16d. Poisson laws

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

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Let us go back now to the general case, where [math]M,N\in\mathbb N[/math] are arbitrary. The problem that we would like to solve is that of finding the good regime, of the following type, where the measure in Theorem 16.16 converges, after some suitable manipulations:

[[math]] M=f(K)\quad,\quad N=g(K)\quad,\quad K\to\infty [[/math]]

As before by following , we will see that this is indeed possible, and that as limiting laws we have some very interesting objects, namely some versions of the Marchenko-Pastur laws, or free Poisson laws, that we met at the end of chapter 13. Let us first recall from there the definition and main properties of these laws, in the general context:

Theorem

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

[[math]] p_t=\lim_{n\to\infty}\left(\left(1-\frac{t}{n}\right)\delta_0+\frac{t}{n}\delta_1\right)^{*n}\quad,\quad \pi_t=\lim_{n\to\infty}\left(\left(1-\frac{t}{n}\right)\delta_0+\frac{t}{n}\delta_1\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!}\quad,\quad \pi_t=\max(1-t,0)\delta_0+\frac{\sqrt{4t-(x-1-t)^2}}{2\pi x}\,dx [[/math]]

with at [math]t=1[/math], the Marchenko-Pastur law being given by the following formula:

[[math]] \pi_1=\frac{1}{2\pi}\sqrt{4x^{-1}-1}\,dx [[/math]]

Moreover, the moments of these laws are given by the 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]]

where [math]|.|[/math] is the number of blocks.

Show Proof

All this is standard probability and free probability theory:

(1) In what regards the classical results, concerning [math]p_t[/math], the standard way of viewing them is by defining the Poisson law [math]p_t[/math] by the formula in the statement, then by establishing the Poisson Limiting Theorem (PLT) via Fourier transform, and finally by working out the moment formula either by recurrence, or from Fourier via cumulants.

(2) In the free case now, in relation with [math]\pi_t[/math], pretty much the same procedure can be used, with however the change that the study of free PLT comes first, by using Voiculescu's [math]R[/math]-transform, which produces then via Stieltjes inversion the formula of [math]\pi_t[/math] in the statement. We refer here to , or to any other free probability book.

■

In order to establish our results, we have to do some combinatorics. We denote by [math]NC(p)[/math] the set of noncrossing partitions of [math]\{1,\ldots,p\}[/math], and for [math]\pi\in P(p)[/math] we denote by [math]|\pi|\in\{1,\ldots,p\}[/math] the number of blocks. We will also use some standard tools from combinatorics, such as the Kreweras complementation, which are well-known in free probability . With these conventions, we have the following result from , regarding the moments [math]c_p[/math] of the measure that we are interested in, computed in Theorem 16.16:

Proposition

With [math]M=\alpha K,N=\beta K[/math], [math]K\to\infty[/math] we have:

[[math]] \frac{c_p}{K^{p-1}}\simeq\sum_{r=1}^p\#\left\{\pi\in NC(p)\Big||\pi|=r\right\}\alpha^{r-1}\beta^{p-r} [[/math]]

In particular, with [math]\alpha=\beta[/math] we have:

[[math]] c_p\simeq\frac{1}{p+1}\binom{2p}{p}(\alpha K)^{p-1} [[/math]]

Show Proof

We use the combinatorial formula in Theorem 16.16. Our claim is that, with [math]\pi=\ker(i_1,\ldots,i_p)[/math], the corresponding contribution to [math]c_p[/math] is:

[[math]] C_\pi\simeq \begin{cases} \alpha^{|\pi|-1}\beta^{p-|\pi|}K^{p-1}&{\rm if}\ \pi\in NC(p)\\ O(K^{p-2})&{\rm if}\ \pi\notin NC(p) \end{cases} [[/math]]

As a first observation, the number of choices for a multi-index [math](i_1,\ldots,i_p)\in X^p[/math] satisfying the condition [math]\ker i=\pi[/math] is:

[[math]] M(M-1)\ldots (M-|\pi|+1)\simeq M^{|\pi|} [[/math]]

Thus, we have the following estimate:

[[math]] C_\pi\simeq M^{|\pi|-1}N^{-1}\#\left\{d_1,\ldots,d_p\in Y\Big|[d_\alpha|\alpha\in b]=[d_{\alpha-1}|\alpha\in b],\forall b\in\pi\right\} [[/math]]

Consider now the following partition:

[[math]] \sigma=\ker d [[/math]]

The contribution of [math]\sigma[/math] to the above quantity [math]C_\pi[/math] is then given by:

[[math]] \Delta(\pi,\sigma)N(N-1)\ldots(N-|\sigma|+1)\simeq\Delta(\pi,\sigma)N^{|\sigma|} [[/math]]

Here the quantities on the right are as follows:

[[math]] \Delta(\pi,\sigma)=\begin{cases} 1&{\rm if}\ |b\cap c|=|(b-1)\cap c|,\forall b\in\pi,\forall c\in\sigma\\ 0&{\rm otherwise} \end{cases} [[/math]]

We use now the standard fact that for [math]\pi,\sigma\in P(p)[/math] satisfying [math]\Delta(\pi,\sigma)=1[/math] we have:

[[math]] |\pi|+|\sigma|\leq p+1 [[/math]]

In addition, the equality case is known to happen when [math]\pi,\sigma\in NC(p)[/math] are inverse to each other, via Kreweras complementation. This shows that for [math]\pi\notin NC(p)[/math] we have:

[[math]] C_\pi=O(K^{p-2}) [[/math]]

Also, this shows that for [math]\pi\in NC(p)[/math] we have:

[[math]] \begin{eqnarray*} C_\pi &\simeq&M^{|\pi|-1}N^{-1}N^{p-|\pi|-1}\\ &=&\alpha^{|\pi|-1}\beta^{p-|\pi|}K^{p-1} \end{eqnarray*} [[/math]]

Thus, we have obtained the result.

■

We denote by [math]D[/math] the dilation operation for probability measures, given by:

[[math]] D_r(law(X))=law(rX) [[/math]]

With this convention, we have the following result, based on Proposition 16.22:

Theorem

With [math]M=\alpha K,N=\beta K[/math], [math]K\to\infty[/math] we have:

[[math]] \mu=\left(1-\frac{1}{\alpha\beta K^2}\right)\delta_0+\frac{1}{\alpha\beta K^2}D_{\frac{1}{\beta K}}(\pi_{\alpha/\beta}) [[/math]]

In particular with [math]\alpha=\beta[/math] we have:

[[math]] \mu=\left(1-\frac{1}{\alpha^2K^2}\right)\delta_0+\frac{1}{\alpha^2K^2}D_{\frac{1}{\alpha K}}(\pi_1) [[/math]]

Show Proof

At [math]\alpha=\beta[/math], this follows from Proposition 16.22. In general now, we have:

[[math]] \begin{eqnarray*} \frac{c_p}{K^{p-1}} &\simeq&\sum_{\pi\in NC(p)}\alpha^{|\pi|-1}\beta^{p-|\pi|}\\ &=&\frac{\beta^p}{\alpha}\sum_{\pi\in NC(p)}\left(\frac{\alpha}{\beta}\right)^{|\pi|}\\ &=&\frac{\beta^p}{\alpha}\int x^pd\pi_{\alpha/\beta}(x) \end{eqnarray*} [[/math]]

When [math]\alpha\geq\beta[/math], where [math]d\pi_{\alpha/\beta}(x)=\varphi_{\alpha/\beta}(x)dx[/math] is continuous, we obtain:

[[math]] \begin{eqnarray*} c_p &=&\frac{1}{\alpha K}\int(\beta Kx)^p\varphi_{\alpha/\beta}(x)dx\\ &=&\frac{1}{\alpha\beta K^2}\int x^p\varphi_{\alpha/\beta}\left(\frac{x}{\beta K}\right)dx \end{eqnarray*} [[/math]]

But this gives the formula in the statement. When [math]\alpha\leq\beta[/math] the computation is similar, with a Dirac mass as 0 dissapearing and reappearing, and gives the same result.

■

Let us state as well an explicit result, regarding densities:

Theorem

With [math]M=\alpha K,N=\beta K[/math], [math]K\to\infty[/math] we have:

[[math]] \mu=\left(1-\frac{1}{\alpha\beta K^2}\right)\delta_0+\frac{1}{\alpha\beta K^2}\cdot\frac{\sqrt{4\alpha\beta K^2-(x-\alpha K-\beta K)^2}}{2\pi x}\,dx [[/math]]

In particular with [math]\alpha=\beta[/math] we have:

[[math]] \mu=\left(1-\frac{1}{\alpha^2K^2}\right)\delta_0+\frac{1}{\alpha^2K^2}\cdot\frac{\sqrt{\frac{4\alpha K}{x}-1}}{2\pi} [[/math]]

Show Proof

According to the formula for the density of the free Poisson law, the density of the continuous part [math]D_{\frac{1}{\beta K}}(\pi_{\alpha/\beta})[/math] is indeed given by:

[[math]] \frac{\sqrt{4\frac{\alpha}{\beta}-(\frac{x}{\beta K}-1-\frac{\alpha}{\beta})^2}} {2\pi\cdot\frac{x}{\beta K}}=\frac{\sqrt{4\alpha\beta K^2-(x-\alpha K-\beta K)^2}}{2\pi x} [[/math]]

With [math]\alpha=\beta[/math] now, we obtain the second formula in the statement, and we are done.

■

Observe that at [math]\alpha=\beta=1[/math], where [math]M=N=K\to\infty[/math], the above measure is:

[[math]] \mu=\left(1-\frac{1}{K^2}\right)\delta_0+\frac{1}{K^2}D_{\frac{1}{K}}(\pi_1) [[/math]]

This measure is supported by [math][0,4K][/math]. On the other hand, since the groups [math]\Gamma_{M,N}[/math] are all amenable, the corresponding measures are supported on [math][0,MN][/math], and so on [math][0,K^2][/math] in the [math]M=N=K[/math] situation. The fact that we do not have a convergence of supports is not surprising, because our convergence is in moments.

The above results are of course not the end of the story, because we have now to understand what happens in the case of non-generic parameters. There has been some technical work here, by Bichon and by myself, and as a sample result here, we have:

Theorem

Given two finite abelian groups [math]G,H[/math], having cardinalities

[[math]] |G|=M\quad,\quad |H|=N [[/math]]

consider the main character [math]\chi[/math] of the quantum group associated to [math]\mathcal F_{G\times H}[/math]. We have then

[[math]] law\left(\frac{\chi}{N}\right)=\left(1-\frac{1}{M}\right)\delta_0+\frac{1}{M}\,\pi_t [[/math]]

in moments, with [math]M=tN\to\infty[/math], where [math]\pi_t[/math] is the free Poisson law of parameter [math]t \gt 0[/math]. In addition, this formula holds for any generic fiber of [math]\mathcal F_{G\times H}[/math].

Show Proof

We already know that the second assertion holds, as explained above. Regarding now the first assertion, our first claim is that for the representation coming from the parametric matrix [math]\mathcal F_{G\times H}[/math] we have the following formula, where [math]M=|G|,N=|H|[/math], and the sets between brackets are sets with repetitions:

[[math]] c_p^r=\frac{1}{M^{r+1}N}\#\left\{\begin{matrix}i_1,\ldots,i_r,a_1,\ldots,a_p\in\{0,\ldots,M-1\},\\ b_1,\ldots,b_p\in\{0,\ldots,N-1\},\\ [(i_x+a_y,b_y),(i_{x+1}+a_y,b_{y+1})|y=1,\ldots,p]\\ =[(i_x+a_y,b_{y+1}),(i_{x+1}+a_y,b_y)|y=1,\ldots,p], \forall x \end{matrix}\right\} [[/math]]

Indeed, by using the general moment formula with [math]K=F_G[/math], [math]L=F_H[/math], we have the following formula for the above numbers:

[[math]] \begin{eqnarray*} &&c_p^r\\ &=&\frac{1}{(MN)^r}\int_{T^r}\sum_{i_1^1\ldots i_p^r}\sum_{b_1^1\ldots b_p^r}\frac{Q^1_{i_1^1b_1^1}Q^1_{i_1^2b_2^1}}{Q^1_{i_1^1b_2^1}Q^1_{i_1^2b_1^1}}\ldots\frac{Q^1_{i_p^1b_p^1}Q^1_{i_p^2b_1^1}}{Q^1_{i_p^1b_1^1}Q^1_{i_p^2b_p^1}}\ldots\ldots\frac{Q^r_{i_1^rb_1^r}Q^r_{i_1^1b_2^r}}{Q^r_{i_1^rb_2^r}Q^r_{i_1^1b_1^r}}\ldots\frac{Q^r_{i_p^rb_p^r}Q^r_{i_p^1b_1^r}}{Q^r_{i_p^rb_1^r}Q^r_{i_p^1b_p^r}}\\ &&\hskip15mm\frac{1}{M^{pr}}\sum_{j_1^1\ldots j_p^r}\frac{K_{i_1^1j_1^1}K_{i_1^2j_2^1}}{K_{i_1^1j_2^1}K_{i_1^2j_1^1}}\ldots\frac{K_{i_p^1j_p^1}K_{i_p^2j_1^1}}{K_{i_p^1j_1^1}K_{i_p^2j_p^1}}\ldots\ldots\frac{K_{i_1^rj_1^r}K_{i_1^1j_2^r}}{K_{i_1^rj_2^r}K_{i_1^1j_1^r}}\ldots\frac{K_{i_p^rj_p^r}K_{i_p^1j_1^r}}{K_{i_p^rj_1^r}K_{i_p^1j_p^r}}\\ &&\hskip15mm\frac{1}{N^{pr}}\sum_{a_1^1\ldots a_p^r}\frac{L_{a_1^1b_1^1}L_{a_1^2b_2^1}}{L_{a_1^1b_2^1}L_{a_1^2b_1^1}}\ldots\frac{L_{a_p^1b_p^1}L_{a_p^2b_1^1}}{L_{a_p^1b_1^1}L_{a_p^2b_p^1}}\ldots\ldots\frac{L_{a_1^rb_1^r}L_{a_1^1b_2^r}}{L_{a_1^rb_2^r}L_{a_1^1b_1^r}}\ldots\frac{L_{a_p^rb_p^r}L_{a_p^1b_1^r}}{L_{a_p^rb_1^r}L_{a_p^1b_p^r}}\,dQ\\ \end{eqnarray*} [[/math]]

Since we are in the Fourier matrix case, [math]K=F_G,L=F_H[/math], we can perform the sums over [math]j,a[/math]. To be more precise, the last two averages appearing above are respectively:

[[math]] \begin{eqnarray*} \Delta(i)&=&\prod_x\prod_y\delta(i^x_y+i^{x+1}_{y-1},i^{x+1}_y+i^x_{y-1})\\ \Delta(b)&=&\prod_x\prod_y\delta(b^x_y+b^{x+1}_{y-1},b^{x+1}_y+b^x_{y-1}) \end{eqnarray*} [[/math]]

We therefore obtain the following formula for the truncated moments of the main character, where [math]\Delta[/math] is the product of Kronecker symbols constructed above:

[[math]] \begin{eqnarray*} &&c_p^r\\ &=&\frac{1}{(MN)^r}\int_{T^r}\sum_{\Delta(i)=\Delta(b)=1}\frac{Q^1_{i_1^1b_1^1}Q^1_{i_1^2b_2^1}}{Q^1_{i_1^1b_2^1}Q^1_{i_1^2b_1^1}}\ldots\frac{Q^1_{i_p^1b_p^1}Q^1_{i_p^2b_1^1}}{Q^1_{i_p^1b_1^1}Q^1_{i_p^2b_p^1}}\ldots\ldots\frac{Q^r_{i_1^rb_1^r}Q^r_{i_1^1b_2^r}}{Q^r_{i_1^rb_2^r}Q^r_{i_1^1b_1^r}}\ldots\frac{Q^r_{i_p^rb_p^r}Q^r_{i_p^1b_1^r}}{Q^r_{i_p^rb_1^r}Q^r_{i_p^1b_p^r}}\,dQ \end{eqnarray*} [[/math]]

Now by integrating with respect to [math]Q\in(\mathbb T^{G\times H})^r[/math], we are led to counting the multi-indices [math]i,b[/math] satisfying several conditions. First, we have the following condition:

[[math]] \Delta(i)=\Delta(b)=1 [[/math]]

We have as well the following conditions, where the sets between brackets are by definition sets with repetitions:

[[math]] \begin{bmatrix} i_1^1b_1^1&\ldots&i_p^1b_p^1&i_1^2b_2^1&\ldots&i_p^2b_1^1 \end{bmatrix}= \begin{bmatrix} i_1^1b_2^1&\ldots&i_p^1b_1^1&i_1^2b_1^1&\ldots&i_p^2b_p^1 \end{bmatrix} [[/math]]

[[math]] \vdots [[/math]]

[[math]] \begin{bmatrix} i_1^rb_1^r&\ldots&i_p^rb_p^r&i_1^1b_2^r&\ldots&i_p^1b_1^r \end{bmatrix} =\begin{bmatrix} i_1^rb_2^r&\ldots&i_p^rb_1^r&i_1^1b_1^r&\ldots&i_p^1b_p^r \end{bmatrix} [[/math]]

In a more compact notation, the moment formula that we obtain in this way is therefore as follows:

[[math]] c_p^r=\frac{1}{(MN)^r}\#\left\{i,b\Big|\Delta(i)=\Delta(b)=1,\ [i^x_yb^x_y,i^{x+1}_yb^x_{y+1}]=[i^x_yb^x_{y+1},i^{x+1}_yb^x_y],\forall x\right\} [[/math]]

Now observe that the above Kronecker type conditions [math]\Delta(i)=\Delta(b)=1[/math] tell us that the arrays of indices [math]i=(i^x_y),b=(b^x_y)[/math] must be of the following special form:

[[math]] \begin{pmatrix}i^1_1&\ldots&i^1_p\\&\ldots\\ i^1_r&\ldots&i^r_p\end{pmatrix}=\begin{pmatrix}i_1+a_1&\ldots&i_1+a_p\\&\ldots\\ i_r+a_1&\ldots&i_r+a_p\end{pmatrix} [[/math]]

[[math]] \begin{pmatrix}b^1_1&\ldots&b^1_p\\&\ldots\\ b^1_r&\ldots&b^r_p\end{pmatrix}=\begin{pmatrix}j_1+b_1&\ldots&j_1+b_p\\&\ldots\\ j_r+b_1&\ldots&j_r+b_p\end{pmatrix} [[/math]]

Here all the new indices [math]i_x,j_x,a_y,b_y[/math] are uniquely determined, up to a choice of [math]i_1,j_1[/math]. Now by replacing [math]i^x_y,b^x_y[/math] with these new indices [math]i_x,j_x,a_y,b_y[/math], with a [math]MN[/math] factor added, which accounts for the choice of [math]i_1,j_1[/math], we obtain the following formula:

[[math]] c_p^r=\frac{1}{(MN)^{r+1}}\#\left\{i,j,a,b\Big|\begin{matrix}[(i_x+a_y,j_x+b_y),(i_{x+1}+a_y,j_x+b_{y+1})]\\ =[(i_x+a_y,j_x+b_{y+1}),(i_{x+1}+a_y,j_x+b_y)],\forall x\end{matrix}\right\} [[/math]]

Now observe that we can delete if we want the [math]j_x[/math] indices, which are irrelevant. Thus, we obtain the announced formula. The continuation is via combinatorics.

■

There are many interesting questions that are still open, regarding the computation of the spectral measure in the case where the parameter matrix [math]Q[/math] is not generic, and also regarding the computation for the deformations of the generalized Fourier matrices, which are not necessarily of Di\c t\u a type. We refer here to and related papers. \begin{exercises} To start with, we have the following exercise from the previous chapter, which is related to the above, and that we reproduce here, in case you have not solved it yet:

This exercise is important, because it is related to the first factorization performed in this chapter, in the context of the Fourier models.

This is actually quite unobvious, but finding the relevant literature and writing up a concise account of what is done there would do.

And that is all. In the hope that you liked the present book, and that we will hear from you soon, with interesting results about the Hadamard matrices. There are just so many things to be done, all interesting. You can't go wrong with these matrices. \begin{thebibliography}{99} \baselineskip=12.4pt \bibitem{aga}S. Agaian, Hadamard matrices and their applications, Springer (1985). \bibitem{arn}V.I. Arnold, Mathematical methods of classical mechanics, Springer (1974). \bibitem{aff}J. Avan, T. Fonseca, L. Frappat, P. Kulish, E. Ragoucy and G. Rollet, Temperley-Lieb R-matrices from generalized Hadamard matrices, Theor. Math. Phys. 178 (2014), 223--240. \bibitem{bac}J. Backelin, Square multiples [math]n[/math] give infinitely many cyclic [math]n[/math]-roots (1989). \bibitem{ba1}T. Banica, Introduction to quantum groups, Springer (2023). \bibitem{ba2}T. Banica, Introduction to modern physics (2024). \bibitem{bbi}T. Banica and J. Bichon, Random walk questions for linear quantum groups, Int. Math. Res. Not. 24 (2015), 13406--13436. \bibitem{bbs}T. Banica, J. Bichon and J.M. Schlenker, Representations of quantum permutation algebras, J. Funct. Anal. 257 (2009), 2864--2910. \bibitem{bcs}T. Banica, B. Collins and J.M. Schlenker, On orthogonal matrices maximizing the 1-norm, Indiana Univ. Math. J. 59 (2010), 839--856. \bibitem{bn1}T. Banica and I. Nechita, Almost Hadamard matrices: the case of arbitrary exponents, Discrete Appl. Math. 161 (2013), 2367--2379. \bibitem{bn2}T. Banica and I. Nechita, Flat matrix models for quantum permutation groups, Adv. Appl. Math. 83 (2017), 24--46. \bibitem{bn3}T. Banica and I. Nechita, Almost Hadamard matrices with complex entries, Adv. Oper. Theory 3 (2018), 149--189. \bibitem{bs1}T. Banica, I. Nechita and J.M. Schlenker, Analytic aspects of the circulant Hadamard conjecture, Ann. Math. Blaise Pascal 21 (2014), 25--59. \bibitem{bs2}T. Banica, I. Nechita and J.M. Schlenker, Submatrices of Hadamard matrices: complementation results, Electron. J. Linear Algebra 27 (2014), 197--212. \bibitem{bnz}T. Banica, I. Nechita and K. \.Zyczkowski, Almost Hadamard matrices: general theory and examples, Open Syst. Inf. Dyn. 19 (2012), 1--26. \bibitem{bop}T. Banica, D. \"Ozteke and L. Pittau, Isolated partial Hadamard matrices and related topics, Open Syst. Inf. Dyn. 25 (2018), 1--27. \bibitem{bsk}T. Banica and A. Skalski, The quantum algebra of partial Hadamard matrices, Linear Algebra Appl. 469 (2015), 364--380. \bibitem{bgh}L.D. Baumert, S.W. Golomb and M. Hall, Discovery of an Hadamard matrix of order 92, Bull. Amer. Math. Soc. 68 (1962), 237--238. \bibitem{ben}K. Beauchamp and R. Nicoara, Orthogonal maximal abelian [math]*[/math]-subalgebras of the [math]6\times 6[/math] matrices, Linear Algebra Appl. 428 (2008), 1833--1853. \bibitem{bbe}I. Bengtsson, W. Bruzda, \AA. Ericsson, J.\AA. Larsson, W. Tadej and K. \.Zyczkowski, Mutually unbiased bases and Hadamard matrices of order six, J. Math. Phys. 48 (2007), 1--33. \bibitem{bzy}I. Bengtsson and K. \.Zyczkowski, Geometry of quantum states, Cambridge Univ. Press (2006). \bibitem{bpa}H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory, Ann. of Math. 149 (1999), 1023--1060. \bibitem{bep}P. Biran, M. Entov and L. Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math. 6 (2004), 793--802. \bibitem{bjo}G. Björck, Functions of modulus [math]1[/math] on [math]{\rm Z}_n[/math] whose Fourier transforms have constant modulus, and cyclic [math]n[/math]-roots, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 315 (1990), 131--140. \bibitem{bjf}G. Björck and R. Fröberg, A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic [math]n[/math]-roots, J. Symbolic Comput. 12 (1991), 329--336. \bibitem{bha}G. Björck and U. Haagerup, All cyclic [math]p[/math]-roots of index 3 found by symmetry-preserving calculations (2008). \bibitem{bur}R. Burstein, Group-type subfactors and Hadamard matrices, Trans. Amer. Math. Soc. 367 (2015), 6783--6807. \bibitem{but}A.T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13 (1962), 894--898. \bibitem{cho}C.H. Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not. 35 (2004), 1803--1843. \bibitem{cdi}C.J. Colbourn and J.H. Dinitz, Handbook of combinatorial designs, CRC Press (2007). \bibitem{con}A. Connes, Noncommutative geometry, Academic Press (1994). \bibitem{ckh}R. Craigen and H. Kharaghani, On the nonexistence of Hermitian circulant complex Hadamard matrices, Australas. J. Combin. 7 (1993), 225--227. \bibitem{del}W. de Launey, On the non-existence of generalized weighing matrices, Ars Combin. 17 (1984), 117--132. \bibitem{dda}W. de Launey and J.E. Dawson, An asymptotic result on the existence of generalised Hadamard matrices, J. Combin. Theory Ser. A 65 (1994), 158--163. \bibitem{dfh}W. de Launey, D.L. Flannery and K.J. Horadam, Cocyclic Hadamard matrices and difference sets, Discrete Appl. Math. 102 (2000), 47--61. \bibitem{dgo}W. de Launey and D.M. Gordon, A comment on the Hadamard conjecture, J. Combin. Theory Ser. A 95 (2001), 180--184. \bibitem{dle}W. de Launey and D.A. Levin, A Fourier-analytic approach to counting partial Hadamard matrices, Cryptogr. Commun. 2 (2010), 307--334. \bibitem{dir}P.A.M. Dirac, Principles of quantum mechanics, Oxford Univ. Press (1930). \bibitem{dit}P. Di\c t\u a, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (2004), 5355--5374. \bibitem{dur}R. Durrett, Probability: theory and examples, Cambridge Univ. Press (1990). \bibitem{deb}T. Durt, B.G. Englert, I. Bengtsson and K. \.Zyczkowski, On mutually unbiased bases, Int. J. Quantum Inf. 8 (2010), 535--640. \bibitem{fau}J.C. Faugère, Finding all the solutions of Cyclic 9 using Gröbner basis techniques, Lecture Notes Ser. Comput. 9 (2001), 1--12. \bibitem{fey}R.P. Feynman, R.B. Leighton and M. Sands, The Feynman lectures on physics III: quantum mechanics, Caltech (1966). \bibitem{fsl}P.C. Fishburn and N.J.A. Sloane, The solution to Berlekamp's switching game, Discrete Math. 74 (1989), 263--290. \bibitem{gri}D.J. Griffiths and D.F. Schroeter, Introduction to quantum mechanics, Cambridge Univ. Press (2018). \bibitem{ha1}U. Haagerup, Orthogonal maximal abelian [math]*[/math]-subalgebras of the [math]n\times n[/math] matrices and cyclic [math]n[/math]-roots, in “Operator algebras and quantum field theory”, International Press (1997), 296--323. \bibitem{ha2}U. Haagerup, Cyclic [math]p[/math]-roots of prime lengths [math]p[/math] and related complex Hadamard matrices (2008). \bibitem{had}J. Hadamard, Résolution d'une question relative aux déterminants, Bull. Sci. Math. 2 (1893), 240--246. \bibitem{hal}M. Hall, Integral matrices [math]A[/math] for which [math]AA^T=mI[/math], in “Number Theory and Algebra”, Academic Press (1977), 119--134. \bibitem{hsc}G. Hiranandani and J.M. Schlenker, Small circulant complex Hadamard matrices of Butson type, European J. Combin. 51 (2016), 306--314. \bibitem{hor}K.J. Horadam, Hadamard matrices and their applications, Princeton Univ. Press (2007). \bibitem{iwo}M. Idel and M.M. Wolf, Sinkhorn normal form for unitary matrices, Linear Algebra Appl. 471 (2015), 76--84. \bibitem{ito}N. Ito, Hadamard Graphs I, Graphs Combin. 1 (1985), 57--64. \bibitem{jo1}V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1--25. \bibitem{jo2}V.F.R. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math. 137 (1989), 311--334. \bibitem{jo3}V.F.R. Jones, Planar algebras I (1999). \bibitem{kar}A. Karabegov, The reconstruction of a unitary matrix from the moduli of its elements and symbols on a finite phase space (1989). \bibitem{kse}H. Kharaghani and J. Seberry, The excess of complex Hadamard matrices, Graphs Combin. 9 (1993), 47--56. \bibitem{kta}H. Kharaghani and B. Tayfeh-Rezaie, A Hadamard matrix of order 428, J. Combin. Des. 13 (2005), 435--440. \bibitem{kms}C. Koukouvinos, M. Mitrouli and J. Seberry, An algorithm to find formulae and values of minors for Hadamard matrices, Linear Algebra Appl. 330 (2001), 129--147. \bibitem{lle}T.Y. Lam and K.H. Leung, On vanishing sums of roots of unity, J. Algebra 224 (2000), 91--109. \bibitem{lax}P. Lax, Functional analysis, Wiley (2002). \bibitem{mpa}V.A. Marchenko and L.A. Pastur, Distribution of eigenvalues in certain sets of random matrices, Mat. Sb. 72 (1967), 507--536. \bibitem{mwe}D. McNulty and S. Weigert, Isolated Hadamard matrices from mutually unbiased product bases, J. Math. Phys. 53 (2012), 1--21. \bibitem{moh}M.T. Mohan, On some p-almost Hadamard matrices, Oper. Matrices 13 (2019), 253--281. \bibitem{nic}R. Nicoara, A finiteness result for commuting squares of matrix algebras, J. Operator Theory 55 (2006), 295--310. \bibitem{nwh}R. Nicoara and J. White, Analytic deformations of group commuting squares and complex Hadamard matrices, J. Funct. Anal. 272 (2017), 3486--3505. \bibitem{nch}M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge Univ. Press (2000). \bibitem{ocn}A. Ocneanu, Quantum symmetry, differential geometry of finite graphs, and classification of subfactors, Univ. of Tokyo Seminary Notes (1990). \bibitem{pal}R. Paley, On orthogonal matrices, J. Math. Phys. 12 (1933), 311--320. \bibitem{pso}K.H. Park and H.Y. Song, Quasi-Hadamard matrices, Proc. ISIT 2010, Austin, TX (2010). \bibitem{pet}M. Petrescu, Existence of continuous families of complex Hadamard matrices of certain prime dimensions and related results, Ph.D. Thesis, UCLA (1997). \bibitem{pol}G. Pòlya, \"Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz, Math. Ann. 84 (1921), 149--160. \bibitem{pop}S. Popa, Orthogonal pairs of [math]*[/math]-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), 253--268. \bibitem{rsh}L.B. Richmond and J. Shallit, Counting abelian squares, Electron. J. Combin. 16 (2009), 1--9. \bibitem{rvi}R. Roth and K. Viswanathan, On the hardness of decoding the Gale-Berlekamp code, IEEE Trans. Inform. Theory 54 (2008), 1050--1060. \bibitem{rys}H.J. Ryser, Combinatorial mathematics, Wiley (1963). \bibitem{sya}J. Seberry and M. Yamada, Hadamard matrices: constructions using number theory and linear algebra, Wiley (2020). \bibitem{sti}D.R. Stinson, Combinatorial designs: constructions and analysis, Springer-Verlag (2006). \bibitem{syl}J.J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tesselated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers, Phil. Mag. 34 (1867), 461--475. \bibitem{sz1}F. Szöllősi, Parametrizing complex Hadamard matrices, European J. Combin. 29 (2008), 1219--1234. \bibitem{sz2}F. Szöllősi, Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun. 2 (2010), 187--198. \bibitem{tz1}W. Tadej and K. \.Zyczkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006), 133--177. \bibitem{tz2}W. Tadej and K. \.Zyczkowski, Defect of a unitary matrix, Linear Algebra Appl. 429 (2008), 447--481. \bibitem{tao}T. Tao, Fuglede's conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), 251--258. \bibitem{tvu}T. Tao and V. Vu, On random [math]\pm 1[/math] matrices: singularity and determinant, Random Structures Algorithms 28 (2006), 1--23. \bibitem{tli}N.H. Temperley and E.H. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. Roy. Soc. London 322 (1971), 251--280. \bibitem{tur}R.J. Turyn, Character sums and difference sets, Pacific J. Math. 15 (1965), 319--346. \bibitem{ver}E. Verheiden, Integral and rational completions of combinatorial matrices, J. Combin. Theory Ser. A 25 (1978) 267--276. \bibitem{vdn}D.V. Voiculescu, K.J. Dykema and A. Nica, Free random variables, AMS (1992). \bibitem{von}J. von Neumann, Mathematical foundations of quantum mechanics, Princeton Univ. Press (1955). \bibitem{wa1}S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195--211. \bibitem{wa2}S. Wang, [math]L_p[/math]-improving convolution operators on finite quantum groups, Indiana Univ. Math. J. 65 (2016), 1609--1637. \bibitem{wat}J. Watrous, The theory of quantum information, Cambridge Univ. Press (2018). \bibitem{wei}S. Weinberg, Lectures on quantum mechanics, Cambridge Univ. Press (2012). \bibitem{wey}H. Weyl, The theory of groups and quantum mechanics, Princeton Univ. Press (1931). \bibitem{wil}J. Williamson, Hadamard's determinant theorem and the sum of four squares, Duke Math. J. 11 (1944), 65--81. \bibitem{win}A. Winterhof, On the non-existence of generalized Hadamard matrices, J. Statist. Plann. Inference 84 (2000), 337--342. \bibitem{wo1}S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613--665. \bibitem{wo2}S.L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35--76. \end{thebibliography} \baselineskip=14pt \printindex \end{document}

General references

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