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Lecture 1 Representation Theory

1.1 Representations

Objects unveil their nature in actions. Groups act on other sets by means of representations. A representation of a group G is a group homomorphism of G in a transformation group of a set. It is a fundamental observation that linear objects are easer to study. Therefore we begin from linear representations of groups.

Definition 1  A linear continuous representation of a group G is a continuous function T(g) on G with values in the group of non-degenerate linear continuous transformation in a linear space H (either finite or infinite dimensional) such that T(g) satisfies to the functional identity:
T(g1 g2) =T(g1)  T(g2). (1)
Remark 2  If we have a representation of a group G by its action on a set X we can use the following linearization procedure. Let us consider a linear space L(X) of functions X→ ℂ which may be restricted by some additional requirements (e.g. integrability, boundedness, continuity, etc.). There is a natural representation of G on L(X) which produced by its action on X:
g: f(x) ↦ ρg f(x)= f(g−1· x),     where g∈ G, x∈ X.  (2)
Clearly this representation is already linear. However in many practical cases the formula for linearization (2) has some additional terms which are required to make it, for example, unitary.
Exercise 3  Show that T(g−1)=T−1(g) and T(e)=I, where I is the identity operator on H.
Exercise 4  Show that these are linear continuous representations of corresponding groups:
  1. Operators T(x) such that [T(x) f](t)=f(t+x) form a representation of ℝ in L2(ℝ).
  2. Operators T(n) such that T(n) ak=ak+n form a representation of ℤ in l2.
  3. Operators T(a,b) defined by
    [T(a,b)  f](x)= √
    a
    		f(ax+b),    a ∈ ℝ+,  b∈ℝ (3)
    form a representation of ax+b group in L2(ℝ).
  4. Operators T(s,x,y) defined by
    [T(s,x,y)  f] (t)=e
    i(2s−√
    2
    		yt+xy)
     
    		 f(t− √
    2
    		x) (4)
    form Schrödinger representation of the Heisenberg group ℍ1 in L2(ℝ).
  5. Operators T(g) defined by
    [T(g) f](t) = 
    1
    ct+d
    		 f⎛
    ⎜
    ⎜
    ⎝
    at+b
    ct+d
    		⎞
    ⎟
    ⎟
    ⎠    		,    where  g−1=
    ⎛
    ⎜
    ⎝
          ab
          cd
    		⎞
    ⎟
    ⎠
    		, (5)
    form a representation of SL2(ℝ) in L2(ℝ).

In the sequel a representation always means linear continuous representation. T(g) is an exact representation (or faithful representation if T(g)=I only for g=e. The opposite case when T(g)=I for all g∈ G is a trivial representation. The space H is representation space and in most cases will be a Hilber space [164]*§ III.5. If dimensionality of H is finite then T is a finite dimensional representation, in the opposite case it is infinite dimensional representation.

We denote the scalar product on H by ⟨ ·,· ⟩. Let {ej} be an (finite or infinite) orthonormal basis in H, i.e.

  ⟨ ej,ej  ⟩=δjk,

where δjk is the Kroneker delta, and linear span of {ej} is dense in H.

Definition 5  The matrix elements tjk(g) of a representation T of a group G (with respect to a basis {ej} in H) are complex valued functions on G defined by
tjk(g) = ⟨ T(g)ej,ek  ⟩. (6)
Exercise 6  Show that [333]*§ 1.1.3
  1. T(g) ek=∑j tjk(g) ej.
  2. tjk(g1g2)=∑n tjn(g1) tnk(g2).

It is typical mathematical questions to determine identical objects which may have a different appearance. For representations it is solved in the following definition.

Definition 7  Two representations T1 and T2 of the same group G in spaces H1 and H2 correspondingly are equivalent representations if there exist a linear operator A: H1 → H2 with the continuous inverse operator A−1 such that:
    T2(g)= A  T1(g)  A−1,    ∀ g∈ G.
Exercise 8  Show that representation T(a,b) of ax+b group in L2(ℝ) from Exercise 3 is equivalent to the representation
[T1(a,b) f] (x)= 
e
i
b
a
 
√
a
  f⎛
⎜
⎜
⎝
x
a
⎞
⎟
⎟
⎠		. (7)

Proof.[Hint] Use the Fourier transform.


The relation of equivalence is reflexive, symmetric, and transitive. Thus it splits the set of all representations of a group G into classes of equivalent representations. In the sequel we study group representations up to their equivalence classes only.

Exercise 9  Show that equivalent representations have the same matrix elements in appropriate basis.
Definition 10  Let T be a representation of a group G in a Hilbert space H The adjoint representation T′(g) of G in H is defined by
    T′(g)=⎛
⎝		T(g−1)⎞
⎠		*,
where * denotes the adjoint operator in H.
Exercise 11  Show that
  1. T′ is indeed a representation.
  2. t′jk(g)=tkj(g−1).

Recall [164]*§ III.5.2 that a bijection U: H → H is a unitary operator if

  ⟨ Ux,Uy  ⟩=⟨ x,y  ⟩,    ∀ x, y ∈ H.
Exercise 12  Show that UU*=I.
Definition 13  T is a unitary representation of a group G in a space H if T(g) is a unitary operator for all g∈ G. T1 and T2 are unitary equivalent representations if T1=UT2U−1 for a unitary operator U.
Exercise 14 
  1. Show that all representations from Exercises 4 are unitary.
  2. Show that representations from Exercises 3 and 8 are unitary equivalent.

Proof.[Hint] Take that the Fourier transform is unitary for granted.


Exercise 15  Show that if a Lie group G is represented by unitary operators in H then its Lie algebra g is represented by selfadjoint (possibly unbounded) operators in H.

The following definition have a sense for finite dimensional representations.

Definition 16  A character of representation T is equal χ(g)= tr(T(g)), where tr is the trace [164, § III.5.2 (Probl.)] of operator.
Exercise 17  Show that
  1. Characters of a representation T are constant on the adjoint elements g−1hg, for all g∈ G.
  2. Character is an algebra homomorphism from an algebra of representations with Kronecker’s (tensor) multiplication [333]*§ 1.9 to complex numbers.

Proof.[Hint] Use that tr(AB)=tr(BA), tr(A+B)=trA + trB, and tr( A ⊗ B)= trA trB.


For infinite dimensional representation characters can be defined either as distributions [159]*§ 11.2 or in infinitesimal terms of Lie algebras [159]*§ 11.3.

The characters of a representation should not be confused with the following notion.

Definition 18  A character of a group G is a one-dimensional representation of G.
Exercise 19 
  1. Let χ be a character of a group G. Show that a character of representation χ coincides with it and thus is a character of G.
  2. A matrix element of a group character χ coincides with χ.
  3. Let χ1 and χ1 be characters of a group G. Show that χ1 ⊗ χ2 =χ1χ2 and χ′(g)=χ1(g−1) are again characters of G. In other words characters of a group form a group themselves.

1.2 Decomposition of Representations

The important part of any mathematical theory is classification theorems on structural properties of objects. Very well known examples are:

  1. The main theorem of arithmetic on unique representation an integer as a product of powers of prime numbers.
  2. Jordan’s normal form of a matrix.

The similar structural results in the representation theory are very difficult. The easiest (but still rather difficult) questions are on classification of unitary representations up to unitary equivalence.

Definition 20  Let T be a representation of G in H. A linear subspace L⊂ H is invariant subspace for T if for any x∈ L and any g∈ G the vector T(g)x again belong to L.

There are always two trivial invariant subspaces: the null space and entire H. All other are non-trivial invariant subspaces.

Definition 21  If there are only two trivial invariant subspaces then T is irreducible representation. Otherwise we have reducible representation.

For any non-trivial invariant subspace we can define the restriction of representation of T on it. In this way we obtain a subrepresentation of T.

Example 22  Let T(a), a∈ℝ+ be defined as follows: [T(a)]f(x)=f(ax). Then spaces of even and odd functions are invariant.
Definition 23  If the closure of liner span of all vectors T(g) v is dense in H then v is called cyclic vector for T.
Exercise 24  Show that for an irreducible representation any non-zero vector is cyclic.

The following important result of representation theory of compact groups is a consequence of the Exercise 33 and we state here it without a proof.

Theorem 25  [159]*§ 9.2
  1. Every topologically irreducible representation of a compact group G is finite-dimensional and unitarizable.
  2. If T1 and T2 are two inequivalent irreducible representations, then every matrix element of T1 is orthogonal in L2(G) to every matrix element of T2.
  3. For a compact group G its dual space Ĝ is discrete.

The important property of unitary representation is complete reducibility.

Exercise 26  Let a unitary representation T has an invariant subspace L⊂ H, then its orthogonal completion L⊥ is also invariant.
Definition 27  A representation on H is called decomposable if there are two non-trivial invariant subspaces H1 and H2 of H such that H=H1⊕ H2.

If a representation is not decomposable then its primary.

Theorem 28  [159]*§ 8.4 Any unitary representation T of a locally compact group G can be decomposed in a (continuous) direct sum irreducible representations: T=∫X Tx  dµ(x).

The necessity of continuous sums appeared in very simple examples:

Exercise 29  Let T be a representation of ℝ in L2(ℝ) as follows: [T(a)f](x)=eiaxf(x). Show that
  1. Any measurable set E⊂ ℝ define an invariant subspace of functions vanishing outside E.
  2. T does not have invariant irreducible subrepresentations.
Definition 30  The set of equivalence classes of unitary irreducible representations of a group G is denoted by Ĝ and called dual object (or dual space) of the group G.
Definition 31  A left regular representation Λ(g) of a group G is the representation by left shifts in the space L2(G) of square-integrable function on G with the left Haar measure
Λg: f(h) ↦ f(g−1h). (8)
The main problem of representation theory is to decompose a left regular representation Λ(g) into irreducible components.

1.3 Schur’s Lemma

It is a pleasant feature of an abstract theory that we obtain important general statements from simple observations. Finiteness of invariant measure on a compact group is one such example. Another example is Schur’s Lemma presented here.

To find different classes of representations we need to compare them each other. This is done by intertwining operators.

Definition 32  Let T1 and T2 are representations of a group G in a spaces H1 and H2 correspondingly. An operator A: H1 → H2 is called an intertwining operator if
    A T1(g) = T2(g) A,    ∀ g∈ G.
If T1=T2=T then A is interntwinig operator or commuting operator for T.
Exercise 33  Let G, H, T(g), and A be as above. Show the following: [333]*§ 1.3.1
  1. Let x∈ H be an eigenvector for A with eigenvalue λ. Then T(g)x for all g∈ G are eigenvectors of A with the same eigenvalue λ.
  2. All eigenvectors of A with a fixed eigenvalue λ for a linear subspace invariant under all T(g), g∈ G.
  3. If an operator A is commuting with irreducible representation T then A=λ I.

Proof.[Hint] Use the spectral decomposition of selfadjoint operators [164]*§ V.2.2.


The next result have very important applications.

Lemma 34 (Schur)  [159]*§ 8.2 If two representations T1 and T2 of a group G are irreducible, then every intertwining operator between them is either zero or invertible.

Proof.[Hint] Consider subspaces kerA⊂ H1 and im A⊂ H2.


Exercise 35  Show that
  1. Two irreducible representations are either equivalent or disjunctive.
  2. All operators commuting with an irreducible representation form a field.
  3. Irreducible representation of commutative group are one-dimensional.
  4. If T is unitary irreducible representation in H and B(·,·) is a bounded semi linear form in H invariant under T: B(T(g)x,T(g)y)=B(x,y) then B(·,·)=λ⟨ ·,· ⟩.

Proof.[Hint] Use that B(·,·)=⟨ A·,· ⟩ for some A [164]*§ III.5.1.


1.4 Induced Representations

The general scheme of induced representations is as follows, see [159]*§ 13.2 [321]*Ch. 5 [105]*Ch. 6 [170]*§ 3.1 and subsection 2.2.2. Let G be a group and let H be its subgroup. Let X=G / H be the corresponding left homogeneous space and s: X → G be a continuous function (section) [159]*§ 13.2 which is a right inverse to the natural projection p:G→ G/H.

Then any g∈ G has a unique decomposition of the form g=s(x)h−1 where x=p(g)∈ X and h∈ H. We define the map r: G→ H:

r(g)=s(x)−1g,     where  x=p(g). (9)

Note that X is a left homogeneous space with the G-action defined in terms of p and s as follows, see Ex. 22:

g: x  ↦ g· x=p(g* s(x)), (10)

where * is the multiplication on G. A useful consequences of the above formulae is:

     
    s(x) = g*s(y)*(r(g*s(y)))−1,        (11)
  r(g−1*s(x)) = r(g*s(y)),    where  y = g−1· x  for  x,y∈ X  and  g∈ G.         (12)

Let χ: H → B(V) be a linear representation of H in a vector space V, e.g. by unitary rotations in the algebra of either complex, dual or double numbers. Then χ induces a linear representation of G, which is known as induced representation in the sense of Mackey [159]*§ 13.2. This representation has the canonical realisation in a space of V-valued functions on X. It is given by the formula (cf. [159]*§ 13.2.(7)–(9)):

  [χ(g) f](x)= χ(r(g−1 * s(x)))  f(g−1· x), (13)

where g∈ G, x∈ X, h∈ H and r: G → H, s: X → G are maps defined above; * denotes multiplication on G and · denotes the action (10) of G on X from the left.

In the case of complex numbers this representation automatically becomes unitary in the space L2(X) of the functions square integrable with respect to a measure dµ if instead of the representation χ one uses the following substitute:

  χ0(h)=χ(h)⎛
⎜
⎜
⎝
dµ(h· x)
dµ(x)
 ⎞
⎟
⎟
⎠
 
1
2
 



 
. (14)

However in our study the unitarity of representations or its proper replacements is a more subtle issue and we will consider it separately.

An alternative construction of induced representations is realised on the space of functions on G which have the following property:

F(gh)=χ(h)F(g),     for all h∈ H.  (15)

This space is invariant under the left shifts. The restriction of the left regular representation to this subspace is equivalent to the induced representation described above.

Exercise 36 
  1. Write the intertwining operator for this equivalence.
  2. Define the corresponding inner product on the space of functions 15 in such a way that the above intertwining operator becomes unitary.

    Proof.[Hint] Use the map s: X → G.


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