SCHUR-WEYL DUALITY

JAMES STEVENS

Contents

Introduction 11. Representation Theory of Finite Groups 21.1. Preliminaries 21.2. Group Algebra 41.3. Character Theory 52. Irreducible Representations of the Symmetric Group 82.1. Specht Modules 82.2. Dimension Formulas 112.3. The RSK-Correspondence 123. Schur-Weyl Duality 133.1. Representations of Lie Groups and Lie Algebras 133.2. Schur-Weyl Duality for GL(V ) 153.3. Schur Functors and Algebraic Representations 163.4. Other Cases of Schur-Weyl Duality 17Appendix A. Semisimple Algebras and Double Centralizer Theorem 19Acknowledgments 20References 21

Introduction. In this paper, we build up to one of the remarkable results inrepresentation theory called Schur-Weyl Duality. It connects the irreducible rep-resentations of the symmetric group to irreducible algebraic representations of thegeneral linear group of a complex vector space. We do so in three sections:

(1) In Section 1, we develop some of the general theory of representations offinite groups. In particular, we have a subsection on character theory. Wewill see that the simple notion of a character has tremendous consequencesthat would be very difficult to show otherwise. Also, we introduce the groupalgebra which will be vital in Section 2.

(2) In Section 2, we narrow our focus down to irreducible representations ofthe symmetric group. We will show that the irreducible representations ofSn up to isomorphism are in bijection with partitions of n via a construc-tion through certain elements of the group algebra. Finally, we mentionthe beautiful Robinson-Schensted correspondence. The correspondence up-grades a formula involving the dimension of the irreducible representationsof Sn to a bijection which has many combinatorial applications.

(3) In Section 3, we prove the classical case of Schur-Weyl Duality by using theDouble Centralizer Theorem. We describe exactly which representations

1

2 JAMES STEVENS

of the general linear group GL(V ) come from this relationship. And weclose the section by showcasing other forms of Schur-Weyl Duality for othermatrix groups and Lie algebras.

1. Representation Theory of Finite Groups

1.1. Preliminaries. A representation of a finite group G on a finite-dimensionalcomplex vector space V is a group homomorphism ρ : G → GL(V ). We will oftenabuse notation by referring to V as the representation of G and write gv for ρ(g)(v)whenever the map ρ is understood from context. We say that a subspace W of arepresentation V is a subrepresentation if W is invariant under the the action ofG, that is, gw ∈ W for all g ∈ G and w ∈ W . Notice that a representation Valways contains at least two subrepresentations, namely, 0 and V itself. We say Vis irreducible if it contains no proper nonzero invariant subspaces, i.e., 0 and V arethe only subrepresentations of V . However, note that we do not consider V = 0 tobe an irreducible representation. We will see that this is analogous to the reason 1is not a prime number.

Examples 1.1. Let G be a finite group. Here are some examples of representations.

(1) The trivial representation C where gv = v for all g ∈ G and v ∈ U . Noticethat the trivial representation is always (trivially) irreducible.

(2) Let G act on a finite set X. The permutation representation is the vectorspace with basis X where ρ(g)(x) = gx for all g ∈ G and x ∈ X. When Gacts on itself by multiplication, we get the regular representation.

We can always build up new vector spaces from old using operations such as thetensor product. Similarly, given representations V and W of G, we can form therepresentations V ⊕W , V ⊗W , Symn V , Altn V , V ∗, and Hom(V,W ) where eachis given their ordinary underlying vector space structure. For the first four, just letG act factorwise, e.g., g(v⊗w) = gv⊗gw in V ⊗W . For the dual V ∗ = Hom(V,C)of V , we require ρ∗(g) be the transpose of ρ(g−1). This forces ρ to respect thenatural pairing of V ∗ and V in the following sense: 〈ρ∗(g)(λ), ρ(g)(v)〉 = 〈λ, v〉 forall g ∈ G, λ ∈ V ∗, and v ∈ V . Having this, the action on Hom(V,W ) is given bythe identification Hom(V,W ) = V ∗ ⊗W .

One of the main goals in representation theory is to classify all representationsof a given group G. For a general (possibly infinite) group, this is hard. And if wework over a field other than C, this can complicate matters even more. However,we will see that this classification is possible for representations over C of a finitegroup. But instead of directly classifying all representations, we can narrow ourfocus down to indecomposable representations, those which cannot be written as adirect sum of others. So we then can build up all other representations from these.At this point, we have introduced two notions of atomicity: irreducible and inde-composable. Clearly, if a representation is irreducible, then it is indecomposable.Luckily, in the case of representations over C of finite groups, the converse holds.

Maschke’s Theorem 1.2. If W is a subrepresentation of V of a finite group G,then there exists a complementary subrepresentation W ′ of V so that V = W ⊕W ′.

SCHUR-WEYL DUALITY 3

Proof. Pick W ′′ to be any subspace so that V = W ⊕W ′′ and let π′ : V → W bethe projection along W ′′ onto W . Now define the projection π : V →W by

π(v) =1

|G|∑g∈G

gπ′(g−1v).

Notice that hπ(v) = π(hv) for all h ∈ G. In particular, W ′ := Kerπ is a comple-mentary G-invariant subspace because we have π(hv) = hπ(w) = 0 for all h ∈ Gand v ∈W ′. In which case, V = W ⊕W ′. �

Corollary 1.3. Every representation over C of a finite group can be decomposedas a direct sum of irreducible representations.

Examples 1.4. Corollary 1.3 holds in further generality. If we work over a differentscalar field or characteristic zero but still require our group to be finite, then theprevious proof of Maschke’s Theorem 1.2 still works. For scalar fields with posi-tive characteristic, we cannot have the characteristic of the scalar field divide theorder of the group (otherwise, the projection cannot be defined). To see why thisgeneralization of Corollary 1.3 fails if we do not assume this:

(1) Let V = F22 and G = Z/2Z where 1 ∈ Z/2Z sends (x, y) ∈ F2

2 to (y, x).Notice the span of (1, 1) ∈ F2

2 is invariant under the action Z/2Z. However,the remaining complementary subspaces—the span of (1, 0) and the spanof (0, 1)—are not invariant. So F2

2 is indecomposable but not irreduciblewith this action of Z/2Z.

To see why the generalization of Corollary 1.3 fails if the group is infinite:

(2) Let V = C2 and G = C where z ∈ C acts on (x, y) ∈ C2 by

z(x, y) :=

(1 z0 1

)(xy

)=

(x+ zyy

).

Then the only 1-dimensional invariant subspace is the span of (1, 0). Again,we have C2 is indecomposable but not irreducible under this action of C.

However, we can adjust the proof of Maschke’s Theorem 1.2 by swapping the sumwith integration with respect to a Haar measure to show a version of Corollary 1.3for compact groups, e.g. S1. More precisely, any finite-dimensional continuous rep-resentation of a compact group can be decomposed into irreducible representations.

So the previous corollary tells us that there always exists a decomposition of arepresentation into a sum of irreducible representations. So the natural question isnow: in what sense (if any) is this decomposition unique? But we have yet to evendescribe a way to distinguish representations.

A G-map f of representations V and W of G is a linear map f : V → W thatcommutes with the action of G, that is, gf(v) = f(gv) for all g ∈ G and v ∈ V .Notice that Ker f and Im f are always subrepresentations of V and W respectively.We say a G-map f : V → W is an isomorphism of representations if it is anisomorphism of vector spaces. In this case, we say that V and W are isomorphicand denoted this by V ' W . Let HomG(V,W ) denote the set of all G-maps fromV to W and define EndG(V ) = HomG(V, V ). The following elementary lemma isperhaps the most useful fact about G-maps.

Schur’s Lemma 1.5. Let f : V →W be a nonzero G-map.

(1) If V is irreducible, then f is injective.

4 JAMES STEVENS

(2) If W is irreducible, then f is surjective.(3) If V = W is irreducible, then f = tI for some nonzero t ∈ C.

Proof. The first two parts boil down to considering the subrepresentations Ker fand Im f . The last part comes from taking any f ∈ EndG(V ) and applying thefirst parts to show f − tI ∈ EndG(V ) is the zero map for any eigenvalue t of f . �

Corollary 1.6. For any representation V of a finite group, we can write

V ' V ⊕n11 ⊕ · · · ⊕ V ⊕nkk

where Vi are distinct irreducible representations. The number of factors k and theVi that occur (up to isomorphism) along with their multiplicities ni are all unique.

Proof. Schur’s Lemma 1.5 tells us that a map in HomG(V,W ) with the decompo-

sitions V =⊕V ⊕nii and W =

⊕W⊕mjj must take the factor V ⊕nii into a factor

W⊕mjj where Vi ' Wj . Applying this to the identity map on V with two different

decompositions will show the stated uniqueness. �

Example 1.7. Consider an abelian group G and an irreducible representation V ofG. For h ∈ G, let fh : V → V so that fh(v) = hv. This is a G-map becausegfh(v) = ghv = hgv = fh(gv) for all g ∈ G. By Schur’s Lemma 1.5, for each h ∈ G,there exists a th ∈ C so that fh = thI. Thus, the span of any nonzero vector vis a nonzero invariant subspace of V and, hence, is V . Therefore, all irreduciblerepresentations of an abelian group are 1-dimensional.

1.2. Group Algebra. We will now introduce a powerful lens though which to viewrepresentations. The group algebra C[G] of a group G is the associative C-algebrawith basis G and where multiplication is inherited from group multiplication, i.e.,∑

p∈Gapp

∑q∈G

bqq

=∑p,q∈G

apbqpq =∑g∈G

cgg

where cg is the sum of all apbq where g = pq.We can generalize the notation of a representation onto other algebraic struc-

tures. A representation of an associative C-algebra A on a complex vector space Vis an algebra homomorphism ρ : A→ End(V ). In particular, V is an A-module. Wewant to stress that throughout this paper, we will only be working with associativealgebras over the scalar field C.

Notice that given a representation of a finite group G, we can extend it linearlyto get a representation of the group algebra C[G]. And conversely, given a repre-sentation of C[G], we can restrict it to G to get back a representation of G. So itmay seem that introducing the group algebra has not given us anything new. Butit gives us a new language and tools which make constructions and definitions moretransparent. For instance, here is a short excerpt of the dictionary between a groupand its corresponding group algebra.

G-representation C[G]-modulesubrepresentation submodule

irreducible representation simple moduleG-map C[G]-homomorphism

We will not yet prove the following proposition on the structure of C[G]. It actu-ally follows from the main theorem that we will show in the in the next subsection.

SCHUR-WEYL DUALITY 5

Proposition 1.8. As algebras,

C[G] =⊕i

End(Vi)

where the sum is over all distinct irreducible representations Vi of G.

But Proposition 1.8 can be proved without the mentioned theorem, using morering theoretic tools as seen in the appendix. In fact, Proposition 1.8 can be usedto prove the mentioned main theorem.

Although the image of g ∈ G under a representation is a map in GL(V ) ⊆End(V ), it is not necessarily in EndG(V ). Clearly, the image of g in End(V ) is aG-map if and only if g is in the center Z(G). Similarly, it can be seen that the imageof f ∈ C[G] in End(V ) is a G-map if and only if f is in the center Z(C[G]). Wewill now show an alternate characterization of elements in Z(C[G]), which involvesfunctions α : G → C so that α(g) = α(hgh−1) for all g, h ∈ G. We call suchfunctions class functions, and they will appear again later on.

Proposition 1.9. Let V be a representation of G and α : G→ C a function. Let

f =∑g∈G

α(g)g ∈ C[G].

The following are equivalent:

(1) f belongs to Z(C[G]).(2) the image of f in End(V ) is in EndG(V ).(3) α is a class function.(4) α is a sum of indicator functions on conjugacy classes of G.

In particular, here is an important such function.

Lemma 1.10. Let V G = {v ∈ V : gv = v for all g ∈ G}. Then the image of

π =1

|G|∑g∈G

g ∈ Z(C[G])

in End(V ) is a G-map and a projection onto V G.

Notice that V G is the sum of all copies of the trivial representation found in V .

1.3. Character Theory. The character of a representation V of G, is the map

χV : G→ C where χV (g) = tr(g|V )

is the trace of g on V . Although this is a simple notion, we will soon see it isabsolutely vital to the study of representations of finite groups.

Since the trace of a linear transformation is the sum of eigenvalues, the identities

(1.11) χV⊕W = χV + χW , χV⊗W = χV χW , χV ∗ = χV

can easily be verified. In addition, by the properties of trace, we have that for anycharacter χ of a representation, χ(g) = χ(hgh−1) for all g, h ∈ G, that is, χ is aclass function on G. Let F(G) denote the set of all class functions G → C. Thenwe can endow F(G) with the inner product

(1.12) 〈α, β〉 =1

|G|∑g∈G

α(g)β(g)

for α, β ∈ F(G). We have the following fundamental theorem of character theory.

6 JAMES STEVENS

Theorem 1.13. The set of characters χV for irreducible V of G form an orthonor-mal basis for F(G) with respect to the inner product in (1.12).

Proof. Let V and W be irreducible representations of G. By the properties (1.11),

〈χV , χW 〉 =1

|G|∑g∈G

χV (g)χW (g) =1

|G|∑g∈G

χHom(V,W )(g) = tr(π|Hom(V,W ))

where π is the projection from Hom(V,W ) onto Hom(V,W )G = HomG(V,W ) inLemma 1.10. Therefore, by Schur’s Lemma 1.5, we have that

〈χV , χW 〉 = tr(π|Hom(V,W )) = dim HomG(V,W ) =

{1, if V 'W0, if V 6'W.

So characters of irreducibles are orthonormal and, thus, are linearly independent.Now we will now show that the linearly independent set of irreducible characters

is in fact maximal and, hence, forms a basis for F(G). Suppose α ∈ F(G) suchthat 〈α, χV 〉 = 0 for all irreducible V . It suffices to show α = 0. Notice that

0 = 〈α, χV 〉 =1

|G|∑g∈G

α(g)χV (g) =1

|G|∑g∈G

α(g) tr(g|V ) = tr(f |V )

where f = 1|G|∑g∈G α(g)g ∈ EndG(V ) by Proposition 1.9. Since f ∈ EndG(V ),

then Schur’s Lemma 1.5 tells us that f = tI for some t ∈ C. Therefore, we havethat t = tr(f |V )/ dimV = 0 and, thus, α = 0. �

Here are some important consequences of this which fall out almost immediately.

Corollary 1.14. We have that V 'W if and only if χV = χW .

Example 1.15. This tells us that all representations V of Sn are self-dual since

χV ∗(g) = χV (g) = χV (g−1) = χV (g)

for all g ∈ Sn because g and g−1 have the same cycle type and, thus, are in thesame conjugacy class. Hence V ' V ∗. This will be a useful fact later on. Note thatfor a group G in general, this does not hold.

Corollary 1.16. A representation V is irreducible if and only if 〈χV , χV 〉 = 1.

Corollary 1.17. If V and W are representations with V being irreducible, thenthe multiplicity of V in the decomposition of W is 〈χV , χW 〉.

Corollary 1.18. For the regular representation R of G, we have that

R =⊕

V ⊕ dimVii

where the sum is over all distinct irreducible representations Vi of G.

In particular, we obtain the beautiful formula

(1.19) |G| = dimR =∑

(dimVi)2.

Notice since this sum is finite, there must be finitely many irreducible represen-tations. But in fact, since the indicator functions of conjugacy classes also form abasis for F(G), we get the following corollary.

Corollary 1.20. The number of distinct irreducible representations of G is equalto the number of conjugacy classes of G.

SCHUR-WEYL DUALITY 7

Interestingly, in general, there is no natural bijection between irreducible repre-sentations and conjugacy classes of a finite group. But we will later see that thereis a somewhat natural bijection in the case of the symmetric group Sn.

The orthogonality in Theorem 1.13 is called row orthogonality for reasons we willsee when constructing character tables. Similarly, there is a corresponding columnorthogonality.

Corollary 1.21. For g, h ∈ G, we have

1

|G|∑i

χVi(g)χVi(h) =

{1/c(g), if g and h are conjugate0, otherwise

where the sum is over all distinct irreducible representations Vi of G and c(g) is thesize of the conjugacy class of g.

Proof. Since characters are class functions, we can define χV : Conj(G) → C byevaluating on a representative. Then the row orthogonality says

(1.22) 〈χVi , χVj 〉 =1

|G|∑k

|Ck|χVi(Ck)χVj (Ck) = δij

where the sum is over all conjugacy classes Ck ∈ Conj(G) and δij is the Kronecker

delta function. Consider the matrix X = (xij) where xij =√|Cj |/|G|χVi(Cj).

Then (1.22) says XXH = I where XH is the conjugate transpose of X. Therefore,we have that XH = X−1 and, thus, XHX = I as well. This says√

|Ci||Cj ||G|

∑k

χVk(Ci)χVk(Cj) = δij

where the sum is over all distinct irreducible representations Vi of G. �

Example 1.23. It would an absolute sin to have a section on character theory andnever mention a character table. So we will compute the character table for thealternating group A4.

In general, if we have a representation G/N → GL(V ) for some normal subgroupN , we can lift to a representation on G by the composition G→ G/N → GL(V ). Sonotice thatN = 〈(12)(34), (14)(23)〉 is a normal subgroup of A4 with A4/N ' Z/3Z.Since A4/N is abelian, then by Example 1.7, A4/N ' Z/3Z has three 1-dimensionalirreducible representations, call them U , U ′, and U ′′. We know that one is thetrivial representation. The other two come from naturally sending 1 ∈ Z/3Z to oneof the two primitive 3rd roots of unity. The lifts of these representations must beirreducible for G as well because they are 1-dimensional. So we have the partialcharacter table

1 4 4 3A4 1 (123) (132) (12)(34)U 1 1 1 1U ′ 1 ω ω2 1U ′′ 1 ω2 ω 1

where the top row indicates sizes, and representatives, for the conjugacy classes ofA4. Now, since there are 4 conjugacy classes of A4, then by Corollary 1.20, weonly need to find one more irreducible representation V to complete the table. By(1.19), we know that dimV = 3 because |A4| = 12 = 12 + 12 + 12 + 32. Using this,we can then compute the rest of the table by using column orthogonality (1.21).

8 JAMES STEVENS

1 4 4 3A4 1 (123) (132) (12)(34)U 1 1 1 1U ′ 1 ω ω2 1U ′′ 1 ω2 ω 1V 3 0 0 −1

The representation V comes from the restriction of a representation of S4 thatwe will later call the standard representation. Also, it turns out that V can berealized geometrically as the rotational symmetries of the tetrahedron.

2. Irreducible Representations of the Symmetric Group

2.1. Specht Modules. Recall that conjugacy classes of Sn are completely deter-mined by cycle type, which can be encoded as a partition. A partition λ of n is aninteger sequence λ = (λ1, . . . , λk) so that λ1 ≥ · · · ≥ λk ≥ 1 and n = λ1 + · · ·+ λk.Let p(n) denote the number of partitions of n. Since the number of conjugacyclasses of Sn is p(n), then by Corollary 1.20, we know that there are also p(n)many distinct irreducible representations of Sn. But, remarkably, we will see thatnot only does each partition λ uniquely give rise to a conjugacy class, but λ alsogives rise to an irreducible representation of Sn.

To a partition λ = (λ1, . . . , λk) of n, we associate a combinatorial object calledthe Young diagram of λ which is a collection of n cells arranged in left-justified rowswith λi cells on the ith row. Given the Young diagram of λ, we number the cells bythe integers 1, . . . , n to form a tableau. In this case, a picture is more illuminatingthan the definition. Here is the tableau for (4, 2, 2, 1) with the canonical labeling.

1 2 3 45 67 89

There is a natural action of the symmetric group on tableaux by permuting thelabels. For a tableau of λ of n, consider the following subgroups of Sn:

Pλ = {g ∈ Sn : g preserves each row of λ}and, similarly,

Qλ = {g ∈ Sn : g preserves each column of λ}.To each of these subgroups, we can associate an element in the group algebra C[Sn]:

aλ =∑g∈Pλ

g and bλ =∑g∈Qλ

sgn(g)g.

Finally, we call their product cλ = aλbλ ∈ C[Sn] the Young symmetrizer of λ.

Theorem 2.1. The image Vλ = C[Sn]cλ is an irreducible representation of Sn(under left multiplication). Furthermore, every irreducible representation of Sn isisomorphic to Vλ for some unique partition λ of n.

We call Vλ a Specht module. Notice that different labellings of the same partitionwill give different elements of C[Sn] for the Young symmetrizer. We will always workwith tableaux with the canonical labeling. However, in this case, these differentYoung symmetrizers will give rise to isomorphic Specht modules.

SCHUR-WEYL DUALITY 9

This Young symmetrizer construction may at first seem to come out of nowhere.But recall that a left module is simple if and only if it is a minimal left ideal, andsuch ideals are generated from primitive idempotents. So a simple left C[Sn]-module(a irreducible representation of Sn) is of the form C[Sn]e for a primitive idempotente. We will later show that these Young symmetrizers are primitive idempotents upto scaling. In particular, recall that from Proposition 1.8, as an algebra

C[Sn] '⊕λ

End(Vλ)

where the sum is over all irreducible representations Vλ of Sn. Fix a partition µof n and pick a basis for Vµ. Notice that the matrix Eii in End(Vµ) of all zerosexcept for a single one in the (i, i)th position i is a primitive idempotent. Moreover,the image c of Eii under the map End(Vµ) ↪→

⊕λ End(Vλ) → C[Sn] then has the

property C[Sn]c ' End(Vµ)Eii ' Vµ. The choice of i is analogous to the choice oflabeling of our tableau.

Example 2.2. We will compute the Young symmetrizers for all partitions of 3 and,thus, all the distinct irreducible representations of S3. Here are all the canonicaltableau.

1 2 31 23

123

For λ = (3), we have that P(3) = S3 and Q(3) = {1}. Hence

c(3) = a(3)b(3) =∑g∈S3

g.

For any h ∈ S3, we have hc(3) = c(3). Therefore, V(3) = C[S3]c(3) is the trivialrepresentation.

Now for λ = (2, 1), we have that P(2,1) = {1, (12)} and Q(2,1) = {1, (13)}. Hence

c(2,1) = a(2,1)b(2,1) = (1 + (12))(1− (13)) = 1 + (12)− (13)− (132).

It turns out that V(2,1) = C[S3]c(2,1) is the span of c(2,1) and (13)c(2,1).For λ = (1, 1, 1), we have that P(1,1,1) = {1} and Q(1,1,1) = S3. Hence

c(3) = a(3)b(3) =∑g∈S3

sgn(g)g.

For any h ∈ S3, we have hc(1,1,1) = sgn(h)c(1,1,1). Therefore, V(1,1,1) = C[S3]c(1,1,1)is the alternating representation.

The general case for Sn is that V(n) is the trivial representation, V(1,...,1) is thealternating representation, and V(n−1,1) is the standard representation. Consider thepermutation representation of the natural action of Sn on {1, 2, . . . , n}. The imageof the element of Sn are the familiar permutation matrices. This representation isnot irreducible because it contains of the proper Sn-invariant subspace U spannedby 1 + 2 + · · ·+ n. But the complementary invariant subspace of U is irreducible.

We will now prove Theorem 2.1 though a sequence of lemmas.

Lemma 2.3. There exists tλ ∈ C[Sn]∗ such that aλgbλ = tλ(g)cλ for all g ∈ C[Sn].

10 JAMES STEVENS

Proof. It suffices to show this for g ∈ Sn. Notice if g ∈ PλQλ, we can uniquelywrite g = pq with p ∈ Pλ and q ∈ Qλ. Thus,

aλgbλ = (aλp)(qbλ) = aλ(sgn(q)bλ) = sgn(q)cλ.

Now if g 6∈ PλQλ, then notice that the coefficient for the identity 1 in aλgbλ is 0because if pgq = 1 for some p ∈ Pλ and q ∈ Qλ, then g = p−1q−1 ∈ PλQλ. Thus,we must show that `λ(g) = 0. So it suffices to show that there exists a transpositionq ∈ Qλ so that p = gqg−1 ∈ Pλ because then

aλgbλ = (aλp)g(sgn(q)qbλ) = −aλ(gqg−1)gqbλ = −aλgbλand, hence, aλgbλ = 0. Consider the tableau T ′ = gT where T is the originaltableau and T ′ is a tableau of the same shape where each entry i in T is replacedby g(i). Equivalently, we want to show that there exists two distinct integers whichlie in the same row in T and in the same column in T ′. (This is because p is a row-preserving transposition in T because p ∈ Pλ, and p = gqg−1 is a column-preservingtransposition in T ′ because q ∈ Qλ.)

So suppose there were not two such integers. We could then find a row-preservingpermutation p1 ∈ Pλ of T and a column-preserving permutation q′1 ∈ gQλg−1 ofT ′ so that p1T and q′1T

′ have the same first row. Continuing this, we could findp ∈ Pλ and q′ = gqg−1 ∈ gQλg

−1 so that pT = q′T ′ = q′gT = gqT . But thenp = gq and, hence, g = pq−1 ∈ PλQλ. �

Notice that under the lexicographical order, partitions are totally ordered.

Lemma 2.4. If λ > µ, then aλgbµ = 0 for all g ∈ C[Sn].

Proof. Again, it suffices to show this for g ∈ Sn. And similarly, it suffices to find atransposition q ∈ Qµ so that p = gqg−1 ∈ Pλ because then

aλgbµ = (aλp)g(sgn(q)qbµ) = −aλ(gqg−1)gqbµ = −aλgbµand, hence, aλgbµ = 0. Let T be the tableau used to construct aλ and T ′ be thetableau used to construct bµ. Thus, we want to show there exists two distinctintegers which lie in the same row of T and in the same column of gT ′.

Notice if λ1 > µ1, then this must be true because of the pigeonhole argumentthat the λ1 many entries of the first row of T have µ1 many possible columns ingT ′ to lie in. So if λ1 = µ1 and two such integers did not exist, we could finda row-preserving permutation p1 ∈ Pλ of T and a column-preserving permutationq′1 ∈ gQµg−1 of gT ′ so that p1T and q′1gT

′ have the same first row.We can continue this for each consecutive row until we have λi > µi. By the

pigeonhole argument, there exists two distinct integers which lie in the same rowof pi−1 · · · p1T and in the same column of q′i−1 · · · q′1gT ′. �

Lemma 2.5. We have that cλcλ = nλcλ where nλ = n!/ dimVλ.

Proof. It follows from Lemma 2.3 that cλcλ = nλcλ for some nλ ∈ C. Now, considerthe map F : C[Sn] → Vλ where x 7→ xcλ. Notice that F multiplies by nλ on Vλ,while F multiplies by 0 on KerF . Therefore, trF = nλ dimVλ.

On the other hand, since the coefficient of 1 ∈ Sn in cλ is 1, the coefficient of gin F (g) = gcλ is also 1. Thus, we also have that trF = dimC[Sn] = n!. �

Lemma 2.6. For each partition λ of n, Vλ is an irreducible representation of Sn.

SCHUR-WEYL DUALITY 11

Proof. Notice that cλVλ = Ccλ by Lemma 2.3 and because cλcλ = nλcλ 6= 0 byLemma 2.5 where cλcλ ∈ cλVλ. Let W ⊆ Vλ be a subrepresentation. There aretwo cases. If cλW = 0, then W ·W ⊆ Vλ ·W = 0 and, hence, W = 0. But ifcλW = Ccλ, then

Vλ = C[Sn]cλ = C[Sn](cλW ) = (C[Sn]cλ)W ⊆Wbecause W is a representation of Sn. Therefore, Vλ is irreducible. �

Lemma 2.7. If λ 6= µ, then Vλ 6' Vµ.

Proof. Without loss of generality, assume λ > µ. Then by the previous proof alongwith Lemma 2.4, we have that cλVλ = Ccλ but cλVµ = cλC[Sn]cµ = 0. �

Now the proof of Theorem 2.1 follows from the previous two lemmas and thefact that partitions of n enumerate all conjugacy classes of Sn, which by Corollary1.20, enumerate all irreducible representations of Sn.

2.2. Dimension Formulas. Although we have given an explicit way to computeall the irreducible representations of Sn, it still takes some work to really see howthese representations behave. It is not even clear what the dimension of the irre-ducibles are. As we saw in the previous section, the most powerful tool for studyingrepresentations of finite groups was characters. It turns out that we can computethe characters χλ := χVλ , but this still takes some work.

Some of the proofs of the statements in this subsection are quite involved and notimportant for our purposes. So we will omit proofs and just focus on the results.All proofs in this subsection can be found in [5].

Frobenius Formula 2.8. Given a partition λ = (λ1, . . . , λk) of n. Let c =(c1, . . . , cn) represent the cycle type of g ∈ Sn, that is, ci is the number of i-cyclesin the disjoint cycle representation of g. Set t = (t1, . . . , tk) where ti = λi + k − i.Then

χλ(g) = [xt]

∆(x) ·∏j

pj(x)ci

where ∆(x) =

∏i<j(xi − xj) is the Vandermonde determinant, pj(x) =

∑ki=1 x

ji

are the power sums, and [xt]f(x) is the coefficient of xt11 · · ·xtkk in f .

Notice that we can specialize the Frobenius formula to the cycle type c =(n, 0, . . . , 0) of the identity, and–with some reinterpretation–extract some formu-las for χλ(1) = dimVλ. The hook length formula is the easiest dimension formulato compute. The hook length of a cell in a Young diagram is the number of boxesthat are either directly to the right or directly below the cell, counting the cell itselfonly once. Here are all the hook lengths for λ = (4, 2, 2, 1).

7 5 2 14 23 11

Hook Length Formula 2.9. Let λ be a partition of n. Then

dimVλ =n!∏h(i, j)

12 JAMES STEVENS

where the product is over all cells (i, j) in the Young diagram of λ, (i, j) denotesthe cell in the ith row and jth column, and h(i, j) is the hook length of cell (i, j).

So for λ = (4, 2, 2, 1), we see that dimVλ = 9!/(7 · 5 · 4 · 3 · 2 · 2) = 216.The next dimensional formula involves the concept of standard Young tableau.

We will see that it has great combinatorial significance, at the cost of being moredifficult to compute. A standard Young tableau has a labeling that is strictlyincreasing to the right across the rows and strictly increasing down the columns.Here are all five of the standard Young tableaux for λ = (3, 2).

1 2 34 5

1 2 43 5

1 2 53 4

1 3 42 5

1 3 52 4

Also, by the Hook Length Formula 2.9, we have dimVλ = 5. This is no coincidence.

Proposition 2.10. Let λ be a partition of n. Then

dimVλ = |SYT(λ)|

where SYT(λ) is the set of standard Young tableaux for λ.

Define fλ = |SYT(λ)|. We have seen that |G| =∑

(dimVi)2 where the sum is

over all irreducible representations Vi of G. For G = Sn, this says

(2.11) n! =∑|λ|=n

f2λ.

2.3. The RSK-Correspondence. In this section will describe a remarkable com-binatorial interpretation for (2.11) known as the Robinson-Schensted correspon-dence. Recall we have defined fλ to count the number of standard Young tableauof λ. Incredibly, we can promote the equation (2.11) to a bijection

SnRSK−−−→

∐|λ|=n

SYT(λ)× SYT(λ).

But permutations and pairs of standard Young tableau are very different objects,so the fact that a nice bijection exists is far from obvious.

The map RSK is an algorithm that takes in permutations g = g1g2 · · · gn inone-line notation and iteratively builds a pair of tableau via a process called row-insertion. An incomplete standard Young tableau of n is a Young diagram of m forsome m ≤ n that is filled by a m-subset of {1, 2, . . . , n} that is increasing to theright across the rows and increasing down the columns. As input, row-insertiontakes in a positive integer k and an incomplete standard Young tableau P notcontaining k. It works as follows:

First, look at the first row and find the smallest j (first entry in row) so thatk < P (1, j). If no such j exists, then add cell containing k at the end of the row andterminate. Otherwise, if j does exist, take k′ = P (1, j) and then replace P (1, j)with k and insert k′ into the second row using the same rules. And so on, continueuntil termination.

For g = g1g2 · · · gn ∈ Sn, find RSK(g) as follows: Start with (P0, Q0) = (∅, ∅).Given a pair of incomplete standard Young tableau (Pi−1, Qi−1). Row-insert giinto Pi−1 to get Pi, and add the same newly formed cell from Pi to Qi−1 and labelit i. In the end, we have RSK(g) = (P,Q) := (Pn, Qn).

SCHUR-WEYL DUALITY 13

Example 2.12. Here is the algorithm for g = 351462 ∈ S6.

P : ∅ 3−−→ 35−−→ 3 5

1−−→ 1 53

4−−→ 1 43 5

6−−→ 1 4 63 5

2−−→1 2 63 45

Q : ∅ −−→ 1 −−→ 1 2 −−→ 1 23

−−→ 1 23 4

−−→ 1 2 53 4

−−→1 2 53 46

Example 2.13. Here are the images for all permutations of S3 under RSK.

123: 132: 213:

1 2 3 1 2 31 23

1 23

1 32

1 32

231: 312: 321:

1 32

1 23

1 23

1 32

123

123

It turns out that RSK exhibits some (non-evident) marvelous symmetry, thathas some unexpected consequences.

Proposition 2.14. If g 7→ (P,Q), then g−1 7→ (Q,P ).

So if g is a involution in Sn, then g 7→ (P, P ). So we obtain the following formula.

Corollary 2.15. We have that

# involutions of Sn =∑|λ|=n

fλ.

The next fact, will be about the longest decreasing subsequence and longestincreasing subsequence of a permutation g in one-line notation, denoted lds(g) andlis(g) respectively.

Proposition 2.16. Let g 7→ (P,Q) and λ be the shape of P (or, equivalently, Q).Then lis(g) = λ1 which is the length of first row of P and, similarly, lds(g) = `(λ)which is the length of first column of P .

Corollary 2.17 (Erdos-Szekeres). If g ∈ Spq+1, then lis(g) > p or lds(g) > q.

Proof. Otherwise, the shape of the image of g would fit in a p× q square. �

3. Schur-Weyl Duality

3.1. Representations of Lie Groups and Lie Algebras. Before we can discussthe main content of this section, we will first have to cover some needed Lie theory.

Recall that a Lie group G is a group that is also a smooth manifold where themultiplication map (g, h) 7→ gh and inverse map g 7→ g−1 are both smooth. A Liegroup homomorphism is group homomorphism that is also smooth. We will reallyonly need the case G = GL(V ) for a finite dimensional complex vector space V .

A Lie algebra g is a vector space with a bilinear map [·, ·] : g× g→ g called theLie bracket with following two properties: [X,X] = 0 for all X ∈ g and

[X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0

14 JAMES STEVENS

for all X,Y, Z ∈ g. Note that g along with multiplication given by [·, ·] forms anonassociative algebra. A Lie algebra homomorphism ρ : g → h of Lie algebras gand h is a linear map that respects the Lie bracket, i.e.,

ρ([X,Y ]g) = [ρ(X), ρ(Y )]h

for all X,Y ∈ g. Like for Lie groups, we are really only interested in the case ofgl(V ) where V is a finite dimensional complex vector space, which we can identifyas End(V ) under the Lie bracket given by the commutator [X,Y ] = XY − Y X.

There is a beautiful connection between Lie groups and Lie algebras. However,we do not have time to fully explain it, and it will not be necessary for later in thesection. For details, see Chapter 8 of [3]. For any Lie group G, there is an associatedLie algebra Lie(G) where as sets, Lie(G) is the tangent space TeG at the identity.And for any Lie algebra g, there is a unique simply connected Lie group G whereLie(G) = g. In fact, for any discrete subgroup Γ ≤ G, we have that Lie(G/Γ) = g.And remarkably, if G is simply connected, then if ρ : Lie(G)→ Lie(H) is a Lie grouphomomorphism, then there exists a unique Lie group homomorphism f : G → Hsuch that ρ = (df)e.

A representation of a Lie group G on a complex vector space W is a Lie grouphomomorphism ρ : G→ GL(W ). Similarly, a representation of a Lie algebra g on acomplex vector space W is a Lie algebra homomorphism ρ : g→ gl(W ). So in thecase where G is simply connected, there is an equivalence between representationsof G and of Lie(G).

Recall that for a finite group G, we had an associative algebra C[G] where rep-resentations of C[G] are representations of G and vice versa. More generally, thegroup algebra of G has the property that for any associative C-algebra A,

HomAlg(C[G], A) ' HomGrp(G,A×)

where A× is the group of units of A. Analogously, for a Lie algebra g, we havean associative algebra Ug called the universal enveloping algebra with the propertythat for any associative C-algebra A,

HomAlg(Ug, A) ' HomLieAlg(g,LA)

where LA is the Lie algebra of the set A with the commutator as the Lie bracket.We can actually construct Ug as a quotient of the tensor algebra T g of which

g is a subset. The tensor algebra T g is the set⊕

n≥0 g⊗n where g⊗0 := C with

multiplication given by the canonical isomorphism g⊗n ⊗ g⊗m → g⊗(n+m). ThenUg = T g/I where I is the ideal generated by elements X ⊗ Y − Y ⊗X − [X,Y ] forall X,Y ∈ g ⊆ T g.

The isomorphism of the above specialized to A = End(V ) says that studyingrepresentations of g is the same as studying representations of Ug. For if we have aLie algebra representation ρ : g→ gl(V ), we can extend to an algebra representationρ′ : Ug→ End(V ) by setting ρ′(X1⊗· · ·⊗Xn) = ρ(X1) · · · ρ(Xn) and then extendingby linearity. Similarly, given an algebra representation ρ′ : Ug → End(V ), we can

SCHUR-WEYL DUALITY 15

restrict to g ⊆ Ug to get a Lie algebra representation ρ : g→ gl(V ) because

ρ([X,Y ]g) = ρ′([X,Y ]g)

= ρ′(X ⊗ Y − Y ⊗X)

= ρ′(X)ρ′(Y )− ρ′(Y )ρ′(X)

= ρ(X)ρ(Y )− ρ(Y )ρ(X)

= [ρ(X), ρ(Y )]gl(V )

for all X,Y ∈ g.

3.2. Schur-Weyl Duality for GL(V ). Consider the space V ⊗n for some complexvector space V . We have two natural actions on this space. Since there are nfactors, there is the natural (right) action of Sn of permuting the factors, that is,where for all π ∈ Sn, we have

(v1 ⊗ v2 ⊗ · · · ⊗ vn)π = vπ(1) ⊗ vπ(2) ⊗ · · · ⊗ vπ(n).And since each factor is V , we have the natural factorwise action of GL(V ) where

g(v1 ⊗ v2 ⊗ · · · ⊗ vn) = g(v1)⊗ g(v2)⊗ · · · ⊗ g(vn)

for all g ∈ GL(V ). Notice that these two actions commute with each other. How-ever, these actions have a stronger connection that we will show. Namely, the spansof the images of Sn and GL(V ) in End(V ⊗n) are centralizers of each other. Wewill see that this fact remarkably connects the representation theory of these twogroups. In order to show this connection, we must prove some lemmas.

Lemma 3.1. The image of U(gl(V )) in End(V ⊗n) is B = EndC[Sn](V⊗n).

Proof. The action of X ∈ gl(V ) on v1 ⊗ · · · ⊗ vn ∈ V ⊗n is

X(v1 ⊗ · · · ⊗ vn) =

n∑i=1

v1 ⊗ · · · ⊗Xvi ⊗ · · · ⊗ vn

and, thus, the image of X in End(V ⊗n) is

Πn(X) := X ⊗ id⊗ · · · ⊗ id + id⊗X ⊗ · · · ⊗ id + · · ·+ id⊗ · · · ⊗ id⊗X.Clearly, the image of gl(V ) and, thus the image of U(gl(V )), is contained in B.

Now since the elementary symmetric polynomial en(x) = x1x2 · · ·xn is express-ible as P (p1(x), p2(x), . . . , pn(x)) for some polynomial P in the power sum sym-

metric polynomials pj(x) = xj1 + xj2 + · · ·+ xjn, we get

X ⊗X ⊗ · · · ⊗X = P (Πn(X),Πn(X2), . . . ,Πn(Xn)).

Thus, elements of the form X⊗n for X ∈ End(V ) are generated by the images ofelements in U(gl(V )). And since elements of the form X⊗n for X ∈ End(V ) span

Symn End(V ) ' (End(V )⊗n)Sn ' (End(V ⊗n))Sn = EndC[Sn](V⊗n),

then the image of U(gl(V )) in End(V ⊗n) is B. �

Proposition 3.2. The images of C[Sn] and U(gl(V )) in End(V ⊗n) are centralizersof each other.

Proof. Let A be the image of C[Sn] in End(V ⊗n). By Maschke’s Theorem 1.2 (andthe fact that quotient of semisimple ring is semisimple), A is semisimple. The restfollows from the Double Centralizer Theorem A.3. �

16 JAMES STEVENS

Lemma 3.3. The span of the image of GL(V ) in End(V ⊗n) is B.

Proof. Since GL(V ) commutes with Sn, then the image of GL(V ), and thus itsspan, is contained in B = EndA(V ⊗n).

Conversely, let X ∈ End(V ) and B′ be the span of the image of the elements g⊗n

for g ∈ G. Since X+tI is invertible for all but finitely many t ∈ C, by interpolation,the polynomial (X + tI)⊗n is in the subalgebra B′ for all but finitely many t and,hence, for all t. In particular, for t = 0, we have that X⊗n ∈ B′. But again, weknow that these in fact span B = EndA(V ⊗n). �

Therefore, by the Double Centralizer Theorem A.3, we have the following.

Schur-Weyl Duality 3.4. We have the decomposition

V ⊗n '⊕|λ|=n

Vλ ⊗ SλV.

as a representation of Sn ×GL(V ) where Vλ runs through all the irreducible repre-sentations of Sn and each SλV := HomSn(Vλ, V

⊗n) is an irreducible representationof GL(V ) or is zero.

In fact, SλV is zero when λd+1 6= 0 where d = dimV , that is, the number ofparts of λ is greater than d.

3.3. Schur Functors and Algebraic Representations. We call the covariantfunctor Sλ : FinVect → FinVect the Schur functor of λ. If f : V → W is a linearmap, we define Sλf : SλV → SλW by (Sλf)(Ψ) = f⊗n ◦Ψ. It can be verified that

Sλ(f ◦ g) = (Sλf) ◦ (Sλg) and Sλ idV = idSλ .

Since representations of Sn are self-dual by Example 1.15, we get the followingmore constructive descriptions of the Schur functor.

SλV = HomSn(Vλ, V⊗n)

' (Vλ)∗ ⊗C[Sn] V⊗n

' Vλ ⊗C[Sn] V⊗n

= C[Sn]cλ ⊗C[Sn] V⊗n

= C[Sn]⊗C[Sn] V⊗ncλ

' V ⊗ncλExample 3.5. For λ = (n), we have that S(n)V ' Symn V . Similiarly, for λ =(1, . . . , 1), we have that S(1,...,1)V ' Altn V . Note that Altn V is zero if dimV < n.So Schur-Weyl Duality tells us that

V ⊗n '⊕|λ|=n

(SλV )⊕fλ

as a representation of GL(V ) where fλ = dimVλ. So, in particular, we have

V ⊗2 ' Sym2 V ⊕Alt2 V and V ⊗3 ' Sym3 V ⊕ (S(2,1)V )⊕2 ⊕Alt3 V.

It turns out that S(2,1)V does not have a description as nice as Symn V or Altn V .But it can be shown using Example 2.2, that

S(2,1)V ' V ⊗3c(2,1) ' Ker(V ⊗Alt2 V → Alt3 V )

where the map V ⊗Alt2 V → Alt3 V the canonical map v1⊗(v2∧v3) 7→ v1∧v2∧v3.

SCHUR-WEYL DUALITY 17

Recall that any representation of Sn could be decomposed into the Specht mod-ules Vλ. However, it is not true that any representation of GL(V ) can be decom-posed into the images SλV of V under the Schur functors. For example, we cannotget the dual of SλV . However, we do essentially get the whole class of algebraicrepresentations. Proofs and details of the following can be found in section 5.23 of[2] and in section 15.5 of [3].

A finite dimensional representation W of GL(V ) is algebraic if the correspondingmap ρ : GL(V )→ GL(W ) is a morphism of algebraic varieties. Concretely, this saysthat if we choose bases, then the coordinates of ρ(A) are rational functions of thecoordinates of A ∈ GL(V ). Any such rational function is, in fact, in C[aij ][1/detA].

It turns out that the Schur functors only give us representations which arepolynomial, that is, the coordinates of ρ(A) are polynomials of the coordinatesof A ∈ GL(V ). Thus, we could never hope to get the algebraic representationdet−1 : GL(V )→ GL(C) where det−1(A) = 1/ detA. But this is essentially it.

Proposition 3.6. Let W be an algebraic representation of GL(V ). Then for all

k > 0 and λ such that SλV 6= 0, we have that SλV ⊗ det−k is an irreduciblealgebraic representation of GL(V ). Moreover, W decomposes as a direct sum of

copies of SλV ⊗ det−k.

3.4. Other Cases of Schur-Weyl Duality. Note that the key to showing Schur-Weyl Duality for GL(V ) was showing that the images of C[Sn] and C[GL(V )]were centralizers of each other in End(V ⊗n). But there are many other groupsthat naturally act on V ⊗n. So if we can realize the centralizer in End(V ⊗n) ofsuch a group, we can learn more about the representations of that group and itscorresponding centralizer. In this section, we will describe a few particularly niceexamples of these. As before, we will focus on the results. The proofs of thefollowing material can be found in [1] and [4].

Recall that when we showed Schur-Weyl Duality for GL(V ), we first showed itfor gl(V ) in Proposition 3.2. That is, as a representation of Sn × gl(V ),

V ⊗n '⊕|λ|=n

Vλ ⊗ SλV.

where Vλ are all the irreducible representations of Sn and each SλV := HomSn(Vλ, V⊗n)

is an irreducible gl(V )-representation or zero.In both these cases, the resulting centralizers have all been the same, the image

of C[Sn]. In fact, if we equipped V with an inner product, then we could showthat C[Sn] is still the centralizer of both the image of u(V ) and the image of U(V )in End(V ⊗n). But now, we will show some examples where this is not the case.For the remainder of the section, we will often be working with a basis for ourunderlying vector space. So let V = Cd with the standard basis.

Let Sd(C) ⊆ GLn(C) denote the group of d×d permutation matrices. This groupis isomorphic to Sd, but we give it different notation because it has a different actionon V ⊗n. It turns out that the corresponding centralizer for Sd(C) in End(V ⊗n) canbe realized as the image of an algebra called the partition algebra C[Pn(d)]. Butwe will first have to do a little work to describe this algebra.

First, we will define the partition monoid Pn. As a set,

Pn = {set partitions of {1, 2, . . . , n, 1′, 2′, . . . , n′}}.

18 JAMES STEVENS

We will represent a set partition S ∈ Pn by a graph written as a top row of nvertices corresponding to 1, 2, . . . , n and bottom row of n vertices corresponding to1′, 2′, . . . , n′ with the property that the collection of path connected componentsis S, that is, the vertices i, j are connected by a path if and only if i, j are in asame block in S. Clearly, such a graph is not unique for a given set partition S,but any representative will work. This freedom will allow us to draw less cluttereddiagrams.

We will define the composition S1◦S2 of two set partitions S1, S2 ∈ Pn as the setpartition corresponding to the diagram formed by stacking a diagram of S1 on topof a diagram of S2 and identifying each vertex on the bottom row of S1 with thecorresponding vertex on the top row of S2. The composition will have the diagramwith the top row as the top row of S1 and the bottom row as the bottom row ofS2, ignoring the middle vertices but not the paths through them.

For example, if S1 = {1354′, 23′, 4, 1′2′, 5′} and S2 = {15′, 22′, 343′4′, 5, 1′}, thenS1 ◦ S2 = {12353′4′, 4, 1′2′, 5′} because

S1 ◦ S2 = =

Now we define the partition algebra C[Pn(d)] as the set C[Pn] with multiplicationon the basis given by S1S2 = d` · (S1 ◦ S2) for S1, S2 ∈ Pn where ` is the numberof middle-only blocks of the 3n set partition (top, middle, and bottom vertices)induced from the composition. In our previous example, we have ` = 1 because5′1 = 52 (identification of 5′ from S1 with 5 from S2) is the only block containingonly middle vertices (namely, just itself). So S1S2 = d · (S1 ◦ S2).

Let S ∈ Pn. For convenience, now label i′ as n + i for 1 ≤ i ≤ n in the setpartition S. For a sequence i1, i2, . . . , i2n with 1 ≤ ij ≤ d define

δ(S)i1,...,inin+1,...,i2n=

1, if ij = ik whenever j and kare in the same block in S,

0, otherwise

Let v1, . . . , vd be the standard basis for V = Cd. We can then define the action ofS on the basis elements vi1 ⊗ vi2 ⊗ · · · ⊗ vin of V ⊗n where 1 ≤ ij ≤ d as

(vi1 ⊗ vi2 ⊗ · · · ⊗ vin)S =∑

id+1,...,i2n

δ(S)i1,...,inin+1,...,i2nvin+1

⊗ vin+2⊗ · · · ⊗ vi2n

where the sum is over all sequences in+1, . . . , i2n with 1 ≤ ij ≤ d. This action ofPn on V ⊗n actually extends to an action of C[Pn(d)].

Proposition 3.7. The spans of the images of C[Pn(d)] and Sd(C) in End(V ⊗n)are centralizers of each other.

Therefore, we can get a version of Schur-Weyl Duality between the algebraC[Pn(d)] and the group Sd(C) ' Sd. It turns out that Pn has some subsets whosegenerated subalgebras give other interesting cases of Schur-Weyl Duality.

SCHUR-WEYL DUALITY 19

Take Sn ⊆ Pn to be the set partitions where each block of is the form {i, j′} wherei, j ∈ {1, 2, . . . , n}. These partitions are identified with bijections on {1, 2, . . . , n}.Moreover, partition composition is exactly function composition and the inducedaction is same as that of the symmetric group. So as the notation suggests, wehave that the subalgebra of C[Pn(d)] generated by Sn is isomorphic to the groupalgebra C[Sn] of the symmetric group. Note d does not matter since ` = 0 for anycomposition. So in this case, we recover the original Schur-Weyl Duality for GL(V ).

Now take Bn ⊆ Pn to be the set partitions where each block is of size two (notnecessarily of the previous form). Then Bn is closed under composition. And, wecall the the subalgebra C[Bn(d)] of C[Pn(d)] generated by Bn the Brauer algebra.This gives rise to two interesting cases of Schur-Weyl Duality.

Proposition 3.8. The spans of the images of C[Bn(d)] and Od(C) in End(V ⊗n)are centralizers of each other where Od(C) is the orthogonal group.

Proposition 3.9. The spans of the images of C[Bn(−2d)] and Sp2d(C) in End(V ⊗n)are centralizers of each other where Sp2d(C) is the symplectic group.

Appendix A. Semisimple Algebras and Double Centralizer Theorem

Recall that a module is said to be simple if it contains no proper nonzero sub-modules. And a module is said to be semisimple if it can be decomposed as a directsum of simple submodules. Note that any module of an algebra over a field k is avector space over k, and hence it makes to talk about the dimension of the module.We say that an algebra is semisimple if all of its finite dimensional modules aresemisimple. The following theorem is Maschke’s Theorem 1.2 reformulated in thelanguage of the group algebra.

Maschke’s Theorem A.1. Let G be a finite group. Then C[G] is semisimple.

The following is an important theorem on the structure of semisimple algebras.However, the proof is quite involved, so we will have to omit it. For full details, seeTheorem 3.5.4. of [2].

Theorem A.2. Let A be a finite dimensional algebra. Then A has finitely manysimple modules Ui up to isomorphism. These simple modules are finite dimensional.Moreover, A is semisimple if and only if as an algebra,

A '⊕i

End(Ui).

where Ui are simple A-modules.

As a corollary of the previous two theorems, we get Proposition 1.8. Now, themain result of this section.

Double Centralizer Theorem A.3. Let V be a finite dimensional vector space,A be a semisimple subalgebra of End(V ), and B = EndA(V ). Then:

(1) B is semisimple.(2) A = EndB(V ).(3) As a module of A⊗B, we have the decomposition

V '⊕i

Ui ⊗Wi

20 JAMES STEVENS

where Ui are all the simple modules of A and Wi := HomA(Ui, V ) are allthe simple modules of B.

Proof. Since A is semisimple, we have the A-module decomposition

(A.4) V '⊕i

Ui ⊗HomA(Ui, V )

where a ∈ A acts on Ui ⊗ HomA(Ui, V ) by a(u ⊗ f) = au ⊗ f . The space Wi :=HomA(Ui, V ) is the multiplicity space of Ui. Then as algebras, we have the naturalisomorphisms

A '⊕i

End(Ui)

and by properties of module homomorphisms,

B = EndA(V )

' HomA

(⊕i

Ui ⊗Wi, V

)'⊕i

HomA(Ui ⊗Wi, V )

'⊕i

HomA(Wi ⊗ Ui, V )

'⊕i

Hom(Wi,HomA(Ui, V ))

=⊕i

End(Wi).

We now just need to show that Wi are simple B-modules. We will show that Bacts transitively on the nonzero maps in HomA(U, V ) where U is a simple A-module.Fix a nonzero u ∈ U . Since U is simple, any map f ∈ HomA(U, V ) is determinedby where it takes u because Au is a nonzero submodule of U and hence Au = U . Sotake f, f ′ ∈ HomA(U, V ) where f(u) = v and f ′(u) = v′. Since Av is an invariantsubspace of V , we can write V = (Av)⊕W for a complementary invariant subspaceW by Maschke’s Theorem. Define T : V → V by T (av) = av′ for av ∈ Av andT (w) = w for w ∈W . This is an A-homomorphism where T ◦ f = f ′.

By Theorem A.2, we see that B is semisimple, and so we have (1). And we cannow view (A.4) as a decomposition of V into simple B-modules Wi where Ui arethe corresponding multiplicity spaces. Then, as B-modules,

V '⊕i

Wi ⊗HomB(Wi, V ) '⊕i

Wi ⊗ Ui

where the sums run over all simple B-modules Wi. This then implies (3). Also, wehave that Ui ' HomB(Wi, V ), and hence (2) follows. �

Acknowledgments. I would like to thank my mentor Minh-Tam Trinh for all ofhis guidance, patience, and encouragement throughout this REU program. I wouldalso like to thank Peter May for running this REU program and giving me theopportunity to learn a lot of fun mathematics.

SCHUR-WEYL DUALITY 21

References

[1] R. Dipper, S. Doty, and J. Hu, Brauer Alegbras, Symplectic Schur Algebras and Schur-WeylDuality, Trans. Amer. Math. Soc., 360, 189-213 (2008).

[2] P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, and E.Yudovina,

Introduction to Representation Theory, AMS, 2011.[3] W. Fulton and J. Harris, Representation Theory: A First Course, Springer, New York, 2004.

[4] T. Halverson, Characters of the Partition Algebras, Journal of Algebra, 238, 502-533 (2001).

[5] R. P. Stanley, Enumerative Combinatorics: Volume 2, Cambridge University Press, 1999.