14a. The correspondence

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We discuss here the construction of the quantum permutation group [math]G\subset S_N^+[/math] associated to a complex Hadamard matrix [math]H\in M_N(\mathbb C)[/math]. Although the construction [math]H\to G[/math] is something very simple, by modern standards, there is a long story with it, as follows:

(1) Everything goes back to an 1983 paper by Popa , who made the key remark that the pairs of maximal abelian subalgebras (MASA) in the simplest von Neumann algebra, namely the matrix algebra [math]M_N(\mathbb C)[/math], are up to conjugation the algebra of diagonal matrices [math]\Delta\subset M_N(\mathbb C)[/math] and its conjugate [math]H\Delta H^*[/math] by an Hadamard matrix [math]H\in M_N(\mathbb C)[/math].

(2) This remark of Popa suggests spending some time in understanding the complex Hadamard matrices [math]H[/math], and among the people involved was notably Jones , , with the far more refined statement, building on Popa's remark, that associated to [math]H[/math] is some sort of abstract “spin model”, whose partition function must be computed.

(3) The Jones finding can be further refined by using quantum groups, somehow in the spirit of the Yang-Baxter equation, with the result that, as announced above, there is a construction [math]H\to G[/math], with the quantum group [math]G[/math] describing the symmetries of the spin model, and with the representation theory of [math]G[/math] computing the partition function.

(4) These latter things go back to work of mine from the late 90s, but took some time to be axiomatized, mainly due to various hesitations in the choice of the formalism, and including a recurrent mistake at [math]N=4[/math] too. All this axiomatization work was done in the 00s, and with several other people, like Bichon, Nicoara, Schlenker involved too.

(5) So, this was for the story, and as a conclusion, we have nowadays a bright, simple construction of type [math]H\to G[/math], that we will explain below, and then all sorts of other more technical things that can be explained afterwards, in relation with the work of Jones, Popa and others, and that we will briefly explain too, in what follows.

Getting started now, as a first observation, the complex Hadamard matrices are related to the quantum permutation groups, via the following simple fact:

Proposition

If [math]H\in M_N(\mathbb C)[/math] is Hadamard, the rank one projections

[[math]] P_{ij}=Proj\left(\frac{H_i}{H_j}\right) [[/math]]

where [math]H_1,\ldots,H_N\in\mathbb T^N[/math] are the rows of [math]H[/math], form a magic unitary.

Show Proof

This is clear, the verification for the rows being as follows:

[[math]] \begin{eqnarray*} \left \lt \frac{H_i}{H_j},\frac{H_i}{H_k}\right \gt &=&\sum_l\frac{H_{il}}{H_{jl}}\cdot\frac{H_{kl}}{H_{il}}\\ &=&\sum_l\frac{H_{kl}}{H_{jl}}\\ &=&N\delta_{jk} \end{eqnarray*} [[/math]]

As for the verification for the columns, this is similar, as follows:

[[math]] \begin{eqnarray*} \left \lt \frac{H_i}{H_j},\frac{H_k}{H_j}\right \gt &=&\sum_l\frac{H_{il}}{H_{jl}}\cdot\frac{H_{jl}}{H_{kl}}\\ &=&\sum_l\frac{H_{il}}{H_{kl}}\\ &=&N\delta_{ik} \end{eqnarray*} [[/math]]

Thus, we have indeed a magic unitary, as claimed.

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The above result suggests the following definition:

Definition

Associated to [math]H\in M_N(\mathbb C)[/math] is the representation

[[math]] \pi:C(S_N^+)\to M_N(\mathbb C)\quad,\quad u_{ij}\to Proj\left(\frac{H_i}{H_j}\right) [[/math]]

where [math]H_1,\ldots,H_N\in\mathbb T^N[/math] are the rows of [math]H[/math].

The representation [math]\pi[/math] constructed above is a “matrix model” for the algebra [math]C(S_N^+)[/math], in the sense that the standard generators [math]u_{ij}\in C(S_N^+)[/math], and more generally any element [math]a\in C(S_N^+)[/math], gets modelled in this way by an explicit matrix [math]\pi(a)\in M_N(\mathbb C)[/math]. And the point now is that, given such a model, we have the following notions:

Definition

Let [math]G[/math] be a compact quantum group, and let [math]\pi:C(G)\to M_N(\mathbb C)[/math] be a matrix model for the associated Woronowicz algebra.

• The Hopf image of [math]\pi[/math] is the smallest quotient Woronowicz algebra [math]C(G)\to C(H)[/math] producing a factorization of type [math]\pi:C(G)\to C(H)\to M_N(\mathbb C)[/math].
• When the inclusion [math]H\subset G[/math] is an isomorphism, i.e. when there is no non-trivial factorization as above, we say that [math]\pi[/math] is inner faithful.

As a first observation, in the case where the model is faithful, in the sense that we have an inclusion [math]\pi:C(G)\subset M_N(\mathbb C)[/math], the Hopf image is the algebra [math]C(G)[/math] itself, and the model is inner faithful as well. However, this situation will not appear often in practice, because the existence of an embedding [math]C(G)\subset M_N(\mathbb C)[/math] forces the algebra [math]C(G)[/math] to be finite dimensional, and so [math]G[/math] to be a finite quantum group, which is something that we cannot expect, in general. At the level of non-trivial examples now, we have:

(1) In the case where [math]G=\widehat{\Gamma}[/math] is a group dual, the model is as follows:

[[math]] \pi:C(G)=C^*(\Gamma)\to M_N(\mathbb C) [[/math]]

Thus, this model must come from a unitary group representation [math]\rho:\Gamma\to U_N[/math], and the minimal factorization of [math]\pi[/math] is then the one obtained by taking the image:

[[math]] \rho:\Gamma\to\Lambda\subset U_N [[/math]]

Also, the model [math]\pi[/math] is inner faithful when [math]\Gamma\subset U_N[/math]. This is the main example for Definition 14.3, which provides intuition, and justifies the terminology as well.

(2) Dually, in the case where [math]G[/math] is a classical compact group, we have a standard construction of a matrix model for [math]C(G)[/math], obtained by taking an arbitrary family of elements [math]g_1,\ldots,g_N\in G[/math], and then constructing the following representation:

[[math]] \pi:C(G)\to\ M_N(\mathbb C)\quad,\quad f\to\begin{pmatrix} f(g_1)\\ &\ddots\\ &&f(g_N) \end{pmatrix} [[/math]]

The minimal factorization of [math]\pi[/math] is then via the algebra [math]C(H)[/math], with:

[[math]] H=\overline{ \lt g_1,\ldots,g_N \gt }\subset G [[/math]]

Also, [math]\pi[/math] is inner faithful precisely when [math]G=H[/math], and so when:

[[math]] G=\overline{ \lt g_1,\ldots,g_N \gt } [[/math]]

This is the second main example for the construction in Definition 14.3, which provides some further intuition, and once again justifies the terminology as well.

In general, the existence and uniqueness of the Hopf image follow by dividing [math]C(G)[/math] by a suitable ideal. We refer to , for more details regarding this construction. In relation now with the complex Hadamard matrices, we can simply combine Definition 14.2 and Definition 14.3, and we are led in this way into the following notion:

Definition

To any Hadamard matrix [math]H\in M_N(\mathbb C)[/math] we associate the quantum permutation group [math]G\subset S_N^+[/math] given by the following Hopf image factorization,

[[math]] \xymatrix{C(S_N^+)\ar[rr]^{\pi}\ar[rd]&&M_N(\mathbb C)\\&C(G)\ar[ur]&} [[/math]]

where [math]\pi(u_{ij})=Proj(H_i/H_j)[/math], with [math]H_1,\ldots,H_N\in\mathbb T^N[/math] being the rows of [math]H[/math].

This was for the general theory. Our claim now is that the construction [math]H\to G[/math] is something really useful, with [math]G[/math] encoding the combinatorics of [math]H[/math], a bit in the same way as [math]\mathbb Z_N[/math] encodes the combinatorics of [math]F_N[/math]. There are several results supporting this, and we will discuss this gradually, in what follows. As a first such result, we have:

Theorem

The construction [math]H\to G[/math] has the following properties:

• For [math]H=F_N[/math] we obtain the group [math]G=\mathbb Z_N[/math], acting on itself.
• More generally, for [math]H=F_G[/math] we obtain the group [math]G[/math] itself, acting on itself.
• For a tensor product [math]H=H'\otimes H''[/math] we obtain a product, [math]G=G'\times G''[/math].

Show Proof

All this is standard, and elementary, as follows:

(1) The rows of the Fourier matrix [math]H=F_N[/math] are given by [math]H_i=\rho^i[/math], where [math]\rho=(1,w,w^2,\ldots,w^{N-1})[/math], with [math]w=e^{2\pi i/N}[/math]. Thus, we have the following formula:

[[math]] \frac{H_i}{H_j}=\rho^{i-j} [[/math]]

It follows that the corresponding rank 1 projections [math]P_{ij}=Proj(H_i/H_j)[/math] form a circulant matrix, all whose entries commute. Since the entries commute, the corresponding quantum group must satisfy [math]G\subset S_N[/math]. Now by taking into account the circulant property of [math]P=(P_{ij})[/math] as well, we are led to the conclusion that we have [math]G=\mathbb Z_N[/math].

(2) In the general case now, where [math]H=F_G[/math], with [math]G[/math] being an arbitrary finite abelian group, the result can be proved either by extending the above proof, of by decomposing [math]G=\mathbb Z_{N_1}\times\ldots\times\mathbb Z_{N_k}[/math] and using (3) below, whose proof is independent from (1,2).

(3) Assume that we have a tensor product [math]H=H'\otimes H''[/math], and let [math]G,G',G''[/math] be the associated quantum permutation groups. We have then a diagram as follows:

[[math]] \xymatrix@R=50pt@C25pt{ C(S_{N'}^+)\otimes C(S_{N''}^+)\ar[r]&C(G')\otimes C(G'')\ar[r]&M_{N'}(\mathbb C)\otimes M_{N''}(\mathbb C)\ar[d]\\ C(S_N^+)\ar[u]\ar[r]&C(G)\ar[r]&M_N(\mathbb C) } [[/math]]

Here all the maps are the canonical ones, with those on the left and on the right coming from [math]N=N'N''[/math]. At the level of standard generators, the diagram is as follows:

[[math]] \xymatrix@R=50pt@C65pt{ u_{ij}'\otimes u_{ab}''\ar[r]&w_{ij}'\otimes w_{ab}''\ar[r]&P_{ij}'\otimes P_{ab}''\ar[d]\\ u_{ia,jb}\ar[u]\ar[r]&w_{ia,jb}\ar[r]&P_{ia,jb} } [[/math]]

Now observe that this diagram commutes. We conclude that the representation associated to [math]H[/math] factorizes indeed through [math]C(G')\otimes C(G'')[/math], and this gives the result.

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Generally speaking, going beyond Theorem 14.5 is a quite difficult question. There are several computations available here, for the most regarding the deformations of the Fourier matrices, and we will be back to this later, in chapter 16 below. At a more abstract level now, one interesting question is that of abstractly characterizing the magic matrices coming from the complex Hadamard matrices, and we have here:

Proposition

Given an Hadamard matrix [math]H\in M_N(\mathbb C)[/math], the vectors

[[math]] \xi_{ij}=\frac{H_i}{H_j} [[/math]]

on which the magic unitary entries [math]P_{ij}[/math] project, have the following properties:

• [math]\xi_{ii}=\xi[/math] is the all-one vector.
• [math]\xi_{ij}\xi_{jk}=\xi_{ik}[/math], for any [math]i,j,k[/math].
• [math]\xi_{ij}\xi_{kl}=\xi_{il}\xi_{kj}[/math], for any [math]i,j,k,l[/math].

Show Proof

All these assertions are trivial, by using the formula [math]\xi_{ij}=H_i/H_j[/math].

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Let us call now magic basis of a given Hilbert space [math]H[/math] any square array of vectors [math]\xi\in M_N(H)[/math], all whose rows and columns are orthogonal bases of [math]H[/math]. With this convention, the above observations lead to the following result, at the magic basis level:

Theorem

The magic bases [math]\xi\in M_N(S^{N-1}_\mathbb C)[/math] coming from the complex Hadamard matrices are those having the following properties:

• We have [math]\xi_{ij}\in\mathbb T^N[/math], after a suitable rescaling.
• The conditions in Proposition 14.6 are satisfied.

Show Proof

By using the multiplicativity conditions (1,2,3) in Proposition 14.6, we conclude that, up to a rescaling, we must have [math]\xi_{ij}=\xi_i/\xi_j[/math], where [math]\xi_1,\ldots,\xi_N[/math] is the first row of the magic basis. Together with our assumption [math]\xi_{ij}\in\mathbb T^N[/math], this gives the result.

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General references

Banica, Teo (2024). "Invitation to Hadamard matrices". arXiv:1910.06911 [math.CO].