15b. Reverse De Finetti

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

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With the above ingredients in hand, we can now investigate invariance questions for the sequences of classical or noncommutative random variables, with respect to the main quantum permutation and rotation groups that we are interested in here, namely:

[[math]] \xymatrix@R=15mm@C=15mm{ S_N^+\ar[r]&O_N^+\\ S_N\ar[r]\ar[u]&O_N\ar[u] } [[/math]]

To be more precise, we first have a reverse De Finetti theorem, from , as follows:

Theorem

Let [math](x_1,\ldots,x_N)[/math] be a sequence in [math]A[/math].

• If [math]x_1,\ldots,x_N[/math] are freely independent and identically distributed with amalgamation over [math]B[/math], then the sequence is [math]S_N^+[/math]-invariant.
• If [math]x_1,\ldots,x_N[/math] are freely independent and identically distributed with amalgamation over [math]B[/math], and have centered semicircular distributions with respect to [math]E[/math], then the sequence is [math]O_N^+[/math]-invariant.
• If [math] \lt B,x_1,\ldots,x_N \gt [/math] is commutative and [math]x_1,\ldots,x_N[/math] are conditionally independent and identically distributed given [math]B[/math], then the sequence is [math]S_N[/math]-invariant.
• If [math] \lt x_1,\ldots,x_N \gt [/math] is commutative and [math]x_1,\ldots,x_N[/math] are conditionally independent and identically distributed given [math]B[/math], and have centered Gaussian distributions with respect to [math]E[/math], then the sequence is [math]O_N[/math]-invariant.

Show Proof

Assume that the joint distribution of [math](x_1,\ldots,x_N)[/math] satisfies one of the conditions in the statement, and let [math]D[/math] be the category of partitions associated to the corresponding easy quantum group. We have then the following computation:

[[math]] \begin{eqnarray*} \sum_{j_1\ldots j_k}tr(x_{j_1}\ldots x_{j_k})v_{j_1i_1}\ldots v_{j_ki_k} &=&\sum_{j_1\ldots j_k}tr(E(x_{j_1}\ldots x_{j_k}))v_{j_1i_1}\ldots v_{j_ki_k}\\ &=&\sum_{j_1\ldots j_k}\sum_{\pi\leq\ker j}tr(\xi^{(\pi)}_E(x_1,\ldots,x_1))v_{j_1i_1}\ldots v_{j_ki_k}\\ &=&\sum_{\pi\in D(k)}tr(\xi^{(\pi)}_E(x_1,\ldots,x_1))\sum_{\ker j\geq\pi}v_{j_1i_1}\ldots v_{j_ki_k} \end{eqnarray*} [[/math]]

Here [math]\xi[/math] denotes the free and classical cumulants in the cases (1,2) and (3,4) respectively. On the other hand, it follows from a direct computation that if [math]\pi\in D(k)[/math] then we have the following formula, in each of the 4 cases in the statement:

[[math]] \sum_{\ker j\geq\pi}v_{j_1i_1}\ldots v_{j_ki_k}= \begin{cases}1&{\rm if}\ \pi\leq\ker i\\ 0&{\rm otherwise} \end{cases} [[/math]]

By using this formula, we can finish our computation, in the following way:

[[math]] \begin{eqnarray*} \sum_{j_1\ldots j_k}tr(x_{j_1}\ldots x_{j_k})v_{j_1i_1}\ldots v_{j_ki_k} &=&\sum_{\pi\in D(k)}tr(\xi^{(\pi)}_E(x_1,\ldots,x_1))\delta_{\pi\leq\ker i}\\ &=&\sum_{\pi\leq\ker i}tr(\xi_E^{(\pi)}(x_1,\ldots,x_1))\\ &=&tr(x_{i_1}\ldots x_{i_k}) \end{eqnarray*} [[/math]]

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

■

Summarizing, we have so far a reverse De Finetti theorem, for the various quantum groups that we are interested in here. Our goal in what follows will be that of proving the corresponding De Finetti theorems, which are converse to the above theorem.

This will be something quite technical, getting us, among others, into certain technical aspects of the Weingarten integration and combinatorics.

Let us begin with some technical results, in view to establish the above-mentioned converse De Finetti theorems. We will use the following standard fact:

Proposition

Assume that a sequence [math](x_1,\ldots,x_N)[/math] is [math]G[/math]-invariant. Then there is a coaction

[[math]] \widetilde{\alpha}:M_N(\mathbb C)\to M_N(\mathbb C)\otimes C(G) [[/math]]

determined by the following formula:

[[math]] \widetilde{\alpha}(p(x))=(ev_x\otimes\pi_N)\alpha(p) [[/math]]

Moreover, the fixed point algebra of [math]\widetilde{\alpha}[/math] is the [math]G[/math]-invariant subalgebra [math]B_N[/math].

Show Proof

This follows indeed after identifying the GNS representation of the algebra [math]\mathbb C \lt t_1,\ldots,t_N \gt [/math] for the state [math]\mu_x[/math] with the morphism [math]ev_x:\mathbb C \lt t_1,\ldots,t_N \gt \to M_N(\mathbb C)[/math].

■

In order to further advance, we use the fact that there is a natural conditional expectation given by integrating the coaction [math]\widetilde{\alpha}[/math] with respect to the Haar state, as follows:

[[math]] E_N:M_N(\mathbb C)\to B_N [[/math]]

[[math]] E_N(m)=\left(id\otimes\int_G\right)\widetilde{\alpha}(m) [[/math]]

The point now is that by using the Weingarten formula, we can give a simple combinatorial formula for the moment functionals with respect to [math]E_N[/math], in the case where [math]G[/math] is one of the easy quantum groups under consideration.

To be more precise, we have the following result, from :

Theorem

Assume that [math](x_1,\ldots,x_N)[/math] is [math]G[/math]-invariant, and that either we have [math]G=O_N^+,S_N^+[/math], or that [math]G=O_N,S_N[/math] and [math](x_1,\ldots,x_N)[/math] commute. We have then

[[math]] E_N^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k)=\frac{1}{N^{|\pi|}}\sum_{\pi\leq\ker i} b_0x_{i_1}\ldots b x_{i_k}b_k [[/math]]

for any [math]\pi[/math] in the partition category [math]D(k)[/math] for [math]G[/math], and any [math]b_0,\ldots,b_k\in B_N[/math].

Show Proof

We prove this result by recurrence on the number of blocks of [math]\pi[/math]. First suppose that [math]\pi=1_k[/math] is the partition with only one block. Then:

[[math]] \begin{eqnarray*} E_N^{(1_k)}(b_0x_1b_1,\ldots,x_1b_k) &=&E_N(b_0x_1\ldots x_1b_k)\\ &=&\sum_{i_1 \ldots i_k}b_0x_{i_1}\ldots x_{i_k}b_k\int_Gv_{i_11}\ldots v_{i_k1} \end{eqnarray*} [[/math]]

Here we have used the fact that the elements [math]b_0,\dotsc,b_k[/math] are fixed by the coaction [math]\widetilde{\alpha}[/math]. Applying now the Weingarten integration formula, we have:

[[math]] \begin{eqnarray*} E_N(b_0x_1\ldots x_1b_k) &=&\sum_{i_1\ldots i_k}b_0x_{i_1}\ldots x_{i_k}b_k\sum_{\pi\leq\ker i}\sum_\sigma W_{kN}(\pi,\sigma)\\ &=&\sum_{\pi\in D(k)}\left(\sum_{\sigma\in D(k)}W_{kN}(\pi,\sigma)\right) \sum_{\pi\leq\ker i}b_0x_{i_1}\ldots x_{i_k}b_k \end{eqnarray*} [[/math]]

Now observe that for any [math]\sigma\in D(k)[/math] we have the following formula:

[[math]] G_{kN}(\sigma,1_k)=N^{|\sigma\vee1_k|}=N [[/math]]

It follows that for any partition [math]\pi \in D(k)[/math], we have:

[[math]] \begin{eqnarray*} N\sum_{\sigma\in D(k)}W_{kN}(\pi,\sigma) &=&\sum_{\sigma \in D(k)}W_{kN}(\pi,\sigma)G_{kN}(\sigma,1_k)\\ &=&\delta_{\pi1_k} \end{eqnarray*} [[/math]]

Applying this in the above context, we find, as desired:

[[math]] \begin{eqnarray*} E_N(b_0x_1\ldots x_1b_k) &=&\sum_{\pi\in D(k)}\frac{1}{N}\,\delta_{\pi1_k}\sum_{\pi\leq\ker i}b_0x_{i_1}\ldots x_{i_k}b_k\\ &=&\frac{1}{N}\sum_{i=1}^Nb_0x_i\ldots x_ib_k \end{eqnarray*} [[/math]]

If the condition (3) or (4) is satisfied, then the general case follows from:

[[math]] E_N^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k)=b_1\ldots b_k\prod_{V\in\pi}E_N(V)(x_1,\ldots,x_1) [[/math]]

Indeed, the one thing that we must check here is that if [math]\pi\in D(k)[/math] and [math]V[/math] is a block of [math]\pi[/math] with [math]s[/math] elements, then [math]1_s\in D(s)[/math]. But this is easily verified, in each case.

Assume now that the condition (1) or (2) is satisfied. Let [math]\pi\in D(k)[/math]. Since [math]\pi[/math] is noncrossing, [math]\pi[/math] contains an interval [math]V=\{l+1,\ldots,l+s+1\}[/math], and we have:

[[math]] \begin{eqnarray*} &&E_N^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k)\\ &=&E_N^{(\pi-V)}(b_0x_1b_1,\ldots,E_N(x_1b_{l+1}\ldots x_1b_{l+s})x_1,\ldots,x_1b_k) \end{eqnarray*} [[/math]]

To apply induction, we must check that we have [math]\pi-V\in D(k-s)[/math] and [math]1_s\in D(s)[/math]. Indeed, this is easily verified for [math]NC,NC_2[/math]. Applying induction, we have:

[[math]] \begin{eqnarray*} &&E_N^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k)\\ &=&\frac{1}{N^{|\pi|-1}}\sum_{\pi-V\leq\ker i}b_0x_{i_1}\ldots b_l\left(E_n(x_1b_{l+1}\ldots x_1b_{l+s})\right)x_{i_{l+s}}\ldots x_{i_k}b_k\\ &=&\frac{1}{N^{|\pi|-1}}\sum_{\pi-V\leq\ker i}b_0x_{i_1}\ldots b_l\left(\frac{1}{N}\sum_{i=1}^Nx_ib_{l+1}\ldots b x_ib_{l+s}\right)x_{i_{l+s}}\ldots x_{i_k}b_k\\ &=&\frac{1}{N^{|\pi|}}\sum_{\pi\leq\ker i}b_0x_{i_1}\ldots x_{i_k}b_k \end{eqnarray*} [[/math]]

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

■

Summarizing, we have so far reverse De Finetti theorems for the quantum groups that we are interested in here, along with some technical results, connecting the corresponding potential De Finetti theorems to the Weingarten function combinatorics.

General references

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