A First Impression of Group Representations

This blog post provides mostly some motivation, basic definitions and examples for group representations, up to Maschke’s theorem. Only familiarity with linear algebra and elementary group theory is required for understanding the main part of this post. However, there are some examples for readers with more background which can be safely ignored. The same is true for all categorical remarks.

This will be the first part of a series.

Introduction

The reason to care about groups is because they act on objects. Group actions arise in many different contexts and can provide insight (into the objects as well as into the groups which act on them).

The basic definition of a group action is an action on a set. A set can be thought of as an object without any structure except size (i.e. cardinality). For finite sets, this is more structure than it might seem: we can use counting and divisibility arguments which leads to results such as strong results on p-groups such as the Sylow theorems, the easy-to-prove but ubiquitous orbit-stabilizer theorem or Burnside’s lemma, which has nontrivial combinatorial implications.

Group actions are everywhere in pure mathematics and frequently the object which is acted upon has more structure than just being a set, e.g. a topological space. In these situations, the natural thing to investigate (or to require, if you’re writing a definition) is some compatibility between the group action and the structure on the object. In the example of a topological space, one would require the action to be continuous.

A common structure are vector spaces. Their usefulness seems to stem from the fact that they are both very well understood and still give one a lot of tools to work with: we have duals, tensor products, traces, determinants, eigenvalues etc. Thus, a technique that is used sometimes is “linearization” in which one tries to reduce a problem to linear algebra or at least gain insight by using linear-algebraic methods. Examples are the tangent space of a smooth manifold or variety (and for smooth maps the derivative) or the linearization of a nonlinear ODE.

Representations of groups can be thought of as a linearization of group theory, or more precisely as a linearization of group actions: One can be define them as actions of groups on vector spaces that respect the linear structure. They arise with an ubiquity comparable to (nonlinear, merely “set-theoretic”) group actions. When they do, they are arguably even more useful, since vector spaces have a lot more structure than sets, as described in the previous paragraph.

Group Representations

Definition 1.1 Let K be a field. Then for a group G, a (K-linear) representation on a K-vector space V is a map \varphi: G \times V \to V that satisfies:

  • \forall v \in V: \varphi(e,v)=v where e\in G is the neutral element.
  • \forall g,h \in G, v \in V: \varphi(gh,v)=\varphi(g,\varphi(h,v))
  • \forall g \in G, v,w \in V : \varphi(g,v+w)=\varphi(g,v)+\varphi(g,w)
  • \forall g \in G, \lambda \in K, v \in V: \varphi(g,\lambda v)=\lambda \varphi(g,v)

Definition 1.2 In the above situation, the degree or dimension of the representation is the dimension of V over K.

Note that the first two axioms state that a representation is a group action and the other two axioms state that for any fixed g \in G, the map V \to V, v \mapsto \varphi(g,v) is K-linear. This map is also invertible, since the axioms imply that v \mapsto \varphi(g^{-1},v) is an inverse. By the second axiom, we also get that the map G \to \mathrm{GL}(V), g \mapsto (v \mapsto \varphi(g,v)) is a group homomorphism. So by using a currying argument, we have seen that every group representation gives rise to a group homomorphism G \to \mathrm{GL}(V).

Conversely, given a group homomorphism \rho:G \to \mathrm{GL}(V), we can uncurry to get a representation on V by setting \varphi(g,v) := \rho(g)(v)

Thus we have another characterization, which allows us to apply concepts defined for group homomorphisms like kernel/image etc, whereas the first characterization allows us to apply notions defined for group actions.

While we’re at it, we may as well add a rephrasing in categorical language of the last characterization and get the following

Lemma 1.3  Let K be a field, then for a group G and a vector space V over K, the following data are equivalent:

  • A representation of G on V as in definition 1.1
  • A group homomorphism G \to \mathrm{GL}(V)
  • A covariant functor from G considered as a one-object category to the category K\textrm{-}\mathbf{Mod} of K-vector spaces that sends the single object in G to V

This is the direct analog of equivalent characterizations of group actions: we can also view them as homomorphisms to the group of permutations of a set or as functors from a group to the category of sets.

(Viewing a group as a category works like this: Let G be a group, then define a category \mathcal{C} with a single object * and set \mathrm{Hom}_{\mathcal{C}}(*,*)=G. Composition \mathrm{Hom}_{\mathcal{C}}(*,*) \times \mathrm{Hom}_{\mathcal{C}}(*,*) \to \mathrm{Hom}_{\mathcal{C}}(*,*), i.e. G \times G \to G is just group multiplication. Associativity and having an identity follows from the group axioms.)

The last characterization allows one to apply constructions from category theory, such as composing representations with other functors, but we will not use it in this post except as an alternative descrition.

Remark 1.4 One can also consider representations of monoids or the case where K is any ring and V is a module.

Remark 1.5 We have defined only left representations. Reversing the chirality in the definitions is straightforward and gives rise to the notion of right representations.

Now for some examples.

As a zeroth example, note that representations of the trivial group are just vector spaces.

Example 1.6 Let G=\langle g \rangle be a finite cyclic group of order n with a fixed generator g, then a homomorphism from G to any group H is determined by where it sends g and we can send g to precisely those elements h \in H such that h^n=1, so a representation of G is just a linear automorphism that satisfies this equation.
For K=\mathbb{R}, one can take for example a rotation matrix \begin{pmatrix} \cos(2\pi/n) & -\sin(2\pi/n) \\ \sin(2\pi/n) & \cos(2\pi/n) \end{pmatrix} to define a two-dimensional representation of G.
For K=\mathbb{C} one can take \zeta_n=\exp{2\pi i/ n} \in \mathbb{C}^\times = \mathrm{GL}_1(\mathbb{C}) to get a one-dimensional representation of G. Both representations correspond to having g^k act by a counterclockwise rotation of 2\pi k/n degrees. The former can be obtained from the latter by restricting scalars from \mathbb{C} to \mathbb{R}.

Example 1.7 Let G=K, the additive group of K, then G acts on V= K^2 via transvections: \phi\left(\lambda,\begin{pmatrix}a\\b\end{pmatrix}\right)=\begin{pmatrix}a+b\lambda\\b\end{pmatrix}

Example 1.8 If X is a G-set (a set equipped with an action from G), then we can consider a vector space V such that the basis elements e_x are indexed by X. We can then define the action on the basis elements by setting \varphi(g,e_x)=e_{gx}. This permutation of the basis elements extends uniquely to a linear automorphism of V and we get a respresentation, called the permutation representation associated to the group action.
From a categorical standpoint, if we view G-sets as functors G \to \mathbf{Set} and representations as functors G \to  K\textrm{-} \mathbf{Mod}, then this construction is just composing a group action with the free module functor \mathbf{Set} \to K\textrm{-} \mathbf{Mod}. (Thus this construction is also functorial with the notion of morphisms of representations to be defined later.)

Example 1.9 Let V be any K vector space. Let G=\mathrm{GL}(V). Then the identity G \to \mathrm{GL}(V) defines a representation. This corresponds to the natural action of G on V that comes from the definition of G as K-linear automorphisms.

Example 1.10 Generalizing the last example, all classical matrix groups such as \mathrm{SL}_n, \mathrm{O}_n, \mathrm{U}_n, \mathrm{Sp}_{2n} etc. are defined as subgroups of some general linear group, so that the subgroup inclusion defines a representation.

Example 1.11 Let G be a finite group and let p be a prime number. Suppose H is a normal abelian subgroup of exponent p. Then H is a K=\mathbb{F}_p vector space and the conjugation action of G on H is \mathbb{F}_p-linear (this is automatic: any group homomorphism between vector spaces over \mathbb{F}_p is \mathbb{F}_p-linear.), thus we obtain a \mathbb{F}_p-linear representation.

Example 1.12 Let L/K be a Galois extension. Then G= \mathrm{Gal}(L/K) acts by definition on L by K-linear field automorphisms. We can just forget the “field automorphism” part and consider V=L just as a K-vector space, then we get a representation \mathrm{Gal}(L/K) \to \mathrm{GL}_K(L).
If L and K are number fields with rings of integers \mathcal{O}_K and \mathcal{O}_L and \mathfrak{p} is a non-zero prime ideal in \mathcal{O}_K, then then G acts on \mathcal{O}_L/\mathfrak{p}\mathcal{O}_L, giving a \mathcal{O}_K/\mathfrak{p}-linear representation.

Example 1.13 Let G=S_n and let W be any vector space over K, then G acts on V = W \otimes_K \dots \otimes_K W where we tensor n copies of W. Then G acts on V by permuting the components of the tensors. Explicitly, if w_1 \otimes w_2 \otimes \dots \otimes w_n is an elementary tensor, then we can define the result of the action of \sigma \in S_n on that to be w_{\sigma(1)} \otimes w_{\sigma(2)} \otimes \dots \otimes w_{\sigma(n)}.

Example 1.14 Let A be a finite-dimensional K-algebra, then the group of units G=A^\times acts on A by conjugation and this action is K-linear, and thus we obtain a representation of the unit group on the underlying vector space of the algebra. (If K=\mathbb{R} and A=\mathbb{H}, then a suitable restriction of the domain and codomain of this representation gives a description of the Hopf fibration.)

Example 1.15 In the same spirit as the last example, let G be an algebraic group over K. Let V= \mathfrak{g}=T_e(G) be the Lie algebra of G. For any g \in G, the conjugation map c_g:G \to G, a \mapsto gag^{-1} is a smooth automorphism of G, so we can take the derivative at the identity and get a linear automorphism \mathrm{ad}(g)=D_e(c_g): V \to V. The map g \mapsto \mathrm{ad}(g) \in \mathrm{GL}(V) is a representation, called the adjoint representation of G. (The same construction works verbatim for Lie groups.)

Example 1.16 Let M be a smooth connected manifold and let \pi:E \to M be a vector bundle with a flat connection. Let x \in M be a base point and set G=\pi_1(M,x) and K=\mathbb{R}.  If we take a smooth loop \gamma: S^1 \to M based at x, parallel transport along that loop defines an automorphism of V=T_xM.
The flatness condition implies that this automorphism depends only on the homotopy class of \gamma and by smooth approximation, every homotopy class of continuous loops may be represented by a smooth loop, thus we obtain the holonomy representation \pi_1(M,x) \to \mathrm{GL}(T_xM).  It turns out that this representation uniquely determines the flat bundle.

As is common practice with group actions, if \rho:G \to \mathrm{GL}(V) is a representation, we also write just gv for \rho(g)v or g for the map \rho(g). By further abuse of notation, we will also just call V a representation of G where the action is clear from the context.

Definition 1.17 If V and W are K-linear representations of a group G for some field K, then a morphism of representations (also called intertwining operator) from V to W is a K-linear map f:V \to W such that \forall g \in G, v \in V: f(gv)=gf(v). (i.e. f is G-equivariant.)

Note that if we consider representations as functors, then a morphism of representations is just a natural transformation. Indeed, for any g \in G, naturality with respect to g as a morphism is precisely the requirement that f(gv) = gf(v) for all v.

Example 1.17 In the situation of example 1.13, let f \in \mathrm{GL}(W), then we can define f^{\otimes n} by acting on each factor: f^{\otimes n}(v_1 \otimes \dots \otimes v_n)=f(v_1) \otimes \dots \otimes f(v_n) for an elementary tensor. Since we act in the same way in each component, this commutes with permutation of the factors, thus f^{\otimes n}: V \to V defines a morphism of the representation of S_n  given by permuting the factors in the tensor product.

Definition 1.18 If V is a representation of G, then a subspace W that is G-invariant (i.e. gW \subset W for all g \in G) defines again a representation of G. These subspaces are called subrepresentations of V.

If W is a subrepresentation of V, then the inclusion is a morphism of representations, which gives a (quite general) family of examples for morphisms.

Example 1.19 In the situation of example 1.7, consider the subspace of K^2 spanned by \begin{pmatrix}1\\0\end{pmatrix}, this is a subrepresentation because \varphi\left(\lambda,\begin{pmatrix}a\\0\end{pmatrix}\right)=\begin{pmatrix}a\\0\end{pmatrix}.

Example 1.20 Given a morphism of representations, the kernel and the image are subrepresentations of the domain and codomain, respectively.

Example 1.21 In the situation of example 1.8, suppose that Y \subset X is a sub G-set, i.e. we have gY \subset Y for all g \in G, then Y is itself a G-set and if we apply the same construction to Y, the resulting vector space is a subspace of V in a canonical way, and so also a subrepresentation. (This is a special case of the mentioned functoriality of this construction.)
If X is finite, another subrepresentaion is given by the span of \sum_{x \in G} e_x.

We now come to the first substantial theorem about representations.

Theorem 1.22 (Maschke) Let G be a finite group and suppose that the order |G| is invertible in K. Then if V is a finite-dimensional representation and W \leq V is a subrepresentation, then there exists another subrepresentation C such that V=W\oplus C,.

Proof By linear algebra, we can find a K-linear projection \pi: V \to W, i.e. we have that \mathrm{im}(\pi)\subset W and \pi is the identity on W. We have that V= W \oplus \mathrm{ker}(\pi), but of course, \pi will not be a morphism of representations in general. The idea is to “average” \pi to get another projection onto W that is a morphism of representations.
Set \pi'(v)=\frac{1}{|G|}\sum_{g \in G}g\pi(g^{-1}v) (Here we use that |G| is invertible in K). This will be K-linear again. This is a morphism of representations, as for h \in G, we have \pi'(hv)=\frac{1}{|G|} \sum_{g \in G}h\pi(g^{-1}hv)=\frac{1}{|G|}\sum_{g \in G} hg\pi(g^{-1}v)=h(\frac{1}{|G|} \sum_{g \in G} g\pi(g^{-1}v))=h\pi'(v). Since W is a subrepresentation and \pi is the identity on W, \pi' is also the identity on W (it is crucial that we divided by |G| for this step.) and the image is also contained in W, so \pi' is still a projection onto W.
Therefore, the kernel is a complement of W and as \pi' is a morphism of representations, the kernel is a subrepresentation.

Example 1.23 To show that the assumptions in Maschke’s theorem are necessary, consider the transvection representation of the additive group of K on K^2 described in example 1.7 and 1.19. Here K acts via \varphi\left(\lambda,\begin{pmatrix}a\\b\end{pmatrix}\right)=\begin{pmatrix}a+\lambda b\\b\end{pmatrix}. As described in example 1.19, the subspace W of vectors in K^2 with second component 0 is a subrepresentation.
But this subrepresentation doesn’t have a complement that is also a subrepresentation: Indeed, if \begin{pmatrix}a\\b \end{pmatrix} is any vector in K^2 such that b \neq 0, then \begin{pmatrix}a\\b \end{pmatrix} and \varphi\left(1,\begin{pmatrix}a\\b \end{pmatrix}\right)=\begin{pmatrix}a+b\\b \end{pmatrix} are linearly independent, as they are clearly not multiples of each other. Thus any subrepresentation that is not contained in W is the whole of V, so W doesn’t have a complement.
This serves as a counterexample in two different ways: if we take K to be a finite field, it shows that the assumption that the order is invertible is necessary. If we take K to be an infinite field (say of characteristic 0), then it shows that even in characteristic 0, the conclusion doesn’t need to hold when the group is infinite.

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