REPRESENTATIONS OF Sn AND GL(n, C)
SEAN MCAFEE
1. outline
For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G. Although numerically
equal, there is no general formula to map the conjugacy classes of a group to its irreducible representations. For the case of the symmetric group Sn , however, there is a
remarkably simple correspondence; we will see that each irreducible representation
of Sn is determined uniquely by a combinatorial object called a Young diagram
corresponding to a given conjugacy class.
In section 2, we will introduce these diagrams and show how they determine a
representation Vλ of Sn . In section 3, we will show that each such representation
is irreducible and, in fact, the collection of Vλ ’s accounts for every irreducible representation of Sn . Finally, in section 4, we will discuss how these representations
can be used to give a large class of irreducible representations of the general linear
group GLk .
For more details, please refer to Chapters 4, 6, and 15 of Representation Theory:
A First Course by William Fulton and Joe Harris.
2. Group algebras, Young diagrams, Young tableaux, and Young
symmetrizers
Our goal is to describe the irreducible representations of Sn . We first define the
group algebra CSn to be the set of all finite formal sums of the form
X
zσ eσ , zσ ∈ C,
σ∈Sn
where the eσ are basis elements indexed by the σ ∈ Sn . The group algebra is
equipped with a natural addition and multiplication which work as follows:
P
P
P
i) Pzi eσi + Pyi eσi =
(zi + yi )eσi
P
ii) ( zi eσi )
yj eσj = zi yj eσi σj .
i,j
For example, in the case of CS3 we have
−3e(12) + 7e(132) + 5e(12) + 2e(132) = 2e(12) + 9e(132) ,
and
(2e1 − 3e(12) )(e(123) + 5e(12) ) = 2e1(123) + 10e1(12) − 3e(12)(123) − 15e(12)(12)
= 2e(123) + 10e(12) − 3e(23) − 15e1 .
We can also view CSn as a complex vector space with basis elements indexed
by the eσ ’s. There is a natural linear action of CSn (the group algebra) on CSn
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SEAN MCAFEE
(the vector space) given by the multiplication rule above. In other words, we have
a representation
ρ : Sn → GL(CSn ) ∼
= GL(Cn! ),
called the regular representation of Sn .
The rest of this section will be devoted to describing certain elements cλ of CSn
(the group algebra), called Young symmetrizers. The images of these elements
in CSn (the vector space) will be subspaces Vλ of CSn that are invariant under the
action of Sn . Thus we will arrive at representations
ρλ : Sn → GL(Vλ )
which, we will see in section 3, are irreducible.
Recall that a partition λ = λ1 + · · · + λk , (λ1 ≥ · · · ≥ λk ) of an integer n gives
rise to a collection of stacked squares called a Young diagram; where the length
of each row is given by λi . For example, the partition 4 + 2 + 2 + 1 of n = 9 induces
the Young diagram
.
By filling in the boxes of a Young diagram in any way with one each of the
numbers 1, ..., n, we get what is known as a Young tableau. For our purposes,
the arrangement of numbers will not matter (this is not obvious and takes some
work to prove), so we will fill in our Young diagrams in the natural way, starting
with 1 in the upper left and increasing by 1 as we move from left to right and top
to bottom:
1 2 3 4
5 6
7 8
9
.
Given such a tableau, we define
Pλ = {σ ∈ Sn | σ preserves each row }
Qλ = {σ ∈ Sn | σ preserves each column } .
We then use these sets to define elements aλ , bλ in the group algebra CSn :
X
aλ :=
eσ
σ∈Pλ
bλ :=
X
sgn(σ)eσ .
σ∈Qλ
Finally, we define the Young symmetrizer:
cλ := aλ · bλ .
For an easy example, consider the partition 3 = 2 + 1; this gives us the following
Young tableau:
1 2
3 .
3
We then have
Pλ = {1, (12)} and Qλ = {1, (13)} ,
so
aλ = e1 + e(12) and bλ = e1 − e(13) ,
and
cλ = (e1 + e(12) )(e1 − e(13) ) = e1 + e(12) − e(13) − e(132) .
As discussed above, cλ acts on the vector space CS3 ; we take this action to be on
the right. The matrix of this action (using the ordered basis {1, (12), (13), (23), (123), (132)})
is


1
1 −1 0
1
0
1
1
0 −1 0 −1


−1 −1 1
0
1
0
.

0
0 −1 1 −1 1 


0
0
1 −1 1 −1
−1 −1 0
1
0
1
The first and third columns form a basis for the column space of this matrix, thus
the image Vλ = CSn cλ of cλ is the span of
e1 + e(12) − e(13) − e(132) and − e1 + e(13) − e(23) + e(123) .
Assuming for the moment that we have proven each such Vλ is irreducible, this
gives us a two dimensional irreducible representation of S3 , i.e. Vλ is equivalent to
the standard representation.
As a final remark, notice that any element d ∈ CSn gives us a subspace CSn d of
CSn which is invariant under the (left) action of CSn . Thus the Young symmetrizers
cλ are not special in this respect; what does make them special is that the CSn invariant subspaces which they determine will turn out to be irreducible under the
CSn -action (and, actually, unique for each partition λ).
Exercise 2.1. Explicitly show that the representation above is equivalent to the
standard representation of S3 , where S3 acts on the subspace of R3 spanned by the
vectors e1 − e2 , e2 − e3 by permuting indices.
Exercise 2.2. Show that the Young diagrams
1 2 3
and
1
2
3
give rise to the trivial and alternating representations of S3 , respectively.
Exercise 2.3. Show that, for Sn , the partitions (n) and (1 + 1 + · · · + 1) give rise
to the trivial and alternating representations of Sn , respectively.
Exercise 2.4. Show that, for arbitrary n, the partition ((n − 1) + 1) gives rise to
the standard representation of Sn .
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SEAN MCAFEE
3. proof that CSn cλ is irreducible
We will prove the following:
Theorem 3.1. Given Sn , let λ be a partition of n. Let Vλ be the subspace of CSn
spanned by the Young symmetrizer cλ (so Vλ = CSn cλ ). Then
i) Vλ is an irreducible CSn -module (i.e. Vλ is an irreducible representation of
Sn ).
ii) If λ, µ are distinct partitions of n, then Vλ ∼
6 Vµ .
=
iii) The Vλ account for all irreducible representations of Sn .
The objects involved are combinatorial in nature, so the proof relies on a couple
unavoidably ugly lemmata. The proofs of these will be omitted, but can be found
in chapter 4 of Fulton and Harris.
Lemma 3.2. For all x ∈ CSn , cλ xcλ is a scalar multiple of cλ .
Lemma 3.3. If λ 6= µ, then cλ CSn cµ = 0.
Assuming the above have been shown, we are ready to prove the theorem:
Proof of theorem 3.1:
i) Let Vλ = CSn cλ for a given Young symmetrizer cλ . By Lemma 3.2, we
have that
cλ Vλ ⊆ Ccλ .
Let W be a nonzero subrepresentation of Vλ . We will show that W = Vλ .
First, we claim that both cλ Vλ , cλ W are nonzero. To see this, suppose that
cλ Vλ = 0. Then
Vλ Vλ = CSn (cλ Vλ ) = 0.
As vector spaces, there exists a projection π : CSn → CSn cλ that commutes
with the action of Sn (this can be obtained by taking the projection induced
by cλ and averaging over the action of Sn ). This projection can be described
as right multiplication on CSn (the group algebra) by an element x ∈ CSn
by letting x := π(1). This x must lie in Vλ , since x = 1·x. By the definition
of a projection,
x = x2 ∈ Vλ Vλ = 0,
thus x = 0, a contradiction since the nonzero cλ itself lies in CSn cλ . Thus
we must have cλ Vλ 6= 0. Similarly, we show cλ W 6= 0.
So we have W is a subspace of Vλ , cλ Vλ ⊆ Ccλ , and cλ W 6= 0; thus we
must have cλ W = Ccλ . Therefore,
Vλ = CSn cλ = CSn (Ccλ ) = CSn (cλ W ) ⊆ W,
where the inclusion on the right follows from the fact that W is a subrepresentation of Vλ , i.e. it is invariant under the action of CSn . Thus, we have
Vλ = W , completing the proof that Vλ is irreducible.
ii) Let λ, µ be distinct partitions of n, and let Vλ , Vµ their corresponding representations. By the above, we have that cλ Vλ = Ccλ 6= 0. By Lemma 3.3,
we have that
cλ Vµ = cλ CSn cµ = 0.
Therefore, Vλ and Vµ cannot be isomorphic as CSn modules, proving ii).
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iii) Each partition λ of n corresponds to a distinct conjugacy class of Sn . We
have shown in ii) that the Vλ determined by such partitions are all inequivalent. Thus, since the number of conjugacy classes of a finite group is equal
to the number of irreducible representations, we have accounted for all of
the irreducible representations of Sn .
4. irreducible representations of GLn (C)
Equipped with our construction of the irreducible representations of Sn , it is
actually fairly simple to describe all irreducible, finite-dimensional, rational representations of the general linear group GLn (C). By rational, we mean the following:
given a finite dimensional representation ρ : GL(n, C) → GL(m, C), we say that
the representation is polynomial if, for each A ∈ GL(n, C), we have that ρ(A) has
entries which are polynomial in the entries of A. We say that the representation
is rational if, for each A ∈ GL(n, C), we have that ρ(A) has entries which are
rational polynomials in the entries of A.
Example 4.1. For any GL(n, C), we have the standard representation, given
by
ρ : GL(n, C) → GL(n, C)
A 7→ A.
This is clearly a polynomial (and thus rational) representation.
Exercise 4.2. Show that the standard representation of GL(n, C) is irreducible.
Exercise 4.3. Show that the representation ρ(g) = (g −1 )T is a representation of
GL(n, C) on C2 which is rational, but not polynomial.
In studying the irreducible representations of Sn , we started by observing that
any σ ∈ Sn acts (by right multiplication) on the group algebra CSn by permuting
the basis elements. We then used linearity to extend this action to a CSn action on
CSn . The image of this action under the element cλ ended up being an irreducible
representation Vλ of Sn .
The process for studying the irreducible representations of GL(n, C) will be
similar. We will start by giving a sketch of the construction, then we will proceed
to fill in some of the details. Consider GL(n, C) acting on the vector space V = Cn
by matrix multiplication. For any positive integer d, this multiplication can be
extended to the dth tensor power of V , written V ⊗d . For such a fixed d, we can
look at a partition λ = (d1 , ..., dk ) and (just like in the Sn case) construct the Young
symmetrizer cλ . This symmetrizer acts on the right of V ⊗d in the same way that
it did on the group algebra CSn : by permuting elements. The image V ⊗d cλ will
turn out to be a GL(n, C)-invariant subspace of V ⊗d . We write Sλ V to denote this
subspace, and call the map V
Sλ V the Schur functor. These Sλ V ’s will turn
out to be precisely the irreducible polynomial representations of GL(n, C).
Let’s begin to unpack the above paragraph by looking at the action of GL(n, C)
on V ⊗d . For a basis vector v1 ⊗ v2 ⊗ · · · ⊗ vd of V ⊗d , and for g ∈ GL(n, C), we
define
g · (v1 ⊗ v2 ⊗ · · · ⊗ vd ) := gv1 ⊗ gv2 ⊗ · · · ⊗ gvd .
By linearity, we can extend this to an action on any linear combination of such
basis vectors.
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SEAN MCAFEE
Exercise 4.4. Check that the above defines a representation of GL(n, C) on V ⊗d .
Now, let’s analyze the right action of Sd on V ⊗d . For σ ∈ Sd and v1 ⊗v2 ⊗· · ·⊗vd
a basis element of V ⊗d , define
(v1 ⊗ v2 ⊗ · · · ⊗ vd ) · σ := vσ(1) ⊗ vσ(2) ⊗ · · · ⊗ vσ(d) .
Again, by linearity we can extend this to a right action of the group algebra CSd on
all of V ⊗d . This action clearly commutes with the left action of GL(n, C) defined
above.
We define two important subspaces of V ⊗d , called the symmetric tensors
Vd
Symd V and the alternating
tensors Altd V (this is often denoted
V , and its
P
elements are written as
v1 ∧ v2 ∧ · · · ∧ vd ):
(
)
X
d
Sym V :=
vσ(1) ⊗ vσ(2) ⊗ · · · ⊗ vσ(d)
σ∈Sd
(
d
Alt V :=
)
X
sgn(σ)vσ(1) ⊗ vσ(2) ⊗ · · · ⊗ vσ(d)
.
σ∈Sd
For example, if d = 2, then (for arbitrary vectors v1 , v2 ∈ V )
v1 ⊗ v2 + v2 ⊗ v1 ∈ Sym2 V,
and
v1 ⊗ v2 − v2 ⊗ v1 ∈ Alt2 V.
Intuitively, we have that Symd V is the set of vectors in V ⊗d that are not changed
by permuting their components, and Altd V is the set of vectors in V ⊗d that change
sign whenever two components are switched (this is due to Sd being generated by
transpositions, which have negative signature).
Exercise 4.5. Verify that Symd V and Altd V are invariant subspaces of V ⊗d under
the action of CSd .
Under the (right) action of CSd , we can redefine Symd V and Altd V as
!
X
d
⊗d
Sym V := V
·
eσ
σ∈Sd
!
d
Alt V := V
⊗d
·
X
sgn(σ)eσ
.
σ∈Sd
Exercise 4.6. Check that the above definitions coincide with our original definitions of Symd V and Altd V .
Exercise 4.7. Check that Symd V and Altd V are invariant under the (left) action
of GL(n, C).
Exercise 4.7 gives two examples of a more general (and extremely useful) phenomenon: for any element c of the group algebra CSd , we have that V ⊗d c is invariant under the action of GL(n, C). Indeed, since the actions of GL(n, C) and CSd
commute, then, for any g ∈ GL(n, C), v ∈ V ⊗d ,
g(vc) = (gv)c ∈ V ⊗d c.
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This gives us an entire family of invariant (and potentially irreducible) subspaces of
V ⊗d obtained by simply choosing an element at random from CSd . As mentioned
before, the only elements we need will be the Young symmetrizers cλ .
There is another thing to point out about Symd V and Altd V : they are the
images V ⊗d cλ = Sλ V of the Young symmetrizers corresponding to the partitions
(d) and (1, 1, ..., 1), respectively. As subspaces of V ⊗d , they have zero intersection,
i.e. they are direct summands in the decomposition of V ⊗d :
V ⊗d = Symd V ⊕ Altd V ⊕ (Symd V ⊕ Altd V )⊥ .
Notice the relation between this and the irreducible spaces V(d) and V(1,1,...,1) in CSd
in our representation of Sd . There is a very simple correspondence between the decomposition of CSd into irreducible representations Vλ of Sd and the decomposition
of V ⊗d into irreducible representations Sλ V of GL(n, C).
The above discussion and more is summarized in the following theorem, which
we will not prove here. For details, refer to pages 84-87 of Fulton and Harris.
Theorem 4.8.
i) Let k = dim V . Then Sλ V = 0 if λk+1 6= 0; that is, if the Young diagram
corresponding to λ has more than k rows.
ii) Let mλ be the dimension of the irreducible representation Vλ of Sd corresponding to a partition λ of d. Then
M
(Sλ V )⊕mλ .
V ⊗d ∼
=
λ
iii) Each Sλ V is an irreducible polynomial representation of GL(n, C).
iv) All rational representations of GL(n, C) can be described by a tensor product
Sλ V ⊗ (det)−k ,
where z ∈ Z>0 and det is the (one dimensional) determinant representation
of GL(n, C).
Example 4.9. Consider GL(2, C) with its natural representation on V = C2 . Let
d = 3, and consider the partition λ = (1, 1, 1). Then we have
c(1,1,1) = e1 − e(12) − e(13) − e(23) + e(123) + e(132) .
Then, for v1 ⊗ v2 ⊗ v3 ∈ V ⊗3 ,
(v1 ⊗ v2 ⊗ v3 ) · c(1,1,1)
= v1 ⊗ v2 ⊗ v3 − v2 ⊗ v1 ⊗ v3 − v3 ⊗ v2 ⊗ v1 − v1 ⊗ v3 ⊗ v2 + v3 ⊗ v1 ⊗ v2 + v2 ⊗ v3 ⊗ v1 .
Since V is two dimensional, we may write v3 = av1 + bv2 ; the above then becomes
v1 ⊗ v2 ⊗ av1 + v1 ⊗ v2 ⊗ bv2 − v2 ⊗ v1 ⊗ av1
− v2 ⊗ v1 ⊗ bv2 − av1 ⊗ v2 ⊗ v1 − bv2 ⊗ v2 ⊗ v1
− v1 ⊗ av1 ⊗ v2 − v1 ⊗ bv2 ⊗ v2 + av1 ⊗ v1 ⊗ v2
+ bv2 ⊗ v1 ⊗ v2 + v2 ⊗ av1 ⊗ v1 + v2 ⊗ bv2 ⊗ v1 .
If we use the properties of tensors to move the constants a, b to the front of each
monomial, we see that the above sum collapses to 0, illustrating part i) of the
theorem.
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SEAN MCAFEE
Example 4.10. We use the setup of the previous example, this time with partition
λ = (2, 1). Then we have
c(2,1) = 1 + e(12) − e(13) − e(132) .
Thus, for v1 ⊗ v2 ⊗ v3 ∈ V ⊗3 ,
(v1 ⊗ v2 ⊗ v3 ) · c(2,1) = v1 ⊗ v2 ⊗ v3 + v2 ⊗ v1 ⊗ v3 − v3 ⊗ v2 ⊗ v1 − v3 ⊗ v1 ⊗ v2 .
We have an embedding of Alt2 V ⊗ V into V ⊗ V ⊗ V given by
(v1 ∧ v3 ) ⊗ v2 7→ v1 ⊗ v2 ⊗ v3 − v3 ⊗ v2 ⊗ v1 .
This realizes the image of c(2,1) as the subspace of Alt2 V ⊗ V spanned by vectors
of the form
(v1 ∧ v3 ) ⊗ v2 + (v2 ∧ v3 ) ⊗ v1 .
These vectors (check this!) are the kernel of the canonical map
φ : Alt2 ⊗V → Alt3 V, (v1 ∧ v2 ) ⊗ v3 7→ v1 ∧ v2 ∧ v3 ,
thus
S(2,1) V = Ker φ.
By part ii) of the theorem, and by our discussion of S3 in section 2, we have the
decomposition
V ⊗V ⊗V ∼
= Sym3 V ⊕ Alt3 V ⊕ Ker φ ⊕ Ker φ,
where the two copies of Ker φ are due to the fact that CS3 c(2,1) is a two-dimensional
(irreducible) representation of S3 , i.e. m(2,1) = 2.
To summarize this section: we can parametrize the irreducible rational representations of GL(n, C) by a triple (d, λd , z), where d is any positive integer, λd is a
partition of d whose Young diagram has no more than n rows, and k is any positive
integer. The resulting representation is the tensor product
Sλd V ⊗ (det)−k .
There is much, much more to say about representations of GL(n, C) than is contained in this paper. For instance, the use of Schur polynomials to describe the
characters of Sλ V , or the decomposition of the tensor product Sλ V ⊗ Sµ V , or the
connection between the representations of GL(n, C) and those of SL(n, C) and
U (n). We encourage the reader to investigate these on their own.