11b. Representation theory

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Let us discuss now another advanced algebraic topic, namely the Peter-Weyl theory for the finite groups, and in particular for the permutation groups. The idea here will be that there are several non-trivial things that can be said about the group actions on graphs, [math]G\curvearrowright X[/math], by using the Peter-Weyl theory for finite groups, according to: \begin{principle} Any finite group action on a finite graph [math]G\curvearrowright X[/math], with [math]|X|=N[/math], produces a unitary representation of [math]G[/math], obtained as

[[math]] G\to S_N\subset U_N [[/math]]

that we can decompose and study by using the Peter-Weyl theory for [math]G[/math]. And with this study leading to non-trivial results about the action [math]G\curvearrowright X[/math], and about [math]X[/math] itself. \end{principle} Getting started now, with this program, we first have to forget about the finite graphs [math]X[/math], and develop the Peter-Weyl theory for the finite groups [math]G[/math]. We first have:

Definition

A representation of a finite group [math]G[/math] is a morphism as follows:

[[math]] u:G\to U_N [[/math]]

The character of such a representation is the function [math]\chi:G\to\mathbb C[/math] given by

[[math]] g\to Tr(u_g) [[/math]]

where [math]Tr[/math] is the usual trace of the [math]N\times N[/math] matrices, [math]Tr(M)=\sum_iM_{ii}[/math].

As a basic example here, for any finite group we always have available the trivial 1-dimensional representation, which is by definition as follows:

[[math]] u:G\to U_1\quad,\quad g\to(1) [[/math]]

At the level of non-trivial examples now, most of the groups that we met so far, in chapter 5, naturally appear as subgroups [math]G\subset U_N[/math]. In this case, the embedding [math]G\subset U_N[/math] is of course a representation, called fundamental representation:

[[math]] u:G\subset U_N\quad,\quad g\to g [[/math]]

In this situation, there are many other representations of [math]G[/math], which are equally interesting. For instance, we can define the representation conjugate to [math]u[/math], as being:

[[math]] \bar{u}:G\subset U_N\quad,\quad g\to\bar{g} [[/math]]

In order to clarify all this, and see which representations are available, let us first discuss the various operations on the representations. The result here is as follows:

Proposition

The representations of a finite group [math]G[/math] are subject to:

• Making sums. Given representations [math]u,v[/math], having dimensions [math]N,M[/math], their sum is the [math]N+M[/math]-dimensional representation [math]u+v=diag(u,v)[/math].
• Making products. Given representations [math]u,v[/math], having dimensions [math]N,M[/math], their tensor product is the [math]NM[/math]-dimensional representation [math](u\otimes v)_{ia,jb}=u_{ij}v_{ab}[/math].
• Taking conjugates. Given a representation [math]u[/math], having dimension [math]N[/math], its complex conjugate is the [math]N[/math]-dimensional representation [math](\bar{u})_{ij}=\bar{u}_{ij}[/math].
• Spinning by unitaries. Given a representation [math]u[/math], having dimension [math]N[/math], and a unitary [math]V\in U_N[/math], we can spin [math]u[/math] by this unitary, [math]u\to VuV^*[/math].

Show Proof

The fact that the operations in the statement are indeed well-defined, among maps from [math]G[/math] to unitary groups, is indeed routine, and this gives the result.

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In relation now with characters, we have the following result:

Proposition

We have the following formulae, regarding characters

[[math]] \chi_{u+v}=\chi_u+\chi_v\quad,\quad \chi_{u\otimes v}=\chi_u\chi_v\quad,\quad \chi_{\bar{u}}=\bar{\chi}_u\quad,\quad \chi_{VuV^*}=\chi_u [[/math]]

in relation with the basic operations for the representations.

Show Proof

All these assertions are elementary, by using the following well-known trace formulae, valid for any two square matrices [math]g,h[/math], and any unitary [math]V[/math]:

[[math]] Tr(diag(g,h))=Tr(g)+Tr(h)\quad,\quad Tr(g\otimes h)=Tr(g)Tr(h) [[/math]]

[[math]] Tr(\bar{g})=\overline{Tr(g)}\quad,\quad Tr(VgV^*)=Tr(g) [[/math]]

Thus, we are led to the conclusions in the statement.

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Assume now that we are given a finite group [math]G\subset U_N[/math]. By using the above operations, we can construct a whole family of representations of [math]G[/math], as follows:

Definition

Given a finite group [math]G\subset U_N[/math], its Peter-Weyl representations are the tensor products between the fundamental representation and its conjugate:

[[math]] u:G\subset U_N\quad,\quad \bar{u}:G\subset U_N [[/math]]

We denote these tensor products [math]u^{\otimes k}[/math], with [math]k=\circ\bullet\bullet\circ\ldots[/math] being a colored integer, with the colored tensor powers being defined according to the rules

[[math]] u^{\otimes\circ}=u\quad,\quad u^{\otimes\bullet}=\bar{u}\quad,\quad u^{\otimes kl}=u^{\otimes k}\otimes u^{\otimes l} [[/math]]

and with the convention that [math]u^{\otimes\emptyset}[/math] is the trivial representation [math]1:G\to U_1[/math].

Here are a few examples of such Peter-Weyl representations, namely those coming from the colored integers of length 2, to be often used in what follows:

[[math]] u^{\otimes\circ\circ}=u\otimes u\quad,\quad u^{\otimes\circ\bullet}=u\otimes\bar{u} [[/math]]

[[math]] u^{\otimes\bullet\circ}=\bar{u}\otimes u\quad,\quad u^{\otimes\bullet\bullet}=\bar{u}\otimes\bar{u} [[/math]]

Observe also that the characters of Peter-Weyl representations are given by the following formula, with the powers [math]\chi[/math] being given by [math]\chi^\circ=\chi[/math], [math]\chi^\bullet=\bar{\chi}[/math] and multiplicativity:

[[math]] \chi_{u^{\otimes k}}=(\chi_u)^k [[/math]]

In order now to advance, let us formulate the following key definition:

Definition

Given a finite group [math]G[/math], and two of its representations,

[[math]] u:G\to U_N\quad,\quad v:G\to U_M [[/math]]

we define the linear space of intertwiners between these representations as being

[[math]] Hom(u,v)=\left\{T\in M_{M\times N}(\mathbb C)\Big|Tu_g=v_gT,\forall g\in G\right\} [[/math]]

and we use the following conventions:

• We use the notations [math]Fix(u)=Hom(1,u)[/math], and [math]End(u)=Hom(u,u)[/math].
• We write [math]u\sim v[/math] when [math]Hom(u,v)[/math] contains an invertible element.
• We say that [math]u[/math] is irreducible, and write [math]u\in Irr(G)[/math], when [math]End(u)=\mathbb C1[/math].

The terminology here is very standard, with Hom and End standing for “homomorphisms” and “endomorphisms”, and with Fix standing for “fixed points”. We have:

Theorem

The following happen:

• The intertwiners are stable under composition:
  [[math]] T\in Hom(u,v)\ ,\ S\in Hom(v,w) \implies ST\in Hom(u,w) [[/math]]
• The intertwiners are stable under taking tensor products:
  [[math]] S\in Hom(u,v)\ ,\ T\in Hom(w,t)\\ \implies S\otimes T\in Hom(u\otimes w,v\otimes t) [[/math]]
• The intertwiners are stable under taking adjoints:
  [[math]] T\in Hom(u,v) \implies T^*\in Hom(v,u) [[/math]]
• Thus, the Hom spaces form a tensor [math]*[/math]-category.

Show Proof

All this is clear from definitions, the verifications being as follows:

(1) This follows indeed from the following computation, valid for any [math]g\in G[/math]:

[[math]] STu_g=Sv_gT=w_gST [[/math]]

(2) Again, this is clear, because we have the following computation:

[[math]] \begin{eqnarray*} (S\otimes T)(u_g\otimes w_g) &=&Su_g\otimes Tw_g\\ &=&v_gS\otimes t_gT\\ &=&(v_g\otimes t_g)(S\otimes T) \end{eqnarray*} [[/math]]

(3) This follows from the following computation, valid for any [math]g\in G[/math]:

[[math]] \begin{eqnarray*} Tu_g=v_gT &\implies&u_g^*T^*=T^*v_g^*\\ &\implies&T^*v_g=u_gT^* \end{eqnarray*} [[/math]]

(4) This is just an abstract conclusion of (1,2,3), with a tensor [math]*[/math]-category being by definition an abstract beast satisfying these conditions (1,2,3). We will be back to tensor categories later on in this book, with more details on all this.

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As a main consequence of Theorem 11.19, we have:

Theorem

Given a representation [math]u:G\to U_N[/math], the linear space

[[math]] End(u)\subset M_N(\mathbb C) [[/math]]

is a [math]*[/math]-algebra, with respect to the usual involution of the matrices.

Show Proof

We know from Theorem 11.19 (1) that [math]End(u)[/math] is a subalgebra of [math]M_N(\mathbb C)[/math], and we know as well from Theorem 11.19 (3) that this subalgebra is stable under the involution [math]*[/math]. Thus, what we have here is a [math]*[/math]-subalgebra of [math]M_N(\mathbb C)[/math], as claimed.

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Our claim now is that Theorem 11.20 gives us everything that we need, in order to have some advanced representation theory started, for our finite groups [math]G[/math]. Indeed, we can combine this result with the following standard fact, from matrix algebra:

Theorem

Let [math]A\subset M_N(\mathbb C)[/math] be a [math]*[/math]-algebra.

• We can write [math]1=p_1+\ldots+p_k[/math], with [math]p_i\in A[/math] being central minimal projections.
• The linear spaces [math]A_i=p_iAp_i[/math] are non-unital [math]*[/math]-subalgebras of [math]A[/math].
• We have a non-unital [math]*[/math]-algebra sum decomposition [math]A=A_1\oplus\ldots\oplus A_k[/math].
• We have unital [math]*[/math]-algebra isomorphisms [math]A_i\simeq M_{n_i}(\mathbb C)[/math], with [math]n_i=rank(p_i)[/math].
• Thus, we have a [math]*[/math]-algebra isomorphism [math]A\simeq M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C)[/math].

Show Proof

This is something standard, whose proof is however quite long, as follows:

(1) Consider an arbitrary [math]*[/math]-algebra of the [math]N\times N[/math] matrices, [math]A\subset M_N(\mathbb C)[/math], as in the statement. Let us first look at the center of this algebra, [math]Z(A)=A\cap A'[/math]. It is elementary to prove that this center, as an algebra, is of the following form:

[[math]] Z(A)\simeq\mathbb C^k [[/math]]

Consider now the standard basis [math]e_1,\ldots,e_k\in\mathbb C^k[/math], and let [math]p_1,\ldots,p_k\in Z(A)[/math] be the images of these vectors via the above identification. In other words, these elements [math]p_1,\ldots,p_k\in A[/math] are central minimal projections, summing up to 1:

[[math]] p_1+\ldots+p_k=1 [[/math]]

The idea is then that this partition of the unity will eventually lead to the block decomposition of [math]A[/math], as in the statement. We prove this in 4 steps, as follows:

(2) We first construct the matrix blocks, our claim here being that each of the following linear subspaces of [math]A[/math] are non-unital [math]*[/math]-subalgebras of [math]A[/math]:

[[math]] A_i=p_iAp_i [[/math]]

But this is clear, with the fact that each [math]A_i[/math] is closed under the various non-unital [math]*[/math]-subalgebra operations coming from the projection equations [math]p_i^2=p_i=p_i^*[/math].

(3) We prove now that the above algebras [math]A_i\subset A[/math] are in a direct sum position, in the sense that we have a non-unital [math]*[/math]-algebra sum decomposition, as follows:

[[math]] A=A_1\oplus\ldots\oplus A_k [[/math]]

As with any direct sum question, we have two things to be proved here. First, by using the formula [math]p_1+\ldots+p_k=1[/math] and the projection equations [math]p_i^2=p_i=p_i^*[/math], we conclude that we have the needed generation property, namely:

[[math]] A_1+\ldots+ A_k=A [[/math]]

As for the fact that the sum is indeed direct, this follows as well from the formula [math]p_1+\ldots+p_k=1[/math], and from the projection equations [math]p_i^2=p_i=p_i^*[/math].

(4) Our claim now, which will finish the proof, is that each of the [math]*[/math]-subalgebras [math]A_i=p_iAp_i[/math] constructed above is a full matrix algebra. To be more precise here, with [math]n_i=rank(p_i)[/math], our claim is that we have isomorphisms, as follows:

[[math]] A_i\simeq M_{n_i}(\mathbb C) [[/math]]

In order to prove this claim, recall that the projections [math]p_i\in A[/math] were chosen central and minimal. Thus, the center of each of the algebras [math]A_i[/math] reduces to the scalars:

[[math]] Z(A_i)=\mathbb C [[/math]]

But this shows, either via a direct computation, or via the bicommutant theorem, that the each of the algebras [math]A_i[/math] is a full matrix algebra, as claimed.

(5) We can now obtain the result, by putting together what we have. Indeed, by using the results from (3) and (4), we obtain an isomorphism as follows:

[[math]] A\simeq M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

In addition to this, a careful look at the isomorphisms established in (4) shows that at the global level, that of the algebra [math]A[/math] itself, the above isomorphism simply comes by twisting the following standard multimatrix embedding, discussed in the beginning of the proof, step (2) above, by a certain unitary matrix [math]U\in U_N[/math]:

[[math]] M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C)\subset M_N(\mathbb C) [[/math]]

Now by putting everything together, we obtain the result.

■

We can now formulate our first Peter-Weyl theorem, as follows:

Theorem (PW1)

Let [math]u:G\to U_N[/math] be a representation, consider the algebra [math]A=End(u)[/math], and write its unit as above, with [math]p_i[/math] being central minimal projections:

[[math]] 1=p_1+\ldots+p_k [[/math]]

The representation [math]u[/math] decomposes then as a direct sum, as follows,

[[math]] u=u_1+\ldots+u_k [[/math]]

with each [math]u_i[/math] being an irreducible representation, obtained by restricting [math]u[/math] to [math]Im(p_i)[/math].

Show Proof

This follows from Theorem 11.20 and Theorem 11.21, as follows:

(1) As a first observation, by replacing [math]G[/math] with its image [math]u(G)\subset U_N[/math], we can assume if we want that our representation [math]u[/math] is faithful, [math]G\subset_uU_N[/math]. However, this replacement will not be really needed, and we will keep using [math]u:G\to U_N[/math], as above.

(2) In order to prove the result, we will need some preliminaries. We first associate to our representation [math]u:G\to U_N[/math] the corresponding action map on [math]\mathbb C^N[/math]. If a linear subspace [math]V\subset\mathbb C^N[/math] is invariant, the restriction of the action map to [math]V[/math] is an action map too, which must come from a subrepresentation [math]v\subset u[/math]. This is clear indeed from definitions, and with the remark that the unitaries, being isometries, restrict indeed into unitaries.

(3) Consider now a projection [math]p\in End(u)[/math]. From [math]pu=up[/math] we obtain that the linear space [math]V=Im(p)[/math] is invariant under [math]u[/math], and so this space must come from a subrepresentation [math]v\subset u[/math]. It is routine to check that the operation [math]p\to v[/math] maps subprojections to subrepresentations, and minimal projections to irreducible representations.

(4) With these preliminaries in hand, let us decompose the algebra [math]End(u)[/math] as in Theorem 11.21, by using the decomposition [math]1=p_1+\ldots+p_k[/math] into minimal projections. If we denote by [math]u_i\subset u[/math] the subrepresentation coming from the vector space [math]V_i=Im(p_i)[/math], then we obtain in this way a decomposition [math]u=u_1+\ldots+u_k[/math], as in the statement.

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Here is now our second Peter-Weyl theorem, complementing Theorem 11.22:

Theorem (PW2)

Given a subgroup [math]G\subset_uU_N[/math], any irreducible representation

[[math]] v:G\to U_M [[/math]]

appears inside a tensor product of the fundamental representation [math]u[/math] and its adjoint [math]\bar{u}[/math].

Show Proof

We define the space of coefficients a representation [math]v:G\to U_M[/math] to be:

[[math]] C_v=span\Big[g\to(v_g)_{ij}\Big] [[/math]]

The construction [math]v\to C_v[/math] is then functorial, in the sense that it maps subrepresentations into linear subspaces. Also, we have an inclusion of linear spaces as follows:

[[math]] C_v\subset \lt g_{ij} \gt [[/math]]

On the other hand, by definition of the Peter-Weyl representations, we have:

[[math]] \lt g_{ij} \gt =\sum_kC_{u^{\otimes k}} [[/math]]

Thus, we must have an inclusion as follows, for certain exponents [math]k_1,\ldots,k_p[/math]:

[[math]] C_v\subset C_{u^{\otimes k_1}\oplus\ldots\oplus u^{\otimes k_p}} [[/math]]

We conclude that we have an inclusion of representations, as follows:

[[math]] v\subset u^{\otimes k_1}\oplus\ldots\oplus u^{\otimes k_p} [[/math]]

Together with Theorem 11.22, this leads to the conclusion in the statement.

■

As a third Peter-Weyl theorem, which is something more advanced, we have:

Theorem (PW3)

We have a direct sum decomposition of linear spaces

[[math]] C(G)=\bigoplus_{v\in Irr(G)}M_{\dim(v)}(\mathbb C) [[/math]]

with the summands being pairwise orthogonal with respect to the scalar product

[[math]] \lt a,b \gt =\int_Gab^* [[/math]]

where [math]\int_G[/math] is the averaging over [math]G[/math].

Show Proof

This is something more tricky, the idea being as follows:

(1) By combining the previous two Peter-Weyl results, from Theorem 11.22 and Theorem 11.23, we deduce that we have a linear space decomposition as follows:

[[math]] C(G) =\sum_{v\in Irr(G)}C_v =\sum_{v\in Irr(G)}M_{\dim(v)}(\mathbb C) [[/math]]

Thus, in order to conclude, it is enough to prove that for any two irreducible corepresentations [math]v,w\in Irr(A)[/math], the corresponding spaces of coefficients are orthogonal:

[[math]] v\not\sim w\implies C_v\perp C_w [[/math]]

(2) We will need the basic fact, whose proof is elementary, that for any representation [math]v[/math] we have the following formula, where [math]P[/math] is the orthogonal projection on [math]Fix(v)[/math]:

[[math]] \left(id\otimes\int_G\right)v=P [[/math]]

(3) We will also need the basic fact, whose proof is elementary too, that for any two representations [math]v,w[/math] we have an isomorphism as follows, called Frobenius isomorphism:

[[math]] Hom(v,w)\simeq Fix(v\otimes\bar{w}) [[/math]]

(4) Now back to our orthogonality question from (1), let us set indeed:

[[math]] P_{ia,jb}=\int_Gv_{ij}w_{ab}^* [[/math]]

Then [math]P[/math] is the orthogonal projection onto the following vector space:

[[math]] Fix(v\otimes\bar{w}) \simeq Hom(v,w) =\{0\} [[/math]]

Thus we have [math]P=0[/math], and this gives the result.

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Finally, we have the following result, completing the Peter-Weyl theory:

Theorem (PW4)

The characters of irreducible representations belong to

[[math]] C(G)_{central}=\left\{f\in C(G)\Big|f(gh)=f(hg),\forall g,h\in G\right\} [[/math]]

called algebra of central functions on [math]G[/math], and form an orthonormal basis of it.

Show Proof

We have several things to be proved, the idea being as follows:

(1) Observe first that [math]C(G)_{central}[/math] is indeed an algebra, which contains all the characters. Conversely, consider a function [math]f\in C(G)[/math], written as follows:

[[math]] f=\sum_{v\in Irr(G)}f_v [[/math]]

The condition [math]f\in C(G)_{central}[/math] states then that for any [math]v\in Irr(G)[/math], we must have:

[[math]] f_v\in C(G)_{central} [[/math]]

But this means precisely that the coefficient [math]f_v[/math] must be a scalar multiple of [math]\chi_v[/math], and so the characters form a basis of [math]C(G)_{central}[/math], as stated.

(2) The fact that we have an orthogonal basis follows from Theorem 11.24.

(3) As for the fact that the characters have norm 1, this follows from:

[[math]] \int_G\chi_v\chi_v^* =\sum_{ij}\int_Gv_{ii}v_{jj}^* =\sum_i\frac{1}{N} =1 [[/math]]

Here we have used the fact that the above integrals [math]\int_Gv_{ij}v_{kl}^*[/math] form the orthogonal projection onto the following vector space:

[[math]] Fix(v\otimes\bar{v}) \simeq End(v) =\mathbb C1 [[/math]]

Thus, the proof of our theorem is now complete.

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So long for Peter-Weyl theory. As a comment, our approach here, which was rather functional analytic, was motivated by what we will be doing later in this book, in relation with quantum groups. For a more standard presentation of the Peter-Weyl theory for finite groups, there are many good books available, such as the book of Serre .

General references

Banica, Teo (2024). "Graphs and their symmetries". arXiv:2406.03664 [math.CO].