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In short, we are not over with our study, which seems to open more questions than it solves. Fortunately, the solution to the question raised by Proposition 8.29 is quite simple. The idea indeed will be that of improving our [math]g_1,\gamma_1,G_1,\Gamma_1[/math] results above with [math]g_t,\gamma_t,G_t,\Gamma_t[/math] results, which will require [math]N\to\infty[/math] in both the classical and free cases, in order to hold at any [math]t[/math]. In practice, the definition that we will need is as follows:

Definition

Given a Woronowicz algebra [math](A,u)[/math], the variable

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

is called truncation of the main character, with parameter [math]t\in(0,1][/math].

Our purpose in what follows will be that of proving that for [math]O_N,O_N^+,U_N,U_N^+[/math], the asymptotic laws of the truncated characters [math]\chi_t[/math] with [math]t\in(0,1][/math] are the laws [math]g_t,\gamma_t,G_t,\Gamma_t[/math]. This is something quite technical, motivated by the findings in Proposition 8.29, and also by a number of more advanced considerations, to become clear later on.

In order to start now, the formula in Theorem 8.26 is not useful in the general [math]t\in(0,1][/math] setting, and we must use instead general integration methods. We first have:

Theorem

The Haar integration of a Woronowicz algebra is given, on the coefficients of the Peter-Weyl corepresentations, by the Weingarten formula

[[math]] \int_Gu_{i_1j_1}^{e_1}\ldots u_{i_kj_k}^{e_k}=\sum_{\pi,\sigma\in D_k}\delta_\pi(i)\delta_\sigma(j)W_k(\pi,\sigma) [[/math]]

valid for any colored integer [math]k=e_1\ldots e_k[/math] and any multi-indices [math]i,j[/math], where:

• [math]D_k[/math] is a linear basis of [math]Fix(u^{\otimes k})[/math].
• [math]\delta_\pi(i)= \lt \pi,e_{i_1}\otimes\ldots\otimes e_{i_k} \gt [/math].
• [math]W_k=G_k^{-1}[/math], with [math]G_k(\pi,\sigma)= \lt \pi,\sigma \gt [/math].

Show Proof

This is something that we know from chapter 3, coming from the fact that integrals in the statement form altogether the orthogonal projection onto [math]Fix(u^{\otimes k})[/math].

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In the easy case, this gives the following formula, from , :

Theorem

For an easy quantum group [math]G\subset U_N^+[/math], coming from a category of partitions [math]D=(D(k,l))[/math], we have the Weingarten integration formula

[[math]] \int_Gu_{i_1j_1}^{e_1}\ldots u_{i_kj_k}^{e_k}=\sum_{\pi,\sigma\in D(k)}\delta_\pi(i)\delta_\sigma(j)W_{kN}(\pi,\sigma) [[/math]]

for any colored integer [math]k=e_1\ldots e_k[/math] and any multi-indices [math]i,j[/math], where [math]D(k)=D(\emptyset,k)[/math], [math]\delta[/math] are usual Kronecker symbols, and

[[math]] W_{kN}=G_{kN}^{-1} [[/math]]

with [math]G_{kN}(\pi,\sigma)=N^{|\pi\vee\sigma|}[/math], where [math]|.|[/math] is the number of blocks.

Show Proof

With notations from Theorem 8.31, the Kronecker symbols are given by:

[[math]] \begin{eqnarray*} \delta_{\xi_\pi}(i) &=& \lt \xi_\pi,e_{i_1}\otimes\ldots\otimes e_{i_k} \gt \\ &=&\delta_\pi(i_1,\ldots,i_k) \end{eqnarray*} [[/math]]

The Gram matrix being as well the correct one, we obtain the result.

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We can use this for truncated characters, and following , we obtain:

Proposition

The moments of truncated characters are given by the formula

[[math]] \int_G(u_{11}+\ldots +u_{ss})^k=Tr(W_{kN}G_{ks}) [[/math]]

and with [math]N\to\infty[/math] this quantity equals [math](s/N)^k|D(k)|[/math].

Show Proof

The first assertion follows from the following computation:

[[math]] \begin{eqnarray*} \int_G(u_{11}+\ldots +u_{ss})^k &=&\sum_{i_1=1}^{s}\ldots\sum_{i_k=1}^s\int u_{i_1i_1}\ldots u_{i_ki_k}\\ &=&\sum_{\pi,\sigma\in D(k)}W_{kN}(\pi,\sigma)\sum_{i_1=1}^{s}\ldots\sum_{i_k=1}^s\delta_\pi(i)\delta_\sigma(i)\\ &=&\sum_{\pi,\sigma\in D(k)}W_{kN}(\pi,\sigma)G_{ks}(\sigma,\pi)\\ &=&Tr(W_{kN}G_{ks}) \end{eqnarray*} [[/math]]

The point now is that we have the following trivial estimates:

[[math]] G_{kN}(\pi,\sigma): \begin{cases} =N^k&(\pi=\sigma)\\ \leq N^{k-1}&(\pi\neq\sigma) \end{cases} [[/math]]

Thus with [math]N\to\infty[/math] we have the following estimate:

[[math]] G_{kN}\sim N^k1 [[/math]]

But this gives the following estimate, for our moment:

[[math]] \begin{eqnarray*} \int_G(u_{11}+\ldots +u_{ss})^k &=&Tr(G_{kN}^{-1}G_{ks})\\ &\sim&Tr((N^k1)^{-1} G_{ks})\\ &=&N^{-k}Tr(G_{ks})\\ &=&N^{-k}s^k|D(k)| \end{eqnarray*} [[/math]]

Thus, we have obtained the formula in the statement. See .

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In order to process the above formula, we will need some more free probability theory. Following Nica-Speicher , given a random variable [math]a[/math], we write:

[[math]] \log F_a(\xi)=\sum_nk_n(a)\xi^n\quad,\quad R_a(\xi)=\sum_n\kappa_n(a)\xi^n [[/math]]

We call the coefficients [math]k_n(a),\kappa_n(a)[/math] cumulants, respectively free cumulants of [math]a[/math]. With this notion in hand, we can define then more general quantities [math]k_\pi(a),\kappa_\pi(a)[/math], depending on partitions [math]\pi\in P(k)[/math], by multiplicativity over the blocks. We have then:

Theorem

We have the classical and free moment-cumulant formulae

[[math]] M_k(a)=\sum_{\pi\in P(k)}k_\pi(a)\quad,\quad M_k(a)=\sum_{\pi\in NC(k)}\kappa_\pi(a) [[/math]]

where [math]k_\pi(a),\kappa_\pi(a)[/math] are the generalized cumulants and free cumulants of [math]a[/math].

Show Proof

These formulae, due to Rota in the classical case, and to Speicher in the free case, are something very standard, obtained by using the formulae of [math]F_a,R_a[/math], or by doing some direct combinatorics, based on the Möbius inversion formula. See .

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Following , we can now improve our results about characters, as follows:

Theorem

With [math]N\to\infty[/math], the laws of truncated characters are as follows:

• For [math]O_N[/math] we obtain the Gaussian law [math]g_t[/math].
• For [math]O_N^+[/math] we obtain the Wigner semicircle law [math]\gamma_t[/math].
• For [math]U_N[/math] we obtain the complex Gaussian law [math]G_t[/math].
• For [math]U_N^+[/math] we obtain the Voiculescu circular law [math]\Gamma_t[/math].

Show Proof

With [math]s=[tN][/math] and [math]N\to\infty[/math], the formula in Proposition 8.33 gives:

[[math]] \lim_{N\to\infty}\int_{G_N}\chi_t^k=\sum_{\pi\in D(k)}t^{|\pi|} [[/math]]

By using now the formulae in Theorem 8.34, this gives the results. See .

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In relation with the above, let us recall now that the Bercovici-Pata bijection is the bijection [math]\{m_t\}\to\{\mu_t\}[/math] between the semigroups [math]\{m_t\}[/math] of infinitely divisible measures and the semigroups [math]\{\mu_t\}[/math] of freely infinitely divisible measures, given by the fact that the classical cumulants of [math]m_t[/math] equal the free cumulants of [math]\mu_t[/math]. Following , we have:

Theorem

The asymptotic laws of truncated characters for the operations

[[math]] O_N\to O_N^+ [[/math]]

[[math]] U_N\to U_N^+ [[/math]]

are in Bercovici-Pata bijection.

Show Proof

This follows indeed from the computations in the proof of Theorem 8.35.

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Let us discuss now the other easy quantum groups that we have. Regarding the half-liberations [math]O_N^*,U_N^*[/math], the situation here is a bit complicated, and we will discuss this later on. But we have the following result that, we can formulate here, at [math]t=1[/math]:

Proposition

The asymptotic laws of characters for [math]O_N^*,U_N^*[/math] are as follows:

• For [math]O_N^*[/math] we obtain a symmetrized Rayleigh variable.
• For [math]U_N^*[/math] we obtain a complexification of this variable.

Show Proof

The idea is to use a projective version trick. Indeed, assuming that [math]G=(G_N)[/math] is easy, coming from a category of pairings [math]D[/math], we have:

[[math]] \lim_{N\to\infty}\int_{PG_N}(\chi\chi^*)^k=\# D((\circ\bullet)^k) [[/math]]

In our case, where [math]G_N=O_N^*,U_N^*[/math], we can therefore use Theorem 8.35 above at [math]t=1[/math], and we are led to the conclusions in the statement. See .

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The above result is of course something quite modest. We will be back to the quantum groups [math]O_N^*,U_N^*[/math] in chapter 16 below, with some better techniques for dealing with them, and more specifically with explicit modelling results using [math]2\times2[/math] matrices, which virtually allow to prove anything that you want, probabilistically, about them.

Next in our lineup, we have the bistochastic quantum groups. We have here:

Proposition

For the bistochastic quantum groups, namely

[[math]] B_N,B_N^+,C_N,C_N^+ [[/math]]

the asymptotic laws of truncated characters appear as modified versions of

[[math]] g_t,\gamma_t,G_t,\Gamma_t [[/math]]

and [math]B_N\to B_N^+[/math] and [math]C_N\to C_N^+[/math] are compatible with the Bercovici-Pata bijection.

Show Proof

This follows indeed by using the same methods as for [math]O_N,O_N^+,U_N,U_N^+[/math], with the verification of the Bercovici-Pata bijection being elementary, and with the computation of the corresponding laws being routine as well. See , .

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Regarding now the twists, we first have here the following general result:

Proposition

The integration over [math]\bar{G}_N[/math] is given by the Weingarten type formula

[[math]] \int_{\bar{G}_N}u_{i_1j_1}\ldots u_{i_kj_k}=\sum_{\pi,\sigma\in D(k)}\bar{\delta}_\pi(i)\bar{\delta}_\sigma(j)W_{kN}(\pi,\sigma) [[/math]]

where [math]W_{kN}[/math] is the Weingarten matrix of [math]G_N[/math].

Show Proof

This follows exactly as in the untwisted case, the idea being that the signs will cancel. Let us recall indeed from the general twisting theory from chapter 7 that the twisted vectors [math]\bar{\xi}_\pi[/math] associated to the partitions [math]\pi\in P_{even}(k)[/math] are as follows:

[[math]] \bar{\xi}_\pi=\sum_{\tau\geq\pi}\varepsilon(\tau)\sum_{i:\ker(i)=\tau}e_{i_1}\otimes\ldots\otimes e_{i_k} [[/math]]

Thus, the Gram matrix of these vectors is given by:

[[math]] \begin{eqnarray*} \lt \xi_\pi,\xi_\sigma \gt &=&\sum_{\tau\geq\pi\vee\sigma}\varepsilon(\tau)^2\left|\left\{(i_1,\ldots,i_k)\Big|\ker i=\tau\right\}\right|\\ &=&\sum_{\tau\geq\pi\vee\sigma}\left|\left\{(i_1,\ldots,i_k)\Big|\ker i=\tau\right\}\right|\\ &=&N^{|\pi\vee\sigma|} \end{eqnarray*} [[/math]]

Thus the Gram matrix is the same as in the untwisted case, and so the Weingarten matrix is the same as well as in the untwisted case, and this gives the result.

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As a consequence of the above result, we have another general result, as follows:

Theorem

The Schur-Weyl twisting operation [math]G_N\leftrightarrow\bar{G}_N[/math] leaves invariant:

• The law of the main character.
• The coamenability property.
• The asymptotic laws of truncated characters.

Show Proof

This follows from Proposition 8.39, as follows:

(1) This is clear indeed from the integration formula.

(2) This follows from (1), and from the Kesten criterion.

(3) This follows once again from the integration formula.

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To summarize, we have asymptotic character results for all the easy quantum groups introduced so far, and in each case we obtain Gaussian laws, and their versions. We will see in chapters 9-12 below that pretty much the same happens in the discrete setting, where we will obtain Poisson laws, and their versions.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].