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We have seen so far the foundations and basic results of classical probability. Before stepping into more complicated things, such as random matrices and free probability, we would like to clarify one important question which appeared several times, namely the computation of integrals over the compact groups of unitary matrices [math]G\subset U_N[/math], and its probabilistic consequences. The precise question that we have in mind is: \begin{question} Given a compact group [math]G\subset U_N[/math], how to compute the integrals

[[math]] I_{ij}^e=\int_Gg_{i_1j_1}^{e_1}\ldots g_{i_kj_k}^{e_k}\,dg [[/math]]

depending on multi-indices [math]i,j[/math], and of a colored integer exponent [math]e=\circ\bullet\bullet\circ\ldots[/math]? Then, how to use this formula in order to compute the laws of variables of type

[[math]] f_P=P\Big(\{g_{ij}\}_{i,j=1,\ldots,N}\Big) [[/math]]

depending on a polynomial [math]P[/math]? What about the [math]N\to\infty[/math] asymptotics of such laws? \end{question} All this is quite subtle, and as a basic illustration for this, we have a fundamental result from chapter 2, stating that for [math]G=S_N[/math] the law of the variable [math]\chi=\sum_ig_{ii}[/math] can be explicitly computed, and becomes Poisson (1) with [math]N\to\infty[/math]. This is something truly remarkable, and it is this kind of result that we would like to systematically have.

We will discuss this in this whole chapter, and later on too. This might seem of course quite long, but believe me, it is worth the effort, because it is quite hard to do any type of advanced probability theory without knowing the answer to Question 4.1. But probably enough advertisement, let us get to work. Following Weyl , we first have:

Definition

A unitary representation of a compact group [math]G[/math] is a continuous group morphism into a unitary group

[[math]] v:G\to U_N\quad,\quad g\to v_g [[/math]]

which can be faithful or not. The character of such a representation is the function

[[math]] \chi:G\to\mathbb C\quad,\quad g\to Tr(v_g) [[/math]]

where [math]Tr[/math] is the usual, unnormalized trace of the [math]N\times N[/math] matrices.

At the level of examples, most of the compact groups that we met so far, finite or continuous, naturally appear as closed 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. In general now, let us first discuss the various operations on the representations. We have here:

Proposition

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

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

Show Proof

The fact that the operations in the statement are indeed well-defined, among morphisms from [math]G[/math] to unitary groups, is indeed clear from definitions.

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

Proposition

We have the following formulae, regarding characters

[[math]] \chi_{v+w}=\chi_v+\chi_w\quad,\quad \chi_{v\otimes w}=\chi_v\chi_w\quad,\quad \chi_{\bar{v}}=\bar{\chi}_v\quad,\quad \chi_{UvU^*}=\chi_v [[/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 square matrices [math]V,W[/math], and any unitary [math]U[/math]:

[[math]] Tr(diag(V,W))=Tr(V)+Tr(W)\quad,\quad Tr(V\otimes W)=Tr(V)Tr(W) [[/math]]

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

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

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Assume now that we are given a closed subgroup [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 closed subgroup [math]G\subset U_N[/math], its Peter-Weyl representations are the various tensor products between the fundamental representation and its conjugate:

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

We denote these tensor products [math]v^{\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]] v^{\otimes\circ}=v\quad,\quad v^{\otimes\bullet}=\bar{v}\quad,\quad v^{\otimes kl}=v^{\otimes k}\otimes v^{\otimes l} [[/math]]

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

Here are a few examples of such representations, namely those coming from the colored integers of length 2, which will often appear in what follows:

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

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

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

Proposition

The characters of the Peter-Weyl representations are given by

[[math]] \chi_{v^{\otimes k}}=(\chi_v)^k [[/math]]

with the colored powers being given by [math]\chi^\circ=\chi[/math], [math]\chi^\bullet=\bar{\chi}[/math] and multiplicativity.

Show Proof

This follows indeed from the additivity, multiplicativity and conjugation formulae from Proposition 4.4, via the conventions in Definition 4.5.

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Getting back now to our motivations, we can see the interest in the above constructions. Indeed, the joint moments of the main character [math]\chi=\chi_v[/math] and its adjoint [math]\bar{\chi}=\chi_{\bar{v}}[/math] are the expectations of the characters of various Peter-Weyl representations:

[[math]] \int_G\chi^k=\int_G \chi_{v^{\otimes k}} [[/math]]

In order to advance, we must develop some general theory. Let us start with:

Definition

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

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

we define the space of intertwiners between these representations as being

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

and we use the following conventions:

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

The terminology here is standard, with Fix, Hom, End standing for fixed points, homomorphisms and endomorphisms. We will see later that irreducible means indecomposable, in a suitable sense. Here are now a few basic results, regarding these spaces:

Proposition

The spaces of intertwiners have the following properties:

• [math]T\in Hom(v,w),S\in Hom(w,z)\implies ST\in Hom(v,z)[/math].
• [math]S\in Hom(v,w),T\in Hom(z,t)\implies S\otimes T\in Hom(v\otimes z,w\otimes t)[/math].
• [math]T\in Hom(v,w)\implies T^*\in Hom(w,v)[/math].

In abstract terms, we say that the Hom spaces form a tensor [math]*[/math]-category.

Show Proof

All the formulae in the statement are indeed clear from definitions, via elementary computations. As for the last assertion, this is something coming from (1,2,3). We will be back to tensor categories later on, with more details on this latter fact.

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As a main consequence of the above result, we have:

Proposition

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

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

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

Show Proof

By definition, [math]End(v)[/math] is a linear subspace of [math]M_N(\mathbb C)[/math]. We know from Proposition 4.8 (1) that this subspace [math]End(v)[/math] is a subalgebra of [math]M_N(\mathbb C)[/math], and then we know as well from Proposition 4.8 (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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In order to exploit the above fact, we will need a basic result from linear algebra, stating that any [math]*[/math]-algebra [math]A\subset M_N(\mathbb C)[/math] decomposes as a direct sum, as follows:

[[math]] A\simeq M_{N_1}(\mathbb C)\oplus\ldots\oplus M_{N_k}(\mathbb C) [[/math]]

Indeed, let us write the unit [math]1\in A[/math] as [math]1=p_1+\ldots+p_k[/math], with [math]p_i\in A[/math] being central minimal projections. Then each of the spaces [math]A_i=p_iAp_i[/math] is a subalgebra of [math]A[/math], and we have a decomposition [math]A=A_1\oplus\ldots\oplus A_k[/math]. But since each central projection [math]p_i\in A[/math] was chosen minimal, we have [math]A_i\simeq M_{N_i}(\mathbb C)[/math], with [math]N_i=rank(p_i)[/math], as desired.

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

Theorem (Peter-Weyl 1)

Let [math]v:G\to U_N[/math] be a representation, consider the algebra [math]A=End(v)[/math], and write its unit [math]1=p_1+\ldots+p_k[/math] as above. We have then

[[math]] v=v_1+\ldots+v_k [[/math]]

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

Show Proof

This basically follows from Proposition 4.9, as follows:

(1) We first associate to our representation [math]v:G\to U_N[/math] the corresponding action map on [math]\mathbb C^N[/math]. If a linear subspace [math]W\subset\mathbb C^N[/math] is invariant, the restriction of the action map to [math]W[/math] is an action map too, which must come from a subrepresentation [math]w\subset v[/math].

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

(3) With these preliminaries in hand, let us decompose the algebra [math]End(v)[/math] as above, by using the decomposition [math]1=p_1+\ldots+p_k[/math] into central minimal projections. If we denote by [math]v_i\subset v[/math] the subrepresentation coming from the vector space [math]V_i=Im(p_i)[/math], then we obtain in this way a decomposition [math]v=v_1+\ldots+v_k[/math], as in the statement.

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Here is now our second Peter-Weyl theorem, complementing Theorem 4.10:

Theorem (Peter-Weyl 2)

Given a closed subgroup [math]G\subset_vU_N[/math], any of its irreducible smooth representations

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

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

Show Proof

Given a representation [math]w:G\to U_M[/math], we define the space of coefficients [math]C_w\subset C(G)[/math] of this representation as being the following linear space:

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

With this notion in hand, the result can be deduced as follows:

(1) The construction [math]w\to C_w[/math] is functorial, in the sense that it maps subrepresentations into linear subspaces. This is indeed something which is routine to check.

(2) A closed subgroup [math]G\subset_vU_N[/math] is a Lie group, and a representation [math]w:G\to U_M[/math] is smooth when we have an inclusion [math]C_w\subset \lt C_v \gt [/math]. This is indeed well-known.

(3) By definition of the Peter-Weyl representations, as arbitrary tensor products between the fundamental representation [math]v[/math] and its conjugate [math]\bar{v}[/math], we have:

[[math]] \lt C_v \gt =\sum_kC_{v^{\otimes k}} [[/math]]

(4) Now by putting together the above observations (2,3) we conclude that we must have an inclusion as follows, for certain exponents [math]k_1,\ldots,k_p[/math]:

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

(5) By using now (1), we deduce that we have an inclusion [math]w\subset v^{\otimes k_1}\oplus\ldots\oplus v^{\otimes k_p}[/math], and by applying Theorem 4.10, this leads to the conclusion in the statement.

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General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].