8b. Laws of characters

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Now back to our quantum group questions, let us start with the following general result, which provides us with motivations for the study of the main character:

Theorem

Given a Woronowicz algebra [math](A,u)[/math], the law of the main character

[[math]] \chi=\sum_{i=1}^Nu_{ii} [[/math]]

with respect to the Haar integration has the following properties:

The moments of [math]\chi[/math] are the numbers [math]M_k=\dim(Fix(u^{\otimes k}))[/math].
[math]M_k[/math] counts as well the lenght [math]p[/math] loops at [math]1[/math], on the Cayley graph of [math]A[/math].
[math]law(\chi)[/math] is the Kesten measure of the associated discrete quantum group.
When [math]u\sim\bar{u}[/math] the law of [math]\chi[/math] is a usual measure, supported on [math][-N,N][/math].
The algebra [math]A[/math] is amenable precisely when [math]N\in supp(law(Re(\chi)))[/math].
Any morphism [math]f:(A,u)\to (B,v)[/math] must increase the numbers [math]M_k[/math].
Such a morphism [math]f[/math] is an isomorphism when [math]law(\chi_u)=law(\chi_v)[/math].

Show Proof

These are things that we already know, the idea being as follows:

(1) This comes from the Peter-Weyl theory, which tells us the number of fixed points of [math]v=u^{\otimes k}[/math] can be recovered by integrating the character [math]\chi_v=\chi_u^k[/math].

(2) This is something true, and well-known, for [math]A=C^*(\Gamma)[/math], with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] being a discrete group. In general, the proof is quite similar.

(3) This is actually the definition of the Kesten measure, in the case [math]A=C^*(\Gamma)[/math], with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] being a discrete group. In general, this follows from (2).

(4) The equivalence [math]u\sim\bar{u}[/math] translates into [math]\chi_u=\chi_u^*[/math], and this gives the first assertion. As for the support claim, this follows from [math]uu^*=1\implies||u_{ii}||\leq1[/math], for any [math]i[/math].

(5) This is the Kesten amenability criterion, which can be established as in the classical case, [math]A=C^*(\Gamma)[/math], with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] being a discrete group.

(6) This is something elementary, which follows from (1) above, and from the fact that the morphisms of Woronowicz algebras increase the spaces of fixed points.

(7) This follows by using (6), and the Peter-Weyl theory, the idea being that if [math]f[/math] is not injective, then it must strictly increase one of the spaces [math]Fix(u^{\otimes k})[/math].

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As a conclusion, computing [math]\mu=law(\chi)[/math] is the main question to be solved, from a mathematical viewpoint. The same goes for physics too, although this is rather folklore. In what follows we will be interested in computing such laws, for the main examples of quantum groups that we have. In the easy quantum group case, we have:

Theorem

For an easy quantum group [math]G=(G_N)[/math], coming from a category of partitions [math]D=(D(k,l))[/math], the asymptotic moments of the main character are given by

[[math]] \lim_{N\to\infty}\int_{G_N}\chi^k=|D(k)| [[/math]]

where [math]D(k)=D(\emptyset,k)[/math], with the limiting sequence on the left consisting of certain integers, and being stationary at least starting from the [math]k[/math]-th term.

Show Proof

This follows indeed from the general formula from Theorem 8.25 (1), by using the linear independence result for partitions from chapter 5.

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Our next purpose will be that of understanding what happens for the basic classes of easy quantum groups. In the orthogonal case, we have:

Theorem

In the [math]N\to\infty[/math] limit, the law of the main character [math]\chi_u[/math] is as follows:

For [math]O_N[/math] we obtain a Gaussian law, namely:
[[math]] g_1=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}dx [[/math]]
For [math]O_N^+[/math] we obtain a Wigner semicircle law, namely:
[[math]] \gamma_1=\frac{1}{2\pi}\sqrt{4-x^2}dx [[/math]]

Show Proof

These are results that we both know, from chapter 5.

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In the unitary case now, we have:

Theorem

In the [math]N\to\infty[/math] limit, the law of the main character [math]\chi_u[/math] is as follows:

For [math]U_N[/math] we obtain the complex Gaussian law [math]G_1[/math].
For [math]U_N^+[/math] we obtain the Voiculescu circular law [math]\Gamma_1[/math].

Show Proof

These are once again results that we know, from chapter 6.

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Summarizing, for [math]O_N,O_N^+,U_N,U_N^+[/math] the asymptotic laws of the main characters are the laws [math]g_1,\gamma_1,G_1,\Gamma_1[/math] coming from the various CLT in classical and free probability. This is certainly nice, but there is still one conceptual problem, coming from:

Proposition

The above convergences [math]law(\chi_u)\to g_1,\gamma_1,G_1,\Gamma_1[/math] are as follows:

They are non-stationary in the classical case.
They are stationary in the free case, starting from [math]N=2[/math].

Show Proof

This is something quite subtle, which can be proved as follows:

(1) Here we can use an amenability argument, based on the Kesten criterion. Indeed, [math]O_N,U_N[/math] being coamenable, the upper bound of the support of the law of [math]Re(\chi_u)[/math] is precisely [math]N[/math], and we obtain from this that the law of [math]\chi_u[/math] itself depends on [math]N\in\mathbb N[/math].

(2) Here the result follows from the computations in chapter 4 above, performed when working out the representation theory of [math]O_N^+,U_N^+[/math], which show that the linear maps [math]T_\pi[/math] associated to the noncrossing pairings are linearly independent, at any [math]N\geq2[/math].

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General references

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