,∀w ∈ C,
this makes V∗ into a G-module, with (gϕ)(v) = ϕ(g−1v).
Definition 7. A representation V is called irreducible if it has no proper
non-zero invariant subspaces.
Example 8. We consider the symmetric group S . We have two one-
d
dimensional representations: the trivial representation and the alternating
representation, which we denote by U and U0 respectively. Since S is a
d
permutation group, we have a natural permutation representation, in which
G acts on Cd by permuting the coordinates. Explicitly, if {e ,··· ,e } is
1 d
the standard basis, then g · e = e , or, equivalently, g · (z ,··· ,z ) =
i g(i) 1 d
(cid:16) (cid:17)
z ,··· ,z .
g−1(1) g−1(d)
This representation, like any permutation representation, is not irre-
ducible: the line spanned by the sum Pd e of the basis vectors is invariant,
i=1 i
with complement subspace V = {(z ,··· ,z ) : z +···+z = 0}.
1 d 1 d
This (d−1)-dimensional representation V is easily seen to be irreducible.
Moreover, we have Cd ∼= U ⊕V.
Definition9. TherepresentationV definedinthepreviousexampleiscalled
the standard representation of S .
d
5
Definition 10. A G-module homomorphism ϕ between two representations
V and W of G is a vector space map ϕ : V → W such that ϕ◦g = g◦ϕ for
every g ∈ G.
ϕ
V W
g g
V W
ϕ
We will also call ϕ a G-linear map, particularly when we want to dis-
tinguish it from an arbitrary linear map between the vector spaces V and
W. As expected, if ϕ is bijective, we say that V and W are isomorphic
representations.
Proposition11. Boththekernelandtheimageofϕaresub-representations
of V and W, respectively.
We have seen that the representations of G can be built up out of other
representations by taking the direct sum. We should focus, then, on rep-
resentations that are ”atomic” with respect to this operation, that is, that
cannot be expressed as a direct sum of others. Happily, a representation is
atomic in this sense if and only if it is irreducible, and every representation
is the direct sum of irreducible ones, in a suitable sense uniquely so.
Proposition 12. If W is a sub-representation of a representation V of a
finite group G, then there is a complementary invariant subspace W0 of V,
so that V = W ⊕W0.
We present two proofs.
Proof. We introduce an inner product H on V which is preserved by each
g ∈ G the following way: if H is any inner product on V, one gets H by
0
averaging over G:
X
H(v,w) = H (gv,gw)
0
g∈G
We choose W0 to be the orthogonal complement of W with respect to
the inner product H.
Proof. Choose an arbitrary space U complementary to W. Let π : V → W
0
be the projection given by the direct sum decomposition V = W ⊕U, and
average the map over G:
π(v) = Xg(π (g−1v))
0
g∈G
This map is a G-linear map from V onto W, which is multiplication by
|G| on W, therefore its kernel is a subspace of V invariant under G and
complementary to W.
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Corollary 13. Any representation is a direct sum of irreducible represen-
tations.
The extent to which the decomposition of an arbitrary representation
into a direct sum of irreducible ones is unique is one of the consequences of
the following:
Lemma 14 (Schur). If V and W are irreducible representations of G and
ϕ : V → W is a G-module homomorphism, then:
• Either ϕ is an isomorphism or ϕ = 0.
• If V = W, then ϕ = λI for some λ ∈ C, where I is the identity.
We can summarize the previous results in:
Proposition 15. For any representation V on G, there is a decomposition
V = V⊕a1⊕···⊕V⊕ak, where the V are distinct irreducible representations.
1 k i
The decomposition of V into a direct sum of the k factors is unique, as are
the V that occur and their multiplicities a .
i i
In both [1] and [2], it is shown that every irreducible representation of
an abelian group has degree one. Moreover, one can also find in [2] as inter-
esting method to determine the irreducible representations of the dihedral
group. However, there is a remarkably effective tool for understanding the
representations of a finite group G, called character theory.
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3 Character theory
Definition 16. If V is a representation of G, its character χ is the
V
complex-valued function on the group defined by χ (g) = Tr(g ), the trace
V |V
of g on V.
In particular, we have χ (hgh−1) = χ (g), so that χ is constant on
V V V
the conjugacy classes of G, and such a function is called a class function.
Note also that χ (1) = dimV and χ (cid:0)g−1(cid:1) = χ (g) for every g ∈ G.
V V V
Proposition 17. Let V and W be representations of G. Then χ =
V⊕W
χ +χ , χ = χ ·χ , χ = χ and χ (g) = χV(g)2−χV(g2).
V W V⊗W V W V∗ V Λ2V 2
Proposition 18 (Theoriginalfixed-pointformula). If V is the permutation
representation associated to the action of a group G on a finite set X, then,
for every g ∈ G, χ (g) is the number of elements of X fixed by g.
V
As we said before, the character of a representation of a group G is a
function on the set of conjugacy classes in G. This suggests expressing the
basic information about the irreducible representations of a group G in the
form of a character table. This is a table with the conjugacy classes [g] of G
listed across the top, usually given by a representative g, with the number
of elements in each conjugacy class over it, the irreducible representations V
of G listed on the left and, in the appropriate box, the value of the character
χ on the conjugacy class [g].
V
Example19. ThesymmetricgroupS hasthreeirreduciblerepresentations:
3
the trivial representation, the alternating representation and the standard
representation. There are no more irreducible representations since S has
3
three conjugacy classes (we will see later that the number of irreducible rep-
resentations of a finite group is always equal to the number of conjugacy
classes). As one can see in [1, Section 1.3], that can also be proved using
only what we learned in section 2, and the character table of S is:
3
1 3 2
S 1 (12) (123)
3
trivial U 1 1 1
alternating U0 1 −1 1
standard V 2 0 −1
Now we start by giving an explicit formula for the projection of a repre-
sentation onto the direct sum of the trivial factors in it. This formula has
tremendous consequences.
Definition 20. For any representation V of a group G, we set
VG := {v ∈ V : gv = v,∀g ∈ G}.
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We ask for a way of finding VG explicitly. We observed before that for
any representation V of G and any g ∈ G, the endomorphism g : V → V is,
in general, not a G-module homomorphism. On the other hand, if we take
the average of all these endomorphisms, that is, we set
1 X
ϕ = g ∈ End(V),
|G|
g∈G
then the endomorphism ϕ will be G-linear. In fact:
Proposition 21. The map ϕ is a projection of V onto VG.
We thus have a way of finding explicitly the direct sum of the trivial
sub-representations of a given representation, although the formula can be
hard to use if it does not simplify. If we just want to know the number m
of copies of the trivial representation appearing in the decomposition of V,
we can do this numerically, since this number will just be the trace of the
projection ϕ. We have
m = dimVG = Tr(ϕ) = 1 XTr(g) = 1 Xχ (g). (2)
V
|G| |G|
g∈G g∈G
In particular, we observe that for an irreducible representation other than
the trivial one, the sum over all g ∈ G of the values of the character is zero.
We can do much more with this idea. If V and W are representations of
G, then Hom(V,W) is a representation of G and Hom(V,W)G is the set of
G-module homomorphisms between V and W. If V is irreducible then, by
Schur’s lemma, dimHom(V,W)G is the multiplicity of V in W. Similarly, if
W is irreducible, dimHom(V,W)G is the multiplicity of W in V, and in the
case where both V and W are irreducible, we have
( ∼
1, if V = W
dimHom(V,W)G =
∼
0, if V 6= W
But now the character χ of the representation Hom(V,W) =
Hom(V,W)
V∗⊗W is given by χ (g) = χ (g)·χ (g).
Hom(V,W) V W
We can now apply formula (2) in this case to obtain the striking
1 X (1, if V ∼= W
χ (g)·χ (g) = (3)
|G| V W 0, if V 6∼= W
g∈G
To express this, let C (G) be the set of class functions on G an define
class
an Hermitian inner product on C (G) by
class
1 X
hα,βi = α(g)·β(g) (4)
|G|
g∈G
Formula (3) then amounts to:
9
Theorem 22. The characters of the irreducible representations are orthog-
onal with respect to the inner product (4).
Example 23. The orthonormality of the three irreducible representations
of S can be read from its character table. The numbers over each conjugacy
3
class indicate how many times to count entries in that column.
Corollary 24. The number of irreducible representations of G is less than
or equal to the number of conjugacy classes.
We will soon show that there are no nonzero class functions orthogonal
to the characters, so equality holds in the previous corollary.
Corollary 25. Any representation is determined by its character.
Corollary 26. A representation V is irreducible if and only if hχ ,χ i = 1.
V V
Corollary 27. The multiplicity a of V in V is the inner product of χ
i i V
with χ , that is, a = hχ ,χ i = 1.
Vi i V Vi
We obtain some further corollaries by applying all of this to the regular
representation R of G. By proposition 18 we know the character of R, it is
simply
(
0 if g 6= e
χ (g) = .
R |G| if g = e
Corollary 28. Any irreducible representation V of G appears in the regular
representation dimV times.
In particular, this proves again that there are only finitely many irre-
ducible representations. As a numerical consequence of this we have the
formula
|G| = dimR = X(dimV )2. (5)
i
i
Also, applying this to the value of the character of the regular represen-
tation on an element g ∈ G other than the identity, we have
X
(dimV )χ (g) = 0.
i Vi
i
These two formulas amount to the Fourier inversion for finite groups.
For example, if all but one of the characters is known, they give a formula
for the unknown character.
Now we give a more general formula for the projection of a general
representationV ontothedirectsumofthefactorsinV isomorphictoagiven
irreducible representation W. The main idea for this is a generalization of
the ”averaging” of endomorphisms g : V → V used in the beginning of this
section, thepointbeingthatinsteadofsimplyaveragingalltheg wecanask
the question: what linear combinations of the endomorphisms g : V → V
are G-linear endomorphisms? The answer is given by:
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