13d. Liberation theory

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

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In order to study [math]S_N^+[/math], and better understand the liberation operation [math]S_N\to S_N^+[/math], we can use representation theory. We have the following version of Tannakian duality:

Theorem

The following operations are inverse to each other:

• The construction [math]A\to C[/math], which associates to any Woronowicz algebra [math]A[/math] the tensor category formed by the intertwiner spaces [math]C_{kl}=Hom(u^{\otimes k},u^{\otimes l})[/math].
• The construction [math]C\to A[/math], which associates to a tensor category [math]C[/math] the Woronowicz algebra [math]A[/math] presented by the relations [math]T\in Hom(u^{\otimes k},u^{\otimes l})[/math], with [math]T\in C_{kl}[/math].

Show Proof

This is something quite deep, going back to Woronowicz's paper in a slightly different form, with the idea being as follows:

-- We have indeed a construction [math]A\to C[/math] as above, whose output is a tensor [math]C^*[/math]-subcategory with duals of the tensor [math]C^*[/math]-category of Hilbert spaces.

-- We have as well a construction [math]C\to A[/math] as above, simply by dividing the free [math]*[/math]-algebra on [math]N^2[/math] variables by the relations in the statement.

Some elementary algebra shows then that [math]C=C_{A_C}[/math] implies [math]A=A_{C_A}[/math], and also that [math]C\subset C_{A_C}[/math] is automatic. Thus we are left with proving [math]C_{A_C}\subset C[/math], and this can be done by doing some algebra, and using von Neumann's bicommutant theorem. See .

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We will need as well, following the classical work of Weyl, Brauer and many others, the notion of “easiness”. Let us start with the following definition:

Definition

Let [math]P(k,l)[/math] be the set of partitions between an upper row of [math]k[/math] points, and a lower row of [math]l[/math] points. A set [math]D=\bigsqcup_{k,l}D(k,l)[/math] with [math]D(k,l)\subset P(k,l)[/math] is called a category of partitions when it has the following properties:

• Stability under the horizontal concatenation, [math](\pi,\sigma)\to[\pi\sigma][/math].
• Stability under the vertical concatenation, [math](\pi,\sigma)\to[^\sigma_\pi][/math].
• Stability under the upside-down turning, [math]\pi\to\pi^*[/math].
• Each set [math]P(k,k)[/math] contains the identity partition [math]||\ldots||[/math].
• The set [math]P(0,2)[/math] contains the semicircle partition [math]\cap[/math].

Observe that this is precisely the definition that we used in chapter 12, with the condition there on the basic crossing [math]\slash\hskip-2.1mm\backslash[/math], which produces commutativity via Tannakian duality, removed. In relation with the quantum groups, we have the following notion:

Definition

A compact quantum matrix group [math]G[/math] is called easy when

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

for any colored integers [math]k,l[/math], for certain sets of partitions [math]D(k,l)\subset P(k,l)[/math], where

[[math]] T_\pi(e_{i_1}\otimes\ldots\otimes e_{i_k})=\sum_{j_1\ldots j_l}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_l} [[/math]]

with the Kronecker type symbols [math]\delta_\pi\in\{0,1\}[/math] depending on whether the indices fit or not.

Again, this is something coming as a continuation of the material from chapter 12. Many things can be said here, but getting now straight to the point, we have:

Theorem

We have the following results:

• [math]S_N[/math] is easy, coming from the category of all partitions [math]P[/math].
• [math]S_N^+[/math] is easy, coming from the category of all noncrossing partitions [math]NC[/math].

Show Proof

This is something quite fundamental, with the proof, using the above Tannakian results and subsequent easiness theory, being as follows:

(1) [math]S_N^+[/math]. We know that this quantum group comes from the magic condition. In order to interpret this magic condition, consider the fork partition:

[[math]] Y\in P(2,1) [[/math]]

By arguing as in chapter 8, we conclude that we have the following equivalence:

[[math]] T_Y\in Hom(u^{\otimes 2},u)\iff u_{ij}u_{ik}=\delta_{jk}u_{ij},\forall i,j,k [[/math]]

The condition on the right being equivalent to the magic condition, we conclude that [math]S_N^+[/math] is indeed easy, the corresponding category of partitions being, as desired:

[[math]] D = \lt Y \gt =NC [[/math]]

(2) [math]S_N[/math]. Here there is no need for new computations, because we have:

[[math]] S_N=S_N^+\cap O_N [[/math]]

At the categorical level means that [math]S_N[/math] is easy, coming from:

[[math]] \lt NC,\slash\hskip-2.2mm\backslash \gt =P [[/math]]

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

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Summarizing, we have now a good understanding of the liberation operation [math]S_N\to S_N^+[/math], the idea being that this comes, via Tannakian duality, from [math]P\to NC[/math].

In order to go further in this direction, we will need the following result, with [math]*[/math] being the classical convolution, and [math]\boxplus[/math] being Voiculescu's free convolution operation :

Theorem

The following Poisson type limits converge, for any [math]t \gt 0[/math],

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

[[math]] \pi_t=\lim_{n\to\infty}\left(\left(1-\frac{1}{n}\right)\delta_0+\frac{1}{n}\delta_t\right)^{\boxplus n} [[/math]]

the limiting measures being the Poisson law [math]p_t[/math], and the Marchenko-Pastur law [math]\pi_t[/math],

[[math]] p_t=\frac{1}{e^t}\sum_{k=0}^\infty\frac{t^k\delta_k}{k!} [[/math]]

[[math]] \pi_t=\max(1-t,0)\delta_0+\frac{\sqrt{4t-(x-1-t)^2}}{2\pi x}\,dx [[/math]]

whose moments are given by the following formulae:

[[math]] M_k(p_t)=\sum_{\pi\in P(k)}t^{|\pi|}\quad,\quad M_k(\pi_t)=\sum_{\pi\in NC(k)}t^{|\pi|} [[/math]]

The Marchenko-Pastur measure [math]\pi_t[/math] is also called free Poisson law.

Show Proof

This is something quite advanced, related to probability theory, free probability theory, and random matrices, the idea being as follows:

(1) The first step is that of finding suitable functional transforms, which linearize the convolution operations in the statement. In the classical case this is the logarithm of the Fourier transform [math]\log F[/math], and in the free case this is Voiculescu's [math]R[/math]-transform.

(2) With these tools in hand, the above limiting theorems can be proved in a standard way, a bit as when proving the Central Limit Theorem. The computations give the moment formulae in the statement, and the density computations are standard as well.

(3) Finally, in order for the discussion to be complete, what still remains to be explained is the precise nature of the “liberation” operation [math]p_t\to\pi_t[/math], as well as the random matrix occurrence of [math]\pi_t[/math]. This is more technical, and we refer here to , , .

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Getting back now to quantum permutations, the results here are as follows:

Theorem

The law of the main character, given by

[[math]] \chi=\sum_iu_{ii} [[/math]]

for [math]S_N/S_N^+[/math] becomes [math]p_1/\pi_1[/math] with [math]N\to\infty[/math]. As for the truncated character

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

for [math]S_N/S_N^+[/math], with [math]t\in(0,1][/math], this becomes [math]p_t/\pi_t[/math] with [math]N\to\infty[/math].

Show Proof

This is again something quite technical, the idea being as follows:

(1) In the classical case this is well-known, and follows by using the inclusion-exclusion principle, and then letting [math]N\to\infty[/math], as explained in chapter 11.

(2) In the free case there is no such simple argument, and we must use what we know about [math]S_N^+[/math], namely its easiness property. We know from easiness that we have:

[[math]] Fix(u^{\otimes k})=span(NC(k)) [[/math]]

On the other hand, a direct computation shows that the partitions in [math]P(k)[/math], and in particular those in [math]NC(k)[/math], implemented as linear maps via the operation [math]\pi\to T_\pi[/math] from Definition 13.34, become linearly independent with [math]N\geq k[/math]. Thus we have, as desired:

[[math]] \begin{eqnarray*} \int_{S_N^+}\chi^k &=&\dim\left(Fix(u^{\otimes k})\right)\\ &=&\dim\left(span\left(T_\pi\Big|\pi\in NC(k)\right)\right)\\ &\simeq&|NC(k)|\\ &=&\sum_{\pi\in NC(k)}1^{|\pi|} \end{eqnarray*} [[/math]]

(3) In the general case now, where our parameter is an arbitrary number [math]t\in(0,1][/math], the above computation does not apply, but we can still get away with Peter-Weyl theory. Indeed, we know from Theorem 13.25 how to compute the Haar integration of [math]S_N^+[/math], out of the knowledge of the fixed point spaces [math]Fix(u^{\otimes k})[/math], and in practice, by using easiness, this leads to the following formula, called Weingarten integration formula:

[[math]] \int_{S_N^+}u_{i_1j_1}\ldots u_{i_kj_k}=\sum_{\pi,\sigma\in NC(k)}\delta_\pi(i)\delta_\sigma(j)W_{kN}(\pi,\sigma) [[/math]]

Here the [math]\delta[/math] symbols are Kronecker type symbols, checking whether the indices fit or not with the partitions, 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. Now by using this formula for computing the moments of [math]\chi_t[/math], we obtain:

[[math]] \begin{eqnarray*} \int_{S_N^+}\chi_t^k &=&\sum_{i_1=1}^{[tN]}\ldots\sum_{i_k=1}^{[tN]}\int u_{i_1i_1}\ldots u_{i_ki_k}\\ &=&\sum_{\pi,\sigma\in NC(k)}W_{kN}(\pi,\sigma)\sum_{i_1=1}^{[tN]}\ldots\sum_{i_k=1}^{[tN]}\delta_\pi(i)\delta_\sigma(i)\\ &=&\sum_{\pi,\sigma\in NC(k)}W_{kN}(\pi,\sigma)G_{k[tN]}(\sigma,\pi)\\ &=&Tr(W_{kN}G_{k[tN]}) \end{eqnarray*} [[/math]]

(4) The point now is that with [math]N\to\infty[/math] the Gram matrix [math]G_{kN}[/math], and so the Weingarten matrix [math]W_{kN}[/math] too, becomes asymptotically diagonal. We therefore obtain:

[[math]] \int_{S_N^+}\chi_t^k\simeq\sum_{\pi\in NC(k)}t^{|\pi|} [[/math]]

Thus, we are led to the conclusion in the statement. For details, see .

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General references

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