Abstract algebra

Section 18.1-2.

In the next 2-3 lectures we will have a lightning introduction to representations of finite groups.

For any vector space V over a field F, denote by L(V ) the algebra of all linear operators on V , and by GL(V ) the group of invertible linear operators. Note that if dim(V ) = n, then L(V ) = Matn(F) is isomorphic to the algebra of n×n matrices over F , and GL(V ) = GLn(F) to its multiplicative group.

Definition 1. A linear representation of a set X in a vector space V is a map φ: X → L(V ), where L(V ) is the set of linear operators on V , the space V is called the space of the rep- resentation. We will often denote the representation of X by (V,φ) or simply by V if the homomorphism φ is clear from the context. If X has any additional structures, we require that the map φ is a homomorphism. For example, a linear representation of a group G is a homomorphism φ: G → GL(V ).

Definition 2. A morphism of representations φ: X → L(V ) and ψ : X → L(U) is a linear map T : V → U, such that

ψ(x) ◦T = T ◦φ(x)

for all x ∈ X. In other words, T makes the following diagram commutative

V φ(x)

//

T ��

V

T ��

U ψ(x)

// U

An invertible morphism of two representation is called an isomorphism, and two representa- tions are called isomorphic (or equivalent ) if there exists an isomorphism between them.

Example.

(1) A representation of a one-element set in a vector space V is simply a linear operator on V . Two such representations are isomorphic if two operators have the same Jordan Normal Form.

(2) For any vector space, identity maps id : GL(V ) → GL(V ) and id : L(V ) → L(V ) give tautological representations of the group GL(V ) and the algebra L(V ) respectively.

(3) For each representation φ: X → L(V ) one can define its dual representation

φ∗ : X → L(V ∗), φ∗(x)(α)(v) = α(φ(x)(v))

for all x ∈ X, v ∈ V , and α ∈ V ∗. (4) The trivial representation of a group G is a homomorphism φ: G → GL(V ), where

V is a 1-dimensional vector space over the field F, and φ(g) = 1 ∈ GL(V ) = F× for all g ∈ G.

(5) Recall the dihedral group Dn = ⟨ r,s |rn = s2 = 1,rs = sr−1

⟩ . The following map

r 7→ (

cos (2π/n) −sin (2π/n) sin (2π/n) cos (2π/n)

) , s 7→

( 0 1 1 0

) defines a 2-dimensional representation φ: Dn → GL(2,R).

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(6) The following map gives a representation of a symmetric group S4:

(12) 7→

 

0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1

  , (23) 7→

 

1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1

  , (34) 7→

 

1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0

  .

(7) There is a natural representations of Sn on the space of polynomials in n variables:

(σf)(x1, . . . ,xn) = f(xσ(1), . . . ,xσ(n)).

Note, that symmetric polynomials stay invariant under this action. (8) Any Galois group Gal(K/F) comes with its representation in K over the field F.

Definition 3. An invariant subspace of a representation φ: X → L(V ) is a subspace W ⊂ V invariant under φ(x) for all x ∈ X. An invariant subspace gives rise to a subrepresentation

φW : X → L(W), φW (x) = φ(x)|W , and a factor representation

φV/W : X → L(V/W), φV/W (x)(v + W) = φ(x)(v).

Remark 4. If one chooses a basis of the space V of a representation φ: X → L(V ) in such a way that the first k vectors span an invariant subspace W ⊂ V , then any operator φ(x) is written in matrix form as

φ(x) =

( φW (x) ∗

0 φV/W (x)

) Example.

(1) Let Sn act on F n by permuting basis vectors, as in example (4) above. Then the

subspace {a,a, . . . ,a) |a ∈ F} ⊂ Fn is a 1-dimensional invariant subspace. There is a complementary invariant subspace of dimension n− 1 defined by

{(a1, . . . ,an) |a1 + · · · + an = 0} . Note that Fn is isomorphic to the direct sum of these two subspaces.

(2) For any positive integer d, the subspace Poldn of polynomials in n variables of degree d in the space Poln of all polynomials of n variables gives a subrepresentation of Sn.

Definition 5.

(1) A representation is irreducible (or simple) if it (or rather the underlying vector space) has no invariant subspaces other than 0 and itself, otherwise it is reducible.

(2) A representation is indecomposable if it can not be written as a direct sum of two invariant subrepresentations.

(3) A representation is completely reducible if it can be written as a direct sum of irre- ducible representations.

Example. The representation of a 1-element set given by a 2-dimensional space V = F2 and an operator

A =

( a b 0 c

) is reducible, since the subspace 〈e1〉 ⊂ V is invariant, however it is indecomposable unless b = 0. If b = 0, the representation V is completly reducible: V = 〈e1〉⊕〈e2〉. Exercise 6. If T : V → U is a morphism of representations of X, then ker(T) ⊂ V and im(T) ⊂ U are invariant subspaces.

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Corollary 7. If V and U are irreducible representations of X, any morphism T : V → U is either 0 or an isomorphism.

Proof. Since V is irreducible, the subrepresentation ker(T) is either 0, in which case T is injective, or coincides with the whole V , in which case T = 0. Since U is irreducible, we must now have im(T) = U unless T = 0, which forces T to be an isomorphism. �

Corollary 8 (Schur’s lemma). Any endomorphism T : V → V of an irreducible represen- tation V of X over an algebraically closed field is a scalar, that is T = λ · Id for some λ ∈ F . Proof. Let T ∈ EndX(V ) = HomX(V,V ) be an endomorphism of V . Then for any λ ∈ F, so is T−λ·Id. Choosing λ to be an eigenvalue of T (which we can always do over an algebraically closed field) we see that ker(T −λ · Id) is a subrepresentation of X in V . Therefore, since V is irreducible, we must have T = λ · Id. � Corollary 9. Let φ: X → L(V ) and ψ : X → L(U) be a pair of irreducible representations over an algebraically closed field. Then any morphisms T,S : V → U differ by a scalar factor. Proof. If one of the two morphisms is zero, the statement is obvious. Otherwise, both mor- phisms are isomorphisms, hence ST−1 is an endomorphism of an irreducible representation U which implies ST−1 = λ Id. �

Corollary 10. Any irreducible representation of an abelian group over an algebraically closed field is 1-dimensional.

Proof. If G is abelian and φ: G → GL(V ) is a representation of G, operators φ(g) commute for all g ∈ G, hence any of them can be viewed as an endomorphism of V . If V is irre- ducible, By Schur’s lemma, all of them must be scalar, therefore any subspace is invariant and representation V can be irreducible if and only if dim(V ) = 1. �

Example. The 2-dimensional plane R2 is an irreducible representation of the abelian group SO(2,R) of plane rotations

G =

{( cos θ −sin θ sin θ cos θ

)∣∣∣∣θ ∈ R } .

However, over the field of complex numbers, we see that( cos θ −sin θ sin θ cos θ

) ∼ ( eiθ 0 0 e−iθ

) .

Proposition 11. Every subrepresentation and factor representation of a completely reducible representation is completely reducible.

Proof. Let V be a completely reducible representation, and U be its subrepresentation. Then for any U1 ⊂ U, there exists an invarinat subspace V ′ ⊂ V such that V ' U1 ⊕ V ′. Setting U2 = V

′ ∩U we see that U2 is invariant and U ' U1 ⊕U2. Now, let π : V → V/W be a canonical projection to a factor representation, and U1 ⊂ V/W

be the subrepresentation. Then V1 = π −1(U1) is an invariant subspace of V , which con-

tains W . Now, we have V = V1 ⊕ V2 for some invariant subspace V2, which implies that V/W ' U1 ⊕U2 where U2 = π(V2) is an invariant subspace in V/W . � Proposition 12. If a representation φ: X → L(V ) is completely reducible, then V can be written as a direct sum of minimal (nonzero) irreducible subspaces. Conversely, if V can be written as a sum of minimal invariant subspaces V = V1 + · · · + Vn, then V is completely reducible.

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Proof. The first statement is immediate: choose any minimal invariant subspace V1 ⊂ V and write V = V1⊕V2, continue the same process for V2. Now, let V = V1 +· · ·+Vn for some minimal invariant subspaces Vi ⊂ V , and U ⊂ V be any invariant subspace. Then U ∩Vj is either 0 or coincides with Vj for any j = 1, . . . ,n. Setting

U ′ = ∑

j:U∩Vj=∅ Vj

we get V = U ⊕U ′. �

Definition 13. A sum of representations φ: X → L(V ) and ψ : X → L(U) is a representa- tion

φ + ψ : X → V ⊕U, (φ + ψ)(x) = (φ(x),ψ(x)).

Remark 14. In matrix form we have

(φ + ψ)(x) =

( φ(x) 0

0 ψ(x)

) Corollary 15. (1) A representation is completely reducible if and only if it is isomorphic

to a sum of irreducible subrepresentations:

V = V1 ⊕···⊕Vn, φ = (φ1, . . . ,φn).

(2) In that case, any subrepresentation and factor representation of V is isomorphic to a sum of some of the representations Vj.

Proof. Left as an exercise. �

Definition 16. Note that in the above corollary, some of the irreducible summands Vj might be isomorphic.

(1) Let S be an irreducible subrepresentation of a completely reducible representation V , then the S-isotypic component of V is the subspace VS ⊂ V defined as

VS = ⊕

j |Vj'S

Vj.

(2) A representation V is isotypic if it has single isotypic component. (3) We will say that the representation V has simple spectrum if it can be written as a

direct sum of pairwise non-isomorphic irreducible representations, equivalently if all of its isotypic components are irreducible.

Corollary 17. Let φ: X → L(V ) be a completely reducible representation with simple spec- trum, and V1, . . . ,Vn be its irreducible subrepresentations. Then the following holds.

(1) The decomposition of V into a sum of irreducible subrepresentations Vj is unique. (2) If the base field F is algebraically closed, then every endomorphism of V takes the

form

T(x) = λjx, for any x ∈ Vj.

Proof. Left as an exercise. �

It is convenient to describe isotypic representations in the following way. Let φ: X → L(V ) be a representation, and Z be any vector space (over the same field). Define a representation

φ̃: L(V ⊗Z), φ̃(x)(v ⊗z) = φ(x)(v) ⊗z.

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If {z1, . . .zn} is a basis of Z we get

V ⊗Z ' (V ⊗z1) ⊕···⊕ (V ⊗zn),

which provides a decomposition of an isotypic representation V ⊗Z into irreducible subrep- resentations.

Proposition 18. Let φ: X → L(U) be an irreducible representation of X over an alge- braically closed field F , Z be a vector space over F , and ψ : X → L(U ⊗ Z) be the isotypic representation of X discussed above. Then every X-invariant subspace of U ⊗ Z is of the form U ⊗Z0, for some subspace Z0 ⊂ Z.

Proof. It is enough to prove the statement for a minimal invariant subspace W ⊂ U ⊗ Z. Any element w ∈ W can be written as

w = S1(w) ⊗z1 + · · · + Sn(w) ⊗zn,

where Sj : W → U are certain morphisms between representations (W,ψ|W ) and (U,φ). We know that any such isomorphisms have to differ by scalar multiples, hence Sj = λjS for some morphism S : W → U and λ1, . . . ,λn ∈ F . This way we obtain

w = S(w) ⊗ (λ1z1 + · · · + λnzn),

and therefore

W = U ⊗ (λ1z1 + · · · + λnzn). �

Theorem 19. Let φ: X → L(V ) be an irreducible representation of X over an algebraically closed field F . Then the subalgebra of L(V ) generated by operators φ(x), x ∈ X coincides with L(V ) unless dim(V ) = 1 and φ = 0.

Proof. Recall the following isomorphism of vector spaces:

V ⊗V ∗ ' L(V ), v ⊗α 7−→ lα,v where `α,v(u) = α(u)v for all α ∈ V ∗ and u,v ∈ V . Note that under this isomorphism, for any operator T ∈ L(V ) we have

T`α,v = `α,T(v) = `T∗(α),v, (∗)

where T∗ ∈ L(V ∗) defined by T∗(α)(v) = α(Tv).

Due to a canonical bijection between subspaces of V and V ∗, which sends each subspace U ⊂ V to its annihilator Ann(U) ⊂ V ∗, we see that the dual representation φ∗ : X → L(V ∗) is also irreducible. Let us now define representations Tl and Tr of X on the space L(V ), precisely

Tl(x)(A) = φ(x)A and Tr(x)A = Aφ(x).

Thanks to the formula (∗), representations Tl and Tr are isotypic (we can think that they act on only one tensor factor in V ⊗V ∗).

Note now that the subspace φ(X) ⊂ L(V ) is invariant under both Tl and Tr. Using the previous Proposition, we conclude that φ(X) can be written simultaneously as V ⊗W0 and as V0 ⊗ V ∗, where V0 and W0 are subspaces of V and V ∗ respectively. The latter can only happen if φ(X) = L(V ) or φ(X) = 0 in which case one has to assume dim(V ) = 1 since otherwise V would not be irreducible. �

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Theorem 20. Any representation of a finite group G over a field F such that char(F) does not divide |G| is completely reducible.

Proof. Let G be a finite group, and φ: G → GL(V ) be its representation over a field F with char(F) not dividing |G|. It is enough to show that for any invariant subspace U ⊂ V there exists a complementary invariant subspace W ⊂ V such that V ' U ⊕ W . The latter is equivalent to finding a projector π from V onto U, which is a morphism of representations of G. Indeed, recall that a projector is a linear operator P : V → U such that P(V ) ⊂ U and P(u) = u for any u ∈ U. Then setting W = ker(P) we obtain an invariant subspace W ⊂ V satisfying V ' U ⊕W.

Note that the space of all projectors P : V → U forms a linear subspace S in the space L(V ) of all linear operators on V . Consider an action of G on L(V ) by conjugation, that is

g ◦A = φ(g)Aφ(g)−1 for all g ∈ G and A ∈ L(V ).

Since, U is a subrepresentation of G we see that the subspace S ⊂ L(V ) is preserved under this action. Let us choose any projector P0 ∈ S and define P to be the “centre of mass” of the orbit of P0 under the action of G:

P = 1

|G| ∑ g∈G

(g ◦P0).

Then the projector P ∈ S is invariant under the adjoint action of G, or equivalently, commutes with all operators φ(g) ∈ L(V ). �

Finally, let us discuss representations of the group algebra FG of a finite group G, where F is a field. Recall that an algebra A over a ring R is an R-module, which is a ring itself, in particular if R = F is a field, then an algebra A over F is a vector space over F, with a ring structure. As a vector space, algebra FG consists of linear combinations∑

g∈G agg where ag ∈ F,

addition on FG is defined component-wise, and multiplication has the form

agg ·ahh = agah(gh).

In other words, FG is an algebra over the field F with basis vectors labelled by elements of the finite group G, and with multiplication defined from that of G.

As we discussed in the beginning of this lecture, a representation of an algebra A (over the ring R) is a homomorphism φ: A → L(V ) which respects the operations on A. In other words, a representation of A is simply an A-module V (which then forced to be an R-module itself). Now, it is easy to see that there is a bijection between representations of the group G and its group algebra FG, moreover this bijection respects subrepresentations, factor representations, etc. Indeed, given a representation φ: G → GL(V ) of the group G, we define (and denote by the same symbol) a representation

φ: FG → L(V ), φ( ∑ g∈G

agg) = ∑ g∈G

agφ(g).

Conversely, every representation of FG yields a representation of G when restricted to the basis vectors of FG.

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Example. (1) The trivial representation of G corresponds to a 1-dimensional represen- tation of FG where

(agg)(v) = agv for any ag ∈ F, g ∈ G, v ∈ V. (2) Considering FG as a (left) module over itself, we obtain a (left) regular representation

of G, which is a representation of dimension |G| defined by

g( ∑ h∈G

ahh) = ∑ h∈G

ah(gh) for all ah ∈ F, g,h ∈ G.

The bijection described above allows us to study representations of finite groups by study- ing those of finite-dimensional associative algebras.