14d. Hypergeometric laws

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

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Following now , let us discuss a remarkable relation of all this with the quantum permutations, and with the free hypergeometric laws. The idea will be that of working out some abstract algebraic results, regarding twists of quantum automorphism groups, which will particularize into results relating quantum rotations and permutations, having no classical counterpart, both at the algebraic and the probabilistic level.

In order to explain this material, from , which is quite technical, requiring good algebraic knowledge, let us begin with some generalities. We first have:

Definition

A finite quantum space [math]X[/math] is the abstract dual of a finite dimensional [math]C^*[/math]-algebra [math]B[/math], according to the following formula:

[[math]] C(X)=B [[/math]]

The number of elements of such a space is [math]|X|=\dim B[/math]. By decomposing the algebra [math]B[/math], we have a formula of the following type:

[[math]] C(X)=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

With [math]n_1=\ldots=n_k=1[/math] we obtain in this way the space [math]X=\{1,\ldots,k\}[/math]. Also, when [math]k=1[/math] the equation is [math]C(X)=M_n(\mathbb C)[/math], and the solution will be denoted [math]X=M_n[/math].

We endow each finite quantum space [math]x[/math] with its counting measure, corresponding as the algebraic level to the integration functional obtained by applying the regular representation, and then the unique normalized trace of the matrix algebra [math]\mathcal L(C(X))[/math]:

[[math]] tr:C(X)\subset\mathcal L(C(X))\to\mathbb C [[/math]]

Now if we denote by [math]\mu,\eta[/math] the multiplication and unit map of the algebra [math]C(X)[/math], we have the following standard result, from , based on some previous work from :

Theorem

Given a finite quantum space [math]X[/math], there is a universal compact quantum group [math]S_X^+[/math] acting on [math]X[/math], leaving the counting measure invariant. We have

[[math]] C(S_X^+)=C(U_N^+)\Big/\Big \lt \mu\in Hom(v^{\otimes2},v),\eta\in Fix(v)\Big \gt [[/math]]

where [math]N=|X|[/math] and where [math]\mu,\eta[/math] are the multiplication and unit maps of [math]C(X)[/math]. Also:

• For [math]X=\{1,\ldots,N\}[/math] we have [math]S_X^+=S_N^+[/math].
• For [math]X=M_n[/math] we have [math]S_X^+=PO_n^+=PU_n^+[/math].

Show Proof

Consider a linear map [math]\Phi:C(X)\to C(X)\otimes C(G)[/math], written as follows, with [math]\{e_i\}[/math] being a linear space basis of the algebra [math]C(X)[/math], orthonormal with respect to [math]tr[/math]:

[[math]] \Phi(e_j)=\sum_ie_i\otimes v_{ij} [[/math]]

Then [math]\Phi[/math] is a coaction precisely when [math]v[/math] is a unitary corepresentation, satisfying:

[[math]] \mu\in Hom(v^{\otimes2},v)\quad,\quad \eta\in Fix(v) [[/math]]

But this gives the first assertion. Regarding now the statement about [math]X=\{1,\ldots,N\}[/math], this is clear. Finally, regarding [math]X=M_2[/math], here we have embeddings as followss:

[[math]] PO_n^+\subset PU_n^+\subset S_X^+ [[/math]]

Now since the fusion rules of all these 3 quantum groups are known to be the same as the fusion rules for [math]SO_3[/math], these inclusions follow to be isomorphisms. See .

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We have as well the following result, also from :

Theorem

The quantum groups [math]S_X^+[/math] have the following properties:

• The associated Tannakian categories are [math]TL(N)[/math], with [math]N=|X|[/math].
• The main character follows the Marchenko-Pastur law [math]\pi_1[/math], when [math]N\geq4[/math].
• The fusion rules for [math]S_X^+[/math] with [math]|F|\geq4[/math] are the same as for [math]SO_3[/math].

Show Proof

This result is from , the idea being as follows:

(1) This follows from the fact that the multiplication and unit of any complex algebra, and in particular of [math]C(X)[/math], can be modeled by the following two diagrams:

[[math]] m=|\cup|\qquad,\qquad u=\cap [[/math]]

(2) The proof here is as for [math]S_N^+[/math], by using moments. To be more precise, according to (1) these moments are the Catalan numbers, which are the moments of [math]\pi_1[/math].

(3) Once again same proof as for [math]S_N^+[/math], by using the fact that the moments of [math]\chi[/math] are the Catalan numbers, which lead to the Clebsch-Gordan rules. See .

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Let us discuss now a number of more advanced twisting aspects, which will eventually lead us into probability, and hypergeometric laws. Following , we have:

Theorem

If [math]G[/math] is a finite group and [math]\sigma[/math] is a [math]2[/math]-cocycle on [math]G[/math], the Hopf algebras

[[math]] C(S_{\widehat{G}}^+)\quad,\quad C(S_{\widehat{G}_\sigma}^+) [[/math]]

are [math]2[/math]-cocycle twists of each other.

Show Proof

This is something quite technical, requiring a good knowledge of algebraic twisting techniques, and for full details here, we refer to .

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As an example, let [math]G=\mathbb Z_n^2[/math], and consider the following map, with [math]w=e^{2\pi i/n}[/math]:

[[math]] \sigma:G\times G\to\mathbb C^*\quad,\quad \sigma_{(ij)(kl)}=w^{jk} [[/math]]

Then [math]\sigma[/math] is a bicharacter, and hence a 2-cocycle on [math]G[/math]. Thus, we can apply our twisting result, to this situation. We obtain a concrete result, also from , as follows:

Theorem

Let [math]n\geq 2[/math] and [math]w=e^{2\pi i/n}[/math]. Then the formula

[[math]] \Theta(u_{ij}u_{kl})=\frac{1}{n}\sum_{ab=0}^{n-1}w^{-a(k-i)+b(l-j)}p_{ia,jb} [[/math]]

defines a coalgebra isomorphism [math]C(PO_n^+)\to C(S_{n^2}^+)[/math], commuting with the Haar integrals.

Show Proof

This follows indeed from our general twisting result from Theorem 14.33, by using as ingredients the group and the cocycle indicated above.

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As a probabilistic consequence now, which is of interest for us, we have:

Theorem

The following families of variables have the same joint law,

• [math]\{v_{ij}^2\}\in C(O_n^+)[/math],
• [math]\{\eta_{ij}=\frac{1}{n}\sum_{ab}p_{ia,jb}\}\in C(S_{n^2}^+)[/math],

where [math]v=(v_{ij})[/math] and [math]p=(p_{ia,jb})[/math] are the corresponding fundamental corepresentations.

Show Proof

This follows from Theorem 14.34. Alternatively, we can use the Weingarten formula for our quantum groups, and the shrinking operation [math]\pi\to\pi'[/math]. Indeed, we have:

[[math]] \int_{O_n^+}v_{ij}^{2k}=\sum_{\pi,\sigma\in NC_2(2k)}W_{2k,n}(\pi,\sigma) [[/math]]

[[math]] \int_{S_{\!n^2}^+}\eta_{ij}^k=\sum_{\pi,\sigma\in NC_2(2k)}n^{|\pi'|+|\sigma'|-k}W_{k,n^2}(\pi',\sigma') [[/math]]

By doing now some standard combinatorics, the summands coincide, and so the moments are equal, as desired. The proof for joint moments is similar. See .

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As an explicit application of the above, also from , we have:

Theorem

The free hyperspherical and hypergeometric variables,

[[math]] z_i^2\in C(S^{N-1}_{\mathbb R,+})\quad,\quad \eta_{ij}=\frac{1}{n}\sum_{a,b=1}^nu_{ia,jb}\in C(S_{n^2}^+) [[/math]]

has the same law.

Show Proof

This follows indeed from Theorem 14.35, particularized to the case of single variables. For details on all this, and for more, we refer to .

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As a conclusion, interesting things happen when doing noncommutative geometry. Needless to say, all this is of interest too in relation with physics. For instance in the Connes interpretation of the Standard Model, coming from , the probabilistic study of the corresponding free gauge group leads to beasts as above.

General references

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