16d. Fourier models

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In what follows we discuss the Hadamard models, which are of particular importance. Let us start with the following well-known definition:

Definition

A complex Hadamard matrix is a square matrix

[[math]] H\in M_N(\mathbb C) [[/math]]

whose entries are on the unit circle, and whose rows are pairwise orthogonal.

Observe that the orthogonality condition tells us that the rescaled matrix [math]U=H/\sqrt{N}[/math] must be unitary. Thus, these matrices form a real algebraic manifold, given by:

[[math]] X_N=M_N(\mathbb T)\cap\sqrt{N}U_N [[/math]]

The basic example is the Fourier matrix, [math]F_N=(w^{ij})[/math] with [math]w=e^{2\pi i/N}[/math]. In standard matrix form, and with indices [math]i,j=0,1,\ldots,N-1[/math], this matrix is as follows:

[[math]] F_N=\begin{pmatrix} 1&1&1&\ldots&1\\ 1&w&w^2&\ldots&w^{N-1}\\ 1&w^2&w^4&\ldots&w^{2(N-1)}\\ \vdots&\vdots&\vdots&&\vdots\\ 1&w^{N-1}&w^{2(N-1)}&\ldots&w^{(N-1)^2} \end{pmatrix} [[/math]]

More generally, we have as example the Fourier coupling of any finite abelian group [math]G[/math], regarded via the isomorphism [math]G\simeq\widehat{G}[/math] as a square matrix, [math]F_G\in M_G(\mathbb C)[/math]:

[[math]] F_G= \lt i,j \gt _{i\in G,j\in\widehat{G}} [[/math]]

Observe that for the cyclic group [math]G=\mathbb Z_N[/math] we obtain in this way the above standard Fourier matrix [math]F_N[/math]. In general, we obtain a tensor product of Fourier matrices [math]F_N[/math].

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

Theorem

Given a finite abelian group [math]G[/math], with dual group [math]\widehat{G}=\{\chi:G\to\mathbb T\}[/math], consider the Fourier coupling [math]\mathcal F_G:G\times\widehat{G}\to\mathbb T[/math], given by [math](i,\chi)\to\chi(i)[/math].

• Via the standard isomorphism [math]G\simeq\widehat{G}[/math], this Fourier coupling can be regarded as a square matrix, [math]F_G\in M_G(\mathbb T)[/math], which is a complex Hadamard matrix.
• In the case of the cyclic group [math]G=\mathbb Z_N[/math] we obtain in this way, via the standard identification [math]\mathbb Z_N=\{1,\ldots,N\}[/math], the Fourier matrix [math]F_N[/math].
• In general, when using a decomposition [math]G=\mathbb Z_{N_1}\times\ldots\times\mathbb Z_{N_k}[/math], the corresponding Fourier matrix is given by [math]F_G=F_{N_1}\otimes\ldots\otimes F_{N_k}[/math].

Show Proof

This follows indeed from some basic facts from group theory:

(1) With the identification [math]G\simeq\widehat{G}[/math] made our matrix is given by [math](F_G)_{i\chi}=\chi(i)[/math], and the scalar products between the rows are computed as follows:

[[math]] \begin{eqnarray*} \lt R_i,R_j \gt &=&\sum_\chi\chi(i)\overline{\chi(j)}\\ &=&\sum_\chi\chi(i-j)\\ &=&|G|\cdot\delta_{ij} \end{eqnarray*} [[/math]]

Thus, we obtain indeed a complex Hadamard matrix.

(2) This follows from the well-known and elementary fact that, via the identifications [math]\mathbb Z_N=\widehat{\mathbb Z_N}=\{1,\ldots,N\}[/math], the Fourier coupling here is as follows, with [math]w=e^{2\pi i/N}[/math]:

[[math]] (i,j)\to w^{ij} [[/math]]

(3) We use here the following well-known formula, for the duals of products:

[[math]] \widehat{H\times K}=\widehat{H}\times\widehat{K} [[/math]]

At the level of the corresponding Fourier couplings, we obtain from this:

[[math]] F_{H\times K}=F_H\otimes F_K [[/math]]

Now by decomposing [math]G[/math] into cyclic groups, as in the statement, and by using (2) for the cyclic components, we obtain the formula in the statement.

■

There are many other examples of Hadamard matrices, with some being fairly exotic, appearing in various branches of mathematics and physics. The idea is that the complex Hadamard matrices can be though of as being “generalized Fourier matrices”, and this is where the interest in these matrices comes from.

In relation with the quantum groups, the starting observation is as follows:

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]]

The verification for the columns 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 obtain the result.

■

We can proceed now exactly in the same way as we did with the Weyl matrices, namely by constructing a model of [math]C(S_N^+)[/math], and performing the Hopf image construction. We are led in this way to the following definition:

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 fact that [math]C(G)[/math] is the Hopf image of

[[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].

Summarizing, we have a construction [math]H\to G[/math], and our claim is that this construction is something really useful, with [math]G[/math] encoding the combinatorics of [math]H[/math]. To be more precise, our claim is that “[math]H[/math] can be thought of as being a kind of Fourier matrix for [math]G[/math]”.

There are several results supporting this claim, with the main evidence coming from the following result, which collects the basic results regarding the construction [math]H\to G[/math]:

Theorem

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

• For a Fourier matrix [math]H=F_G[/math] we obtain the group [math]G[/math] itself, acting on itself.
• For [math]H\not\in\{F_G\}[/math], the quantum group [math]G[/math] is not classical, nor a group dual.
• For a tensor product [math]H=H'\otimes H''[/math] we obtain a product, [math]G=G'\times G''[/math].

Show Proof

All this material is standard, and elementary, as follows:

(1) Let us first discuss the cyclic group case, where our Hadamard matrix is a usual Fourier matrix, [math]H=F_N[/math]. Here the rows of [math]H[/math] are given by [math]H_i=\rho^i[/math], where:

[[math]] \rho=(1,w,w^2,\ldots,w^{N-1}) [[/math]]

Thus, we have the following formula, for the associated magic basis:

[[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]]

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 the rest.

(2) This is something more tricky, needing some general study of the representations whose Hopf images are commutative, or cocommutative.

(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=45pt@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=45pt@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.

■

Going beyond the above result is an interesting question, and we refer here to , and follow-up papers. There are several computations available here, for the most regarding the deformations of the Fourier models. We believe that the unification of all this with the Weyl matrix models is a very good question, related to many interesting things.

And this is all. In the hope that you liked the present book, and that we will see you soon doing some research on the quantum groups. With things here being however a bit tricky, and here is some advice on this, research matters, to finish with:

(1) Generally speaking, quantum groups have been around since the late 70s, and the work by Faddeev and others , and so, many things are known about them. The whole area is quite advanced, and if you want to come up with some truly original, interesting new things, you need to know well mathematics and physics. No less than that.

(2) So this would be my advice, learn some mathematics and physics. And be aware that you'll have to do that alone, with your love for mathematics and physics being the only thing that you can rely upon. Of course, some things can be learned from various communities, but community basically means specialization, so wrong way.

(3) Getting to mathematics, besides Rudin which is the Bible, you can learn all sorts of useful things from Arnold , Atiyah , Connes , Drinfeld , Jones , Voiculescu , von Neumann , Witten . These are all people knowing well both mathematics and physics, and reading their writings is certainly a good idea.

(4) As for physics, for some general learning here, rather quantum mechanics oriented, you have Feynman , , , or Griffiths , , , or Weinberg , . But, and importantly, if needed complete with some classical mechanics, say from Kibble , and some thermodynamics, say from Schroeder or Huang .

Finally, in what concerns the closed subgroups [math]G\subset U_N^+[/math] from this book, as a good continuation, you can read various standard papers on easiness, such as , , , , , , , , , all from the 00s, and also various standard papers on quantum permutations, such as , , , , , , , for the most from the late 00s and early 10s. This is certainly something quite time-consuming, but with this, you can virtually read afterwards anything that you want to, on quantum groups. \begin{exercises} The matrix modelling problematics from this chapter is something quite exciting, and we have several exercises here. To start with, we have the following question:

This is something quite theoretical, the problem being that of proving that the universal model space [math]T_K[/math] in the above is indeed compact.

Here the first question is something more or less trivial, and so the exercise is about finding counterexamples at [math]K\geq2[/math], in the quantum group case.

This is something that we already discussed in the above, but with some standard functional analysis details missing. The problem is that of working out these details.

This is something quite tricky, and it is of course possible to cheat a bit here, by using product operations. The exercise asks for a high-quality counterexample.

This is something that we discussed above, but the problem now is that of doing the thing, and writing down a concise, self-contained proof for the faithfulness of [math]\pi[/math]. \begin{thebibliography}{99} \bibitem{abe}E. Abe, Hopf algebras, Cambridge Univ. Press (1980). \bibitem{agz}G.W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices, Cambridge Univ. Press (2010). \bibitem{arn}V.I. Arnold, Mathematical methods of classical mechanics, Springer (1974). \bibitem{ati}M.F. Atiyah, The geometry and physics of knots, Cambridge Univ. Press (1990). \bibitem{ba1}T. Banica, The free unitary compact quantum group, Comm. Math. Phys. 190 (1997), 143--172. \bibitem{ba2}T. Banica, Symmetries of a generic coaction, Math. Ann. 314 (1999), 763--780. \bibitem{ba3}T. Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005), 243--280. \bibitem{ba4}T. Banica, Liberation theory for noncommutative homogeneous spaces, Ann. Fac. Sci. 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General references

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