Reflection groups

[math] \newcommand{\mathds}{\mathbb}[/math]

12a. Real reflections

We have seen in the previous chapter that some interesting phenomena, in relation with the law of the main character, appear for the symmetric group [math]S_N[/math], in the [math]N\to\infty[/math] limit. Moreover, we have seen as well that the same kind of conclusions, with the law of the main character becoming Poisson (1), and the laws of the truncated characters becoming Poisson ([math]t[/math]), in the [math]N\to\infty[/math] limit, happen for the alternating subgroup [math]A_N\subset S_N[/math].

All this suggests systematically looking at the general series of complex reflection groups [math]H_N^{sd}[/math]. However, things are quite technical here, and we will do this slowly. First we will study the hyperoctahedral group [math]H_N[/math]. Then we will develop some more general probabilistic theory, in relation with the Poisson laws and their versions. Then we will discuss the groups [math]H_N^s[/math], generalizing both [math]S_N,H_N[/math]. And finally, we will comment on the groups [math]H_N^{sd}[/math], and we will do some new computations for the continuous groups as well.

Summarizing, a lot of things to be done. And you might perhaps ask, is this obsession with computing laws of characters justified? And our answer is that yes it is, the precise reasons behind our obsession, and motivations in general, being as follows:

(1) First, we will do this for the fun. Most of the material here will be research-grade, based on the papers , , , written in the late 00s. And isn't this nice, to read about what researchers are doing, not long after learning what a [math]2\times2[/math] matrix is.

(2) All these character computations will be an excellent introduction to the somewhat heavy representation theory methods to be developed in chapters 13-16 below, following Weyl , , , and then Brauer , Weingarten and others.

(3) And finally, again back to research, all this will be as well an introduction to all sorts of exciting things, such as random matrices and free probability following Wigner , Marchenko-Pastur and Voiculescu , quantum groups, and more.

In short, trust us, we will be doing some first-class mathematics in this chapter, of rather applied and computational type. And for abstractions and everything, don't worry, they will come back in chapters 13-16 below, inspired by what we will be doing here.

Getting started now, let us begin by discussing the hyperoctahedral group [math]H_N[/math]. We first recall, from chapter 9, that we have the following result:

Theorem

Consider the hyperoctahedral group [math]H_N[/math], which appears as the symmetry group of the [math]N[/math]-cube, or the symmetry group of the [math]N[/math] coordinate axes of [math]\mathbb R^N[/math]:

[[math]] S_N\subset H_N\subset O_N [[/math]]

In matrix terms, [math]H_N[/math] consists of the permutation-type matrices having [math]\pm1[/math] as nonzero entries, and we have a wreath product decomposition as follows:

[[math]] H_N=\mathbb Z_2\wr S_N [[/math]]

In this picture, the main character counts the signed number of fixed points, among the coordinate axes, and its truncations count the truncations of such numbers.

Show Proof

This is something that was discussed before, the idea being that the first assertions are clear, and that the wreath product decomposition in the statement comes from a crossed product decomposition [math]H_N=\mathbb Z_2^N\rtimes S_N[/math]. As for the assertions regarding the main character and its truncations, once again these are clear, as for [math]S_N[/math].

■

Regarding now the character laws, we can compute them by using the same method as for the symmetric group [math]S_N[/math], namely inclusion-exclusion, and we have:

Theorem

For the hyperoctahedral group [math]H_N\subset O_N[/math], the law of the variable

[[math]] \chi_t=\sum_{i=1}^{[tN]}g_{ii} [[/math]]

becomes in the [math]N\to\infty[/math] limit the measure

[[math]] b_t=e^{-t}\sum_{k=-\infty}^\infty\delta_k\sum_{p=0}^\infty \frac{(t/2)^{|k|+2p}}{(|k|+p)!p!} [[/math]]

where [math]\delta_k[/math] is the Dirac mass at [math]k\in\mathbb Z[/math].

Show Proof

We follow . We regard [math]H_N[/math] as being the symmetry group of the graph [math]I_N=\{I^1,\ldots ,I^N\}[/math] formed by [math]N[/math] segments. The diagonal coefficients are given by:

[[math]] u_{ii}(g)=\begin{cases} \ 0\ \mbox{ if $g$ moves \ltmath\gtI^i[[/math]]

}\\ \ 1\ \mbox{ if [math]g[/math] fixes [math]I^i[/math]}\\ -1\mbox{ if [math]g[/math] returns [math]I^i[/math]} \end{cases} </math>

We denote by [math]\uparrow g,\downarrow g[/math] the number of segments among [math]\{I^1,\ldots ,I^s\}[/math] which are fixed, respectively returned by an element [math]g\in H_N[/math]. With this notation, we have:

[[math]] u_{11}+\ldots+u_{ss}=\uparrow g-\downarrow g [[/math]]

Let us denote by [math]P_N[/math] probabilities computed over the group [math]H_N[/math]. The density of the law of [math]u_{11}+\ldots+u_{ss}[/math] at a point [math]k\geq 0[/math] is then given by the following formula:

[[math]] \begin{eqnarray*} D(k) &=&P_N(\uparrow g-\downarrow g=k)\\ &=&\sum_{p=0}^\infty P_N(\uparrow g=k+p, \downarrow g=p) \end{eqnarray*} [[/math]]

Assume first that we have [math]t=1[/math]. We use the fact, that we know well from chapter 11, that the probability of [math]\sigma\in S_N[/math] to have no fixed points is asymptotically given by:

[[math]] P_0=\frac{1}{e} [[/math]]

Thus the probability of [math]\sigma\in S_N[/math] to have [math]m[/math] fixed points is asymptotically given by:

[[math]] P_m=\frac{1}{em!} [[/math]]

In terms of probabilities over [math]H_N[/math], we obtain from this, as desired:

[[math]] \begin{eqnarray*} \lim_{N\to\infty}D(k) &=&\lim_{N\to\infty}\sum_{p=0}^\infty(1/2)^{k+2p}\begin{pmatrix}k+2p\\ k+p\end{pmatrix} P_N(\uparrow g+\downarrow g=k+2p)\\ &=&\sum_{p=0}^\infty(1/2)^{k+2p}\begin{pmatrix}k+2p\\ k+p\end{pmatrix}\frac{1}{e(k+2p)!}\\ &=&\frac{1}{e}\sum_{p=0}^\infty \frac{(1/2)^{k+2p}}{(k+p)!p!} \end{eqnarray*} [[/math]]

As for the general case [math]0 \lt t\leq 1[/math], here the result follows by performing some modifications in the above computation. The asymptotic density is computed as follows:

[[math]] \begin{eqnarray*} \lim_{N\to\infty}D(k) &=&\lim_{N\to\infty}\sum_{p=0}^\infty(1/2)^{k+2p}\begin{pmatrix}k+2p\\ k+p\end{pmatrix} P_N(\uparrow g+\downarrow g=k+2p)\\ &=&\sum_{p=0}^\infty(1/2)^{k+2p}\begin{pmatrix}k+2p\\ k+p\end{pmatrix}\frac{t^{k+2p}}{e^t(k+2p)!}\\ &=&e^{-t}\sum_{p=0}^\infty \frac{(t/2)^{k+2p}}{(k+p)!p!} \end{eqnarray*} [[/math]]

Together with [math]D(-k)=D(k)[/math], this gives the formula in the statement.

■

The above result is quite interesting, because the densities there are the Bessel functions of the first kind. Due to this fact, the limiting measures are called Bessel laws:

Definition

The Bessel law of parameter [math]t \gt 0[/math] is the measure

[[math]] b_t=e^{-t}\sum_{k=-\infty}^\infty\delta_k\,f_k(t/2) [[/math]]

with the density being the Bessel function of the first kind:

[[math]] f_k(t)=\sum_{p=0}^\infty \frac{t^{|k|+2p}}{(|k|+p)!p!} [[/math]]

Let us study now these Bessel laws. We first have the following result:

Theorem

The Bessel laws [math]b_t[/math] have the property

[[math]] b_s*b_t=b_{s+t} [[/math]]

so they form a truncated one-parameter semigroup with respect to convolution.

Show Proof

We follow . We use the formula in Definition 12.3, namely:

[[math]] b_t=e^{-t}\sum_{k=-\infty}^\infty\delta_k\,f_k(t/2) [[/math]]

The Fourier transform of this measure is given by:

[[math]] Fb_t(y)=e^{-t}\sum_{k=-\infty}^\infty e^{ky}\,f_k(t/2) [[/math]]

We compute now the derivative with respect to [math]t[/math]:

[[math]] Fb_t(y)'=-Fb_t(y)+\frac{e^{-t}}{2}\sum_{k=-\infty}^\infty e^{ky}\,f_k'(t/2) [[/math]]

On the other hand, the derivative of [math]f_k[/math] with [math]k\geq 1[/math] is given by:

[[math]] \begin{eqnarray*} f_k'(t) &=&\sum_{p=0}^\infty \frac{(k+2p)t^{k+2p-1}}{(k+p)!p!}\\ &=&\sum_{p=0}^\infty \frac{(k+p)t^{k+2p-1}}{(k+p)!p!}+\sum_{p=0}^\infty\frac{p\,t^{k+2p-1}}{(k+p)!p!}\\ &=&\sum_{p=0}^\infty \frac{t^{k+2p-1}}{(k+p-1)!p!}+\sum_{p=1}^\infty\frac{t^{k+2p-1}}{(k+p)!(p-1)!}\\ &=&\sum_{p=0}^\infty \frac{t^{(k-1)+2p}}{((k-1)+p)!p!}+\sum_{p=1}^\infty\frac{t^{(k+1)+2(p-1)}}{((k+1)+(p-1))!(p-1)!}\\ &=&f_{k-1}(t)+f_{k+1}(t) \end{eqnarray*} [[/math]]

This computation works in fact for any [math]k[/math], so we get:

[[math]] \begin{eqnarray*} Fb_t(y)' &=&-Fb_t(y)+\frac{e^{-t}}{2} \sum_{k=-\infty}^\infty e^{ky} (f_{k-1}(t/2)+f_{k+1}(t/2))\\ &=&-Fb_t(y)+\frac{e^{-t}}{2} \sum_{k=-\infty}^\infty e^{(k+1)y}f_{k}(t/2)+e^{(k-1)y}f_{k}(t/2)\\ &=&-Fb_t(y)+\frac{e^{y}+e^{-y}}{2}\,Fb_t(y)\\ &=&\left(\frac{e^{y}+e^{-y}}{2}-1\right)Fb_t(y) \end{eqnarray*} [[/math]]

Thus the log of the Fourier transform is linear in [math]t[/math], and we get the assertion.

■

In order to further discuss all this, we will need a number of probabilistic preliminaries. We recall that, conceptually speaking, the Poisson laws are the laws appearing via the Poisson Limit Theorem (PLT). In order to generalize this construction, as to cover for instance for Bessel laws that we found in connection with the hyperoctahedral group [math]H_N[/math], we have the following notion, extending the Poisson limit theory from chapter 11:

Definition

Associated to any compactly supported positive measure [math]\nu[/math] on [math]\mathbb{R}[/math] is the probability measure

[[math]] p_\nu=\lim_{n\to\infty}\left(\left(1-\frac{c}{n}\right)\delta_0+\frac{1}{n}\nu\right)^{*n} [[/math]]

where [math]c=mass(\nu)[/math], called compound Poisson law.

In other words, what we are doing here is to generalize the construction in the Poisson Limit Theorem, by allowing the only parameter there, which was the positive real number [math]t \gt 0[/math], to be replaced by a certain probability measure [math]\nu[/math], of arbitrary mass [math]c \gt 0[/math].

In what follows we will be interested in the case where [math]\nu[/math] is discrete, as is for instance the case for the measure [math]\nu=t\delta_1[/math] with [math]t \gt 0[/math], which produces via the above procedure the Poisson laws. To be more precise, we will be mainly interested in the case where [math]\nu[/math] is a multiple of the uniform measure on the [math]s[/math]-th roots of unity. More on this later.

The following result allows us to detect compound Poisson laws:

Proposition

For a discrete measure, [math]\nu=\sum_{i=1}^sc_i\delta_{z_i}[/math] with [math]c_i \gt 0[/math] and [math]z_i\in\mathbb R[/math], we have the formula

[[math]] F_{p_\nu}(y)=\exp\left(\sum_{i=1}^sc_i(e^{iyz_i}-1)\right) [[/math]]

where [math]F[/math] denotes as usual the Fourier transform.

Show Proof

Let [math]\mu_n[/math] be the measure appearing in Definition 12.5, namely:

[[math]] \mu_n=\left(1-\frac{c}{n}\right)\delta_0+\frac{1}{n}\nu [[/math]]

We have the following computation, in the context of Definition 12.5:

[[math]] \begin{eqnarray*} &&F_{\mu_n}(y)=\left(1-\frac{c}{n}\right)+\frac{1}{n}\sum_{i=1}^sc_ie^{iyz_i}\\ &\implies&F_{\mu_n^{*n}}(y)=\left(\left(1-\frac{c}{n}\right)+\frac{1}{n}\sum_{i=1}^sc_ie^{iyz_i}\right)^n\\ &\implies&F_{p_\nu}(y)=\exp\left(\sum_{i=1}^sc_i(e^{iyz_i}-1)\right) \end{eqnarray*} [[/math]]

Thus, we have obtained the formula in the statement.

■

We have as well the following result, providing an alternative to Definition 12.5, and which will be our formulation of the Compound Poisson Limit Theorem (CPLT):

Theorem

For a discrete measure, written as

[[math]] \nu=\sum_{i=1}^sc_i\delta_{z_i} [[/math]]

with [math]c_i \gt 0[/math] and [math]z_i\in\mathbb R[/math], we have the formula

[[math]] p_\nu={\rm law}\left(\sum_{i=1}^sz_i\alpha_i\right) [[/math]]

where the variables [math]\alpha_i[/math] are Poisson [math](c_i)[/math], independent.

Show Proof

Let [math]\alpha[/math] be the sum of Poisson variables in the statement:

[[math]] \alpha=\sum_{i=1}^sz_i\alpha_i [[/math]]

By using some well-known Fourier transform formulae, we have:

[[math]] \begin{eqnarray*} F_{\alpha_i}(y)=\exp(c_i(e^{iy}-1) &\implies&F_{z_i\alpha_i}(y)=\exp(c_i(e^{iyz_i}-1))\\ &\implies&F_\alpha(y)=\exp\left(\sum_{i=1}^sc_i(e^{iyz_i}-1)\right) \end{eqnarray*} [[/math]]

Thus we have the same formula as in Proposition 12.6, as desired.

■

Getting back now to the Bessel laws, we have the following result:

Theorem

The Bessel laws [math]b_t[/math] are compound Poisson laws, given by

[[math]] b_t=p_{t\varepsilon} [[/math]]

where [math]\varepsilon=\frac{1}{2}(\delta_{-1}+\delta_1)[/math] is the uniform measure on [math]\mathbb Z_2[/math].

Show Proof

This follows indeed by comparing the formula of the Fourier transform of [math]b_t[/math], from the proof of Theorem 12.4, with the formula in Proposition 12.6.

■

As a conclusion to this, when discussing the asymptotic character law for the basic finite subgroups [math]G\subset U_N[/math], such as [math]G=S_N,H_N[/math], it is all about compound Poisson laws.

12b. Complex reflections

Our next task will be that of generalizing the results that we have for [math]S_N,H_N[/math]. For this purpose, consider the following family of groups, that we met in chapter 10:

Definition

The complex reflection group [math]H_N^s\subset U_N[/math], depending on parameters

[[math]] N\in\mathbb N\quad,\quad s\in\mathbb N\cup\{\infty\} [[/math]]

are the groups of permutation-type matrices with [math]s[/math]-th roots of unity as entries,

[[math]] H_N^s=M_N(\mathbb Z_s\cup\{0\})\cap U_N [[/math]]

with the convention [math]\mathbb Z_\infty=\mathbb T[/math], at [math]s=\infty[/math].

Observe that at [math]s=1,2[/math] we obtain the symmetric and hyperoctahedral groups:

[[math]] H_N^1=S_N\quad,\quad H_N^2=H_N [[/math]]

Another important particular case is [math]s=\infty[/math], where we obtain a compact group which is actually not finite, but is of key importance, that we will denote as follows:

[[math]] H_N^\infty=K_N [[/math]]

In order to do now the character computations for [math]H_N^s[/math], in general, we need a number of further probabilistic preliminaries. Let us start with the following definition:

Definition

The Bessel law of level [math]s\in\mathbb N\cup\{\infty\}[/math] and parameter [math]t \gt 0[/math] is

[[math]] b_t^s=p_{t\varepsilon_s} [[/math]]

with [math]\varepsilon_s[/math] being the uniform measure on the [math]s[/math]-th roots of unity.

Observe that at [math]s=1,2[/math] we obtain the Poisson and real Bessel laws:

[[math]] b^1_t=p_t\quad,\quad b^2_t=b_t [[/math]]

Another important particular case is [math]s=\infty[/math], where we obtain a measure which is actually not discrete, that we will denote as follows:

[[math]] b^\infty_t=B_t [[/math]]

As a basic result on these laws, generalizing those before about [math]p_t,b_t[/math], we have:

Theorem

The generalized Bessel laws [math]b^s_t[/math] have the property

[[math]] b^s_t*b^s_{t'}=b^s_{t+t'} [[/math]]

so they form a truncated one-parameter semigroup with respect to convolution.

Show Proof

This follows indeed from the Fourier transform formula from Proposition 12.6, because for the Bessel laws, the log of this Fourier transform is linear in [math]t[/math].

■

Regarding now the moments, the result here is as follows:

Theorem

The moments of the Bessel law [math]b^s_t[/math] are the numbers

[[math]] M_k=|P^s(k)| [[/math]]

where [math]P^s(k)[/math] is the set of partitions of [math]\{1,\ldots,k\}[/math] satisfying

[[math]] \#\circ=\#\bullet(s) [[/math]]

as a weighted sum, in each block.

Show Proof

We already know that the formula in the statement holds indeed at [math]s=1[/math], where [math]b^1_t=p_t[/math] is the Poisson law of parameter [math]t \gt 0[/math], and where [math]P^1=P[/math] is the set of all partitions. At [math]s=2[/math] we have [math]P^2=P_{even}[/math], and the result is elementary as well, from what we have in the above. In general, this follows by doing some combinatorics. See .

■

We can go back now to the reflection groups, and we have the following result:

Theorem

For the complex reflection group

[[math]] H_N^s=\mathbb Z_s\wr S_N [[/math]]

we have, with [math]N\to\infty[/math], the main character law estimate

[[math]] \chi_t\sim b^s_t [[/math]]

where [math]b_t^s=p_{t\varepsilon_s}[/math], with [math]\varepsilon_s[/math] being the uniform measure on the [math]s[/math]-th roots of unity.

Show Proof

The best is to proceed in two steps, as follows:

(1) Let us first work out the case [math]t=1[/math]. Since the limit probability for a random permutation to have exactly [math]k[/math] fixed points is [math]e^{-1}/k![/math], we get:

[[math]] \lim_{N\to\infty}law(\chi_1)=e^{-1}\sum_{k=0}^\infty \frac{1}{k!}\,\varepsilon_s^{*k} [[/math]]

On the other hand, we get from the definition of the Bessel law [math]b^s_1[/math]:

[[math]] \begin{eqnarray*} b^s_1 &=&\lim_{N\to\infty}\left(\left(1-\frac{1}{N}\right)\delta_0+\frac{1}{N}\,\varepsilon_s\right)^{*N}\\ &=&\lim_{N\to\infty}\sum_{k=0}^N\begin{pmatrix}N\\ k\end{pmatrix}\left(1-\frac{1}{N}\right)^{N-k}\frac{1}{N^k}\,\varepsilon_s^{*k}\\ &=&e^{-1}\sum_{k=0}^\infty\frac{1}{k!}\,\varepsilon_s^{*k} \end{eqnarray*} [[/math]]

But this gives the assertion for [math]t=1[/math], as desired.

(2) Now in the case where [math]t \gt 0[/math] is arbitrary, we can use the same method, by performing the following modifications to the above computation:

[[math]] \begin{eqnarray*} \lim_{N\to\infty}law(\chi_t) &=&e^{-t}\sum_{k=0}^\infty\frac{t^k}{k!}\,\varepsilon_s^{*k}\\ &=&\lim_{N\to\infty}\left(\left(1-\frac{1}{N}\right)\delta_0+\frac{1}{N}\,\varepsilon_s\right)^{*[tN]}\\ &=&b^s_t \end{eqnarray*} [[/math]]

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

■

Here is now some more theory for the Bessel laws, following . First, it is convenient to introduce as well “modified” versions of the Bessel laws, as follows:

Definition

The Bessel and modified Bessel laws are given by

[[math]] b^s_t={\rm law}\left(\sum_{k=1}^sw^ka_k\right) [[/math]]

[[math]] \tilde{b}^s_t={\rm law}\left(\sum_{k=1}^sw^ka_k\right)^s [[/math]]

where [math]a_1,\ldots,a_s[/math] are independent random variables, each of them following the Poisson law of parameter [math]t/s[/math], and [math]w=e^{2\pi i/s}[/math].

As a first remark, at [math]s=1[/math] we get the Poisson law of parameter [math]t[/math]:

[[math]] b^1_t=\tilde{b}^1_t=e^{-t}\sum_{r=0}^\infty \frac{t^r}{r!}\,\delta_r [[/math]]

We will need in our computations the level [math]s[/math] exponential function, given by:

[[math]] \exp_sz=\sum_{k=0}^\infty\frac{z^{sk}}{(sk)!} [[/math]]

We have the following formula, in terms of root of unity [math]w=e^{2\pi i/s}[/math]:

[[math]] \exp_sz=\frac{1}{s}\sum_{k=1}^s\exp(w^kz) [[/math]]

Observe also that at [math]s=1,2[/math] we have the following formulae:

[[math]] \exp_1=\exp\quad,\quad \exp_2=\cosh [[/math]]

We have the following result, regarding both the plain and modified Bessel laws, which is a more explicit version of Proposition 12.6, for the Bessel laws:

Theorem

The Fourier transform of [math]b^s_t[/math] is given by

[[math]] \log F^s_t(z)=t\left(\exp_sz-1\right) [[/math]]

so in particular the measures [math]b^s_t[/math] are additive with respect to [math]t[/math].

Show Proof

Consider, as in Definition 12.14, the following variable:

[[math]] a=\sum_{k=1}^sw^ka_k [[/math]]

We have the following computation, for the corresponding Fourier transform:

[[math]] \begin{eqnarray*} \log F_a(z) &=&\sum_{k=1}^s\log F_{a_k}(w^kz)\\ &=&\sum_{k=1}^s\frac{t}{s}\left(\exp(w^kz)-1\right) \end{eqnarray*} [[/math]]

But this gives the following formula, in terms of the above function [math]\exp_s[/math]:

[[math]] \begin{eqnarray*} \log F_a(z) &=&t\left(\left(\frac{1}{s}\sum_{k=1}^s\exp(w^kz)\right)-1\right)\\ &=&t\left(\exp_s(z)-1\right) \end{eqnarray*} [[/math]]

Now since [math]b^s_t[/math] is the law of [math]a[/math], this gives the formula in the statement.

■

Let us study now the densities of [math]b^s_t,\tilde{b}^s_t[/math]. We have here the following result:

Theorem

We have the formulae

[[math]] b^s_t=e^{-t}\sum_{p_1=0}^\infty\ldots\sum_{p_s=0}^\infty\frac{1}{p_1!\ldots p_s!}\,\left(\frac{t}{s}\right)^{p_1+\ldots+p_s}\delta\left(\sum_{k=1}^sw^kp_k\right) [[/math]]

[[math]] \tilde{b}^s_t=e^{-t}\sum_{p_1=0}^\infty\ldots\sum_{p_s=0}^\infty\frac{1}{p_1!\ldots p_s!}\,\left(\frac{t}{s}\right)^{p_1+\ldots+p_s}\delta\left(\sum_{k=1}^sw^kp_k\right)^s [[/math]]

where [math]w=e^{2\pi i/s}[/math], and the [math]\delta[/math] symbol is a Dirac mass.

Show Proof

It is enough to prove the formula for [math]b^s_t[/math]. For this purpose, we compute the Fourier transform of the measure on the right. This is given by:

[[math]] \begin{eqnarray*} F(z) &=&e^{-t}\sum_{p_1=0}^\infty\ldots\sum_{p_s=0}^\infty\frac{1}{p_1!\ldots p_s!}\left(\frac{t}{s}\right)^{p_1+\ldots+p_s}F\delta\left(\sum_{k=1}^sw^kp_k\right)(z)\\ &=&e^{-t}\sum_{p_1=0}^\infty\ldots\sum_{p_s=0}^\infty\frac{1}{p_1!\ldots p_s!}\left(\frac{t}{s}\right)^{p_1+\ldots+p_s}\exp\left(\sum_{k=1}^sw^kp_kz\right)\\ &=&e^{-t}\sum_{r=0}^\infty\left(\frac{t}{s}\right)^r\sum_{\Sigma p_i=r}\frac{\exp\left(\sum_{k=1}^sw^kp_kz\right)}{p_1!\ldots p_s!} \end{eqnarray*} [[/math]]

We multiply by [math]e^t[/math], and we compute the derivative with respect to [math]t[/math]:

[[math]] \begin{eqnarray*} (e^tF(z))' &=&\sum_{r=1}^\infty\frac{r}{s}\left(\frac{t}{s}\right)^{r-1}\sum_{\Sigma p_i=r}\frac{\exp\left(\sum_{k=1}^sw^kp_kz\right)}{p_1!\ldots p_s!}\\ &=&\frac{1}{s}\sum_{r=1}^\infty\left(\frac{t}{s}\right)^{r-1}\sum_{\Sigma p_i=r}\left(\sum_{l=1}^sp_l\right)\frac{\exp\left(\sum_{k=1}^sw^kp_kz\right)}{p_1!\ldots p_s!}\\ &=&\frac{1}{s}\sum_{r=1}^\infty\left(\frac{t}{s}\right)^{r-1}\sum_{\Sigma p_i=r}\sum_{l=1}^s\frac{\exp\left(\sum_{k=1}^sw^kp_kz\right)}{p_1!\ldots p_{l-1}!(p_l-1)!p_{l+1}!\ldots p_s!}\\ \end{eqnarray*} [[/math]]

By using the variable [math]u=r-1[/math], we get:

[[math]] \begin{eqnarray*} (e^tF(z))' &=&\frac{1}{s}\sum_{u=0}^\infty\left(\frac{t}{s}\right)^u\sum_{\Sigma q_i=u}\sum_{l=1}^s\frac{\exp\left(w^lz+\sum_{k=1}^sw^kq_kz\right)}{q_1!\ldots q_s!}\\ &=&\left(\frac{1}{s}\sum_{l=1}^s\exp(w^lz)\right)\left(\sum_{u=0}^\infty\left(\frac{t}{s}\right)^u\sum_{\Sigma q_i=u}\frac{\exp\left(\sum_{k=1}^sw^kq_kz\right)}{q_1!\ldots q_s!}\right)\\ &=&(\exp_sz)(e^tF(z)) \end{eqnarray*} [[/math]]

On the other hand, consider the following function:

[[math]] \Phi(t)=\exp(t\exp_sz) [[/math]]

This function satisfies as well the equation found above, namely:

[[math]] \Phi'(t)=(\exp_sz)\Phi(t) [[/math]]

We conclude from this that we have the following equality of functions:

[[math]] e^tF(z)=\Phi(t) [[/math]]

But this gives the following formula, for the logarithm of the Fourier transform:

[[math]] \begin{eqnarray*} \log F &=&\log(e^{-t}\exp(t\exp_sz))\\ &=&\log(\exp(t(\exp_sz-1)))\\ &=&t(\exp_sz-1) \end{eqnarray*} [[/math]]

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

■

Let us discuss now the [math]s=\infty[/math] particular case of all the above, which will be of interest in what follows, along with the particular cases [math]s=1,2[/math]. First, we have:

Theorem

The full complex reflection group, denoted

[[math]] K_N\subset U_N [[/math]]

and formed of the permutation-type matrices with numbers in [math]\mathbb T[/math] as nonzero entries,

[[math]] K_N=M_N(\mathbb T\cup\{0\})\cap U_N [[/math]]

has a wreath product decomposition, as follows,

[[math]] K_N=\mathbb T\wr S_N [[/math]]

with [math]S_N[/math] acting on [math]\mathbb T^N[/math] in the standard way, by permuting the factors.

Show Proof

This is something that we already know from the above, as the [math]s=\infty[/math] particular case of the results established for the complex reflection groups [math]H_N^s[/math].

■

At the probabilistic level, we have the following result:

Theorem

For the full reflection group [math]K_N\subset U_N[/math], the law of the variable

[[math]] \chi_t=\sum_{i=1}^{[tN]}g_{ii} [[/math]]

is in the [math]N\to\infty[/math] limit the purely complex Bessel law,

[[math]] B_t=p_{t\varepsilon} [[/math]]

where [math]\varepsilon[/math] is the uniform measure on [math]\mathbb T[/math].

Show Proof

Once again, this is something that we already know from the above, at the [math]s=\infty[/math] particular case of the results established for the complex reflection groups [math]H_N^s[/math].

■

Let us end this presentation with some philosophical considerations, in connection with abstract probability theory. We have 4 main limiting results in probability, which can be real or complex, and discrete or continuous, which are as follows:

[[math]] \xymatrix@R=45pt@C=35pt{ CPLT_\mathbb C\ar@{-}[r]\ar@{-}[d]&CCLT\ar@{-}[d]\\ CPLT_\mathbb R\ar@{-}[r]&CLT } [[/math]]

Here the terminology is more or less self-explanatory, and to be more precise, the corresponding limiting laws are the real and complex Gaussian and Bessel laws:

[[math]] \xymatrix@R=48pt@C=48pt{ B_t\ar@{-}[r]\ar@{-}[d]&G_t\ar@{-}[d]\\ b_t\ar@{-}[r]&g_t } [[/math]]

Moreover, we have seen in the above that at the level of the moments of these limiting laws, these come from certain collections of partitions, as follows:

[[math]] \xymatrix@R=45pt@C=45pt{ \mathcal P_{even}\ar@{-}[r]\ar@{-}[d]&\mathcal P_2\ar@{-}[d]\\ P_{even}\ar@{-}[r]&P_2 } [[/math]]

Finally, we have seen as well that there are some Lie groups behind all this, namely the basic real and complex rotation and reflection groups, which are as follows:

[[math]] \xymatrix@R=45pt@C=45pt{ K_N\ar@{-}[r]\ar@{-}[d]&U_N\ar@{-}[d]\\ H_N\ar@{-}[r]&O_N } [[/math]]

All this is quite interesting, and we will be back to this, with more explanations, in chapters 13-16 below, by using more advanced representation theory tools.

Finally, in order for our discussion to be complete, we still have to talk about the general series of complex reflection groups [math]H_N^{sd}[/math]. Here the determinant does not really contribute, in the [math]N\to\infty[/math] limit, and we obtain the same asymptotic laws as for the groups [math]H_N^s[/math]. This is something that we have already seen in chapter 11, in relation with the alternating group [math]A_N\subset S_N[/math], and the proof in general is similar.

As before with other more advanced questions, we will leave all this for further discussion in chapters 13-16 below, after developing more powerful tools for dealing with such questions, and more specifically, advanced representation theory tools.

12c. Hyperspherical laws

In the continuous group case now, as a continuation of the above investigations, an interesting input comes from the various computations done long ago in chapter 6. In order to discuss all this, let us first recall some useful formulae from that chapter 6. One of the key results there, which is very useful in practice, was as follows:

Proposition

We have the following formula,

[[math]] \int_0^{\pi/2}\cos^pt\sin^qt\,dt=\left(\frac{\pi}{2}\right)^{\varepsilon(p)\varepsilon(q)}\frac{p!!q!!}{(p+q+1)!!} [[/math]]

where [math]\varepsilon(p)=1[/math] if [math]p[/math] is even, and [math]\varepsilon(p)=0[/math] if [math]p[/math] is odd, and where

[[math]] m!!=(m-1)(m-3)(m-5)\ldots [[/math]]

with the product ending at [math]2[/math] if [math]m[/math] is odd, and ending at [math]1[/math] if [math]m[/math] is even.

Show Proof

This is something that follows by partial integration, and then a double recurrence on [math]p,q\in\mathbb N[/math], worked out in chapter 6.

■

More generally now, we can in fact compute the polynomial integrals over the unit sphere in arbitrary dimensions, the result here being as follows:

Theorem

The polynomial integrals over the unit sphere [math]S^{N-1}_\mathbb R\subset\mathbb R^N[/math], with respect to the normalized, mass [math]1[/math] measure, are given by the following formula,

[[math]] \int_{S^{N-1}_\mathbb R}x_1^{k_1}\ldots x_N^{k_N}\,dx=\frac{(N-1)!!k_1!!\ldots k_N!!}{(N+\Sigma k_i-1)!!} [[/math]]

valid when all exponents [math]k_i[/math] are even. If an exponent is odd, the integral vanishes.

Show Proof

As before this is something that we know from chapter 6, the idea being that the [math]N=2[/math] case is solved by Proposition 12.19, and that the general case, [math]N\in\mathbb N[/math], follows from this, by using spherical coordinates and the Fubini theorem.

■

As an application of the above formula, we can investigate the hyperspherical laws and their asymptotics, and we have the following result:

Theorem

The moments of the hyperspherical variables are

[[math]] \int_{S^{N-1}_\mathbb R}x_i^kdx=\frac{(N-1)!!k!!}{(N+k-1)!!} [[/math]]

and the normalized hyperspherical variables

[[math]] y_i=\frac{x_i}{\sqrt{N}} [[/math]]

become normal and independent with [math]N\to\infty[/math].

Show Proof

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

(1) The formula in the statement follows from the general integration formula over the sphere, established above, which is as follows:

[[math]] \int_{S^{N-1}_\mathbb R}x_{i_1}\ldots x_{i_k}\,dx=\frac{(N-1)!!l_1!!\ldots l_N!!}{(N+\sum l_i-1)!!} [[/math]]

Indeed, with [math]i_1=\ldots=i_k=i[/math], we obtain from this:

[[math]] \int_{S^{N-1}_\mathbb R}x_i^kdx=\frac{(N-1)!!k!!}{(N+k-1)!!} [[/math]]

Now observe that with [math]N\to\infty[/math] we have the following estimate:

[[math]] \begin{eqnarray*} \int_{S^{N-1}_\mathbb R}x_i^kdx &=&\frac{(N-1)!!}{(N+k-1)!!}\times k!!\\ &\simeq&N^{k/2}k!!\\ &=&N^{k/2}M_k(g_1) \end{eqnarray*} [[/math]]

Thus, the variables [math]y_i=\frac{x_i}{\sqrt{N}}[/math] become normal with [math]N\to\infty[/math].

(2) As for the asymptotic independence result, this is standard as well, once again by using Theorem 12.20, for computing mixed moments, and taking the [math]N\to\infty[/math] limit.

■

Now back to groups, the point is that we can talk as well about rotations, as follows:

Theorem

We have the integration formula

[[math]] \int_{O_N}U_{ij}^kdU=\frac{(N-1)!!k!!}{(N+k-1)!!} [[/math]]

and the normalized coordinates on [math]O_N[/math], constructed as follows,

[[math]] V_{ij}=\frac{U_{ij}}{\sqrt{N}} [[/math]]

become normal and independent with [math]N\to\infty[/math].

Show Proof

We use the well-known fact that we have an embedding as follows, for any [math]i[/math], which makes correspond the respective integration functionals:

[[math]] C(S^{N-1}_\mathbb R)\subset C(O_N)\quad,\quad x_i\to U_{1i} [[/math]]

With this identification made, the result follows from Theorem 12.21.

■

We have similar results in the unitary case. First, we have:

Theorem

We have the following integration formula over the complex sphere [math]S^{N-1}_\mathbb C\subset\mathbb R^N[/math], with respect to the normalized measure,

[[math]] \int_{S^{N-1}_\mathbb C}|z_1|^{2l_1}\ldots|z_N|^{2l_N}\,dz=4^{\sum l_i}\frac{(2N-1)!l_1!\ldots l_n!}{(2N+\sum l_i-1)!} [[/math]]

valid for any exponents [math]l_i\in\mathbb N[/math]. As for the other polynomial integrals in [math]z_1,\ldots,z_N[/math] and their conjugates [math]\bar{z}_1,\ldots,\bar{z}_N[/math], these all vanish.

Show Proof

As before, this is something that we know from chapter 6, and which can be proved either directly, or by using the formula in Theorem 12.20.

■

We can talk about complex hyperspherical laws, and we have:

Theorem

The rescaled coordinates on the complex sphere [math]S^{N-1}_\mathbb C[/math],

[[math]] w_i=\frac{z_i}{\sqrt{N}} [[/math]]

become complex Gaussian and independent with [math]N\to\infty[/math].

Show Proof

This follows indeed by using Theorem 12.22 and Theorem 12.23.

■

In relation now with Lie groups, the result that we obtain is as follows:

Theorem

For the unitary group [math]U_N[/math], the normalized coordinates

[[math]] V_{ij}=\frac{U_{ij}}{\sqrt{N}} [[/math]]

become complex Gaussian and independent with [math]N\to\infty[/math].

Show Proof

We use the well-known fact that we have an embedding as follows, for any [math]i[/math], which makes correspond the respective integration functionals:

[[math]] C(S^{N-1}_\mathbb C)\subset C(U_N)\quad,\quad x_i\to U_{1i} [[/math]]

With this identification made, the result follows from Theorem 12.21.

■

Our claim now is that the above results can be reformulated in terms of the truncated characters introduced in chapter 11. Let us recall indeed from there that we have:

Definition

Given a closed subgroup [math]G\subset U_N[/math], the function

[[math]] \chi:G\to\mathbb C\quad,\quad \chi_t(g)=\sum_{i=1}^{[tN]}g_{ii} [[/math]]

is called main truncated character of [math]G[/math], of parameter [math]t\in(0,1][/math].

We refer to chapter 11 for more on this notion. Also, we will be back to all this in chapters 13-16 below, with more details about all this, and motivations.

In connection now with the present considerations, the point is that with the above notion in hand, our results above reformulate as follows:

Theorem

For the orthogonal and unitary groups [math]O_N,U_N[/math], the rescalings

[[math]] \chi=\frac{\chi_{1/N}}{\sqrt{N}} [[/math]]

become respectively real and complex Gaussian, in the [math]N\to\infty[/math] limit.

Show Proof

According to our conventions, given a closed subgroup [math]G\subset U_N[/math], the main character truncated at [math]t=1/N[/math] is simply the first coordinate:

[[math]] \chi_{1/N}(g)=g_{11} [[/math]]

With this remark made, the conclusions from the statement follow from the computations performed above, for the laws of coordinates on [math]O_N,U_N[/math].

■

It is possible to get beyond such results, by using advanced representation theory methods, with full results about all the truncated characters, and in particular about the main characters. We will be back to this in chapters 13-16 below.

Also, it is possible to compute as well the laws of individual coordinates for some of the remaining continuous groups. To be more precise, it is possible to work out results for the bistochastic groups [math]B_N,C_N[/math], as well as for the symplectic groups [math]Sp_N[/math], the computations being not very complicated. However, we prefer here to defer the discussion to chapters 13-16 below, after developing some appropriate tools, for dealing with such questions.

12d. Wigner laws

As a last topic now for this chapter, let us discuss the case where [math]N[/math] is fixed. Things are quite complicated here, and as a main goal, we would like to find the law of the main character for our favorite rotation groups, namely [math]SU_2[/math] and [math]SO_3[/math].

In order to do so, we will need some combinatorial preliminaries. We first have the following well-known result, which is the cornerstone of all modern combinatorics:

Theorem

The Catalan numbers, which are by definition given by

[[math]] C_k=|NC_2(2k)| [[/math]]

satisfy the following recurrence formula,

[[math]] C_{k+1}=\sum_{a+b=k}C_aC_b [[/math]]

and their generating series, given by definition by

[[math]] f(z)=\sum_{k\geq0}C_kz^k [[/math]]

satisfies the following degree [math]2[/math] equation,

[[math]] zf^2-f+1=0 [[/math]]

and we have the following explicit formula for these numbers:

[[math]] C_k=\frac{1}{k+1}\binom{2k}{k} [[/math]]

Numerically, these numbers are [math]1,1,2,5,14,42,132,429,1430,4862,16796,\ldots[/math]

Show Proof

We must count the noncrossing pairings of [math]\{1,\ldots,2k\}[/math]. But such a pairing appears by pairing 1 to an odd number, [math]2a+1[/math], and then inserting a noncrossing pairing of [math]\{2,\ldots,2a\}[/math], and a noncrossing pairing of [math]\{2a+2,\ldots,2l\}[/math]. We conclude from this that we have the following recurrence formula for the Catalan numbers:

[[math]] C_k=\sum_{a+b=k-1}C_aC_b [[/math]]

In terms of the generating series [math]f[/math], the above recurrence gives:

[[math]] \begin{eqnarray*} zf^2 &=&\sum_{a,b\geq0}C_aC_bz^{a+b+1}\\ &=&\sum_{k\geq1}\sum_{a+b=k-1}C_aC_bz^k\\ &=&\sum_{k\geq1}C_kz^k\\ &=&f-1 \end{eqnarray*} [[/math]]

Thus the generating series [math]f[/math] satisfies the following degree 2 equation:

[[math]] zf^2-f+1=0 [[/math]]

By choosing the solution which is bounded at [math]z=0[/math], we obtain:

[[math]] f(z)=\frac{1-\sqrt{1-4z}}{2z} [[/math]]

By using now the Taylor formula for [math]\sqrt{x}[/math], we obtain the following formula:

[[math]] f(z)=\sum_{k\geq0}\frac{1}{k+1}\binom{2k}{k}z^k [[/math]]

It follows that the Catalan numbers are given by the formula the statement.

■

The Catalan numbers are central objects in probability as well, and we have the following key result here, complementing the formulae from Theorem 12.28:

Theorem

The normalized Wigner semicircle law, which is by definition

[[math]] \gamma_1=\frac{1}{2\pi}\sqrt{4-x^2}dx [[/math]]

has the Catalan numbers as even moments. As for the odd moments, these all vanish.

Show Proof

The even moments of the Wigner law can be computed with the change of variable [math]x=2\cos t[/math], and we are led to the following formula:

[[math]] \begin{eqnarray*} M_{2k} &=&\frac{1}{\pi}\int_0^2\sqrt{4-x^2}x^{2k}dx\\ &=&\frac{1}{\pi}\int_0^{\pi/2}\sqrt{4-4\cos^2t}\,(2\cos t)^{2k}2\sin t\,dt\\ &=&\frac{4^{k+1}}{\pi}\int_0^{\pi/2}\cos^{2k}t\sin^2t\,dt\\ &=&\frac{4^{k+1}}{\pi}\cdot\frac{\pi}{2}\cdot\frac{(2k)!!2!!}{(2k+3)!!}\\ &=&2\cdot 4^k\cdot\frac{(2k)!/2^kk!}{2^{k+1}(k+1)!}\\ &=&C_k \end{eqnarray*} [[/math]]

As for the odd moments, these all vanish, because the density of [math]\gamma_1[/math] is an even function. Thus, we are led to the conclusion in the statement.

■

We can now formulate our result regarding [math]SU_2[/math], as follows:

Theorem

The main character of [math]SU_2[/math], given by

[[math]] \chi\begin{pmatrix}a&b\\ -\bar{b}&\bar{a}\end{pmatrix}=2Re(a) [[/math]]

follows a Wigner semicircle law [math]\gamma_1[/math].

Show Proof

The idea is that this follows by identifying [math]SU_2[/math] with the sphere [math]S^3_\mathbb R\subset\mathbb R^4[/math], and the uniform measure on [math]SU_2[/math] with the uniform measure on this sphere. Indeed, in terms of the standard parametrization of [math]SU_2[/math], from chapter 10, written in real form, we have the following formula, for the main character of [math]SU_2[/math]:

[[math]] \chi\begin{pmatrix}x+iy&z+it\\ -z+it&x-iy\end{pmatrix}=2x [[/math]]

We are therefore left with computing the law of the following variable:

[[math]] x\in C(S^3_\mathbb R) [[/math]]

But for this purpose, we can use the moment method. Indeed, by using the trigonometric integral formulae from chapter 6, we obtain:

[[math]] \begin{eqnarray*} \int_{S^3_\mathbb R}x^{2k} &=&\frac{3!!(2k)!!}{(2k+3)!!}\\ &=&2\cdot\frac{3\cdot5\cdot7\ldots (2k-1)}{2\cdot4\cdot6\ldots (2k+2)}\\ &=&2\cdot\frac{(2k)!}{2^kk!2^{k+1}(k+1)!}\\ &=&\frac{1}{4^k}\cdot\frac{1}{k+1}\binom{2k}{k}\\ &=&\frac{C_k}{4^k} \end{eqnarray*} [[/math]]

Thus the variable [math]2x\in C(S^3_\mathbb R)[/math] has the Catalan numbers as even moments, and so by Theorem 12.29 its distribution is the Wigner semicircle law [math]\gamma_1[/math], as claimed.

■

In order to do the computation for [math]SO_3[/math], we will need some more probabilistic preliminaries, which are standard random matrix theory material. Let us start with:

Proposition

We have a bijection between noncrossing partitions and pairings

[[math]] NC(k)\simeq NC_2(2k) [[/math]]

constructed as follows:

• The application [math]NC(k)\to NC_2(2k)[/math] is the “fattening” one, obtained by doubling all the legs, and doubling all the strings as well.
• Its inverse [math]NC_2(2k)\to NC(k)[/math] is the “shrinking” application, obtained by collapsing pairs of consecutive neighbors.

Show Proof

The fact that the two operations in the statement are indeed inverse to each other is clear, by computing the corresponding two compositions, with the remark that the construction of the fattening operation requires the partitions to be noncrossing.

■

As a consequence of the above result, we have a new look on the Catalan numbers, which is more adapted to our present [math]SO_3[/math] considerations, as follows:

Proposition

The Catalan numbers [math]C_k=|NC_2(2k)|[/math] appear as well as

[[math]] C_k=|NC(k)| [[/math]]

where [math]NC(k)[/math] is the set of all noncrossing partitions of [math]\{1,\ldots,k\}[/math].

Show Proof

This follows indeed from Proposition 12.31.

■

Let us formulate now the following definition:

Definition

The standard Marchenko-Pastur law [math]\pi_1[/math] is given by:

[[math]] f\sim\gamma_1\implies f^2\sim\pi_1 [[/math]]

That is, [math]\pi_1[/math] is the law of the square of a variable following the semicircle law [math]\gamma_1[/math].

Here the fact that [math]\pi_1[/math] is indeed well-defined comes from the fact that a measure is uniquely determined by its moments. More explicitely now, we have:

Proposition

The density of the Marchenko-Pastur law is

[[math]] \pi_1=\frac{1}{2\pi}\sqrt{4x^{-1}-1}\,dx [[/math]]

and the moments of this measure are the Catalan numbers.

Show Proof

There are several proofs here, either by using Definition 12.33, or by Stieltjes inversion, of just by cheating. Whis this latter method, the point is that the moments of the law in the statement can be computed with the following change of variables:

[[math]] x=4\cos^2t [[/math]]

To be more precise, the moments of the law in the statement can be computed with the change of variable [math]x=4\cos^2t[/math], and we are led to the following formula:

[[math]] \begin{eqnarray*} M_k &=&\frac{1}{2\pi}\int_0^4\sqrt{4x^{-1}-1}\,x^kdx\\ &=&\frac{1}{2\pi}\int_0^{\pi/2}\frac{\sin t}{\cos t}\cdot(4\cos^2t)^k\cdot 2\cos t\sin t\,dt\\ &=&\frac{4^{k+1}}{\pi}\int_0^{\pi/2}\cos^{2k}t\sin^2t\,dt\\ &=&\frac{4^{k+1}}{\pi}\cdot\frac{\pi}{2}\cdot\frac{(2k)!!2!!}{(2k+3)!!}\\ &=&2\cdot 4^k\cdot\frac{(2k)!/2^kk!}{2^{k+1}(k+1)!}\\ &=&C_k \end{eqnarray*} [[/math]]

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

■

We can do now the character computation for [math]SO_3[/math], as follows:

Theorem

The main character of [math]SO_3[/math], modified by adding [math]1[/math] to it, given in standard Euler-Rodrigues coordinates by

[[math]] \chi=3x^2-y^2-z^2-t^2 [[/math]]

follows a squared semicircle law, or Marchenko-Pastur law [math]\pi_1[/math].

Show Proof

The idea is that this follows by using the canonical quotient map [math]SU_2\to SO_3[/math], and the result for [math]SU_2[/math] from Theorem 12.30. To be more precise, let us recall from chapter 10 that the elements of [math]SU_2[/math] can be parametrized as follows:

[[math]] U=\begin{pmatrix}x+iy&z+it\\ -z+it&x-iy\end{pmatrix} [[/math]]

As for the elements of [math]SO_3[/math], these can be parametrized as follows:

[[math]] V=\begin{pmatrix} x^2+y^2-z^2-t^2&2(yz-xt)&2(xz+yt)\\ 2(xt+yz)&x^2+z^2-y^2-t^2&2(zt-xy)\\ 2(yt-xz)&2(xy+zt)&x^2+t^2-y^2-z^2 \end{pmatrix} [[/math]]

The point now is that, by using the above two formulae, in the context of the computation from Theorem 12.30, the main character of [math]SO_3[/math] is given by:

[[math]] \chi=4Re(a)^2 [[/math]]

Now recall from the proof of Theorem 12.30 that we have:

[[math]] 2Re(a)\sim\gamma_1 [[/math]]

On the other hand, a quick comparison between the moment formulae for the Wigner and Marchenko-Pastur laws, which are very similar, shows that we have:

[[math]] f\sim\gamma_1\implies f^2\sim\pi_1 [[/math]]

Thus, with [math]f=2Re(a)[/math], we obtain the result in the statement.

■

As an interesting question now, appearing from the above, and which is quite philosophical, we have the problem of understanding how the Wigner and Marchenko-Pastur laws [math]\gamma_1,\pi_1[/math] fit in regards with the main limiting laws from classical probability.

The answer here is quite tricky, the idea being that, with a suitable formalism for freeness, [math]\gamma_1,\pi_1[/math] can be thought of as being “free analogues” of the Gaussian and Poisson laws [math]g_1,p_1[/math]. This is something quite subtle, requiring some further knowledge, and we will be back to this in chapters 13-16 below, when doing representation theory.

In any case, as a conclusion, we have now a pretty decent level in probability. And if you want to learn right away a bit more about the modern topics here, such as random matrices and free probability, from Marchenko-Pastur , Mehta , Voiculescu , Wigner , you can go for it. As for more groups and algebra, stay with us.

General references

Banica, Teo (2024). "Linear algebra and group theory". arXiv:2206.09283 [math.CO].