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Let us keep discussing what happens at the general level. We will need the following result, valid in the general context of the Hopf image construction:

Theorem

Given a matrix model [math]\pi:C(G)\to M_N(\mathbb C)[/math], the fundamental corepresentation [math]v[/math] of its Hopf image is subject to the Tannakian conditions

[[math]] Hom(v^{\otimes k},v^{\otimes l})=Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

where [math]U_{ij}=\pi(u_{ij})[/math], and where the spaces on the right are taken in a formal sense.

Show Proof

This is something which follows directly from the definition of the Hopf image, without computations needed, the idea being as follows:

(1) Since the morphisms increase the intertwining spaces, when defined either in a representation theory sense, or just formally, we have inclusions as follows:

[[math]] Hom(u^{\otimes k},u^{\otimes l})\subset Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

More generally, we have such inclusions when replacing [math](G,u)[/math] with any pair producing a factorization of [math]\pi[/math]. Thus, by Tannakian duality , the Hopf image must be given by the fact that the intertwining spaces must be the biggest, subject to these inclusions.

(2) On the other hand, since [math]u[/math] is biunitary, so is [math]U[/math], and it follows that the spaces on the right form a Tannakian category. Thus, we have a quantum group [math](H,v)[/math] given by:

[[math]] Hom(v^{\otimes k},v^{\otimes l})=Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

By the above discussion, [math]C(H)[/math] follows to be the Hopf image of [math]\pi[/math], as claimed.

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With the above result in hand, we can now compute the Tannakian category of the Hopf image, in the context of our Hadamard matrix construction. We are led in this way to the following technical statement, going back to Jones in an equivalent form, and which reminds a bit the transfer matrices in statistical mechanics:

Theorem

The Tannakian category of the quantum group [math]G\subset S_N^+[/math] associated to a complex Hadamard matrix [math]H\in M_N(\mathbb C)[/math] is given by

[[math]] T\in Hom(u^{\otimes k},u^{\otimes l})\iff T^\circ G^{k+2}=G^{l+2}T^\circ [[/math]]

where the objects on the right are constructed as follows:

• [math]T^\circ=id\otimes T\otimes id[/math].
• [math]G_{ia}^{jb}=\sum_kH_{ik}\bar{H}_{jk}\bar{H}_{ak}H_{bk}[/math].
• [math]G^k_{i_1\ldots i_k,j_1\ldots j_k}=G_{i_ki_{k-1}}^{j_kj_{k-1}}\ldots G_{i_2i_1}^{j_2j_1}[/math].

Show Proof

With the notations in Theorem 14.8, we have the following formula:

[[math]] Hom(u^{\otimes k},u^{\otimes l})=Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

Here, according to our conventions, the vector space on the right consists by definition of the complex [math]N^l\times N^k[/math] matrices [math]T[/math], satisfying the following relation:

[[math]] TU^{\otimes k}=U^{\otimes l}T [[/math]]

If we denote this equality by [math]L=R[/math], the left term [math]L[/math] is given by:

[[math]] \begin{eqnarray*} L_{ij} &=&(TU^{\otimes k})_{ij}\\ &=&\sum_aT_{ia}U^{\otimes k}_{aj}\\ &=&\sum_aT_{ia}U_{a_1j_1}\ldots U_{a_kj_k} \end{eqnarray*} [[/math]]

As for the right term [math]R[/math], this is given by the following formula:

[[math]] \begin{eqnarray*} R_{ij} &=&(U^{\otimes l}T)_{ij}\\ &=&\sum_bU^{\otimes l}_{ib}T_{bj}\\ &=&\sum_bU_{i_1b_1}\ldots U_{i_lb_l}T_{bj} \end{eqnarray*} [[/math]]

Consider now the vectors [math]\xi_{ij}=H_i/H_j[/math]. Since these vectors span the ambient Hilbert space, the equality [math]L=R[/math] is equivalent to the following equality:

[[math]] \lt L_{ij}\xi_{pq},\xi_{rs} \gt = \lt R_{ij}\xi_{pq},\xi_{rs} \gt [[/math]]

We use now the following well-known formula, expressing a product of rank one projections [math]P_1,\ldots,P_k[/math] in terms of the corresponding image vectors [math]\xi_1,\ldots,\xi_k[/math]:

[[math]] \lt P_1\ldots P_kx,y \gt = \lt x,\xi_k \gt \lt \xi_k,\xi_{k-1} \gt \ldots\ldots \lt \xi_2,\xi_1 \gt \lt \xi_1,y \gt [[/math]]

This gives the following formula for [math]L[/math]:

[[math]] \begin{eqnarray*} \lt L_{ij}\xi_{pq},\xi_{rs} \gt &=&\sum_aT_{ia} \lt P_{a_1j_1}\ldots P_{a_kj_k}\xi_{pq},\xi_{rs} \gt \\ &=&\sum_aT_{ia} \lt \xi_{pq},\xi_{a_kj_k} \gt \ldots \lt \xi_{a_1j_1},\xi_{rs} \gt \\ &=&\sum_aT_{ia}G_{pa_k}^{qj_k}G_{a_ka_{k-1}}^{j_kj_{k-1}}\ldots G_{a_2a_1}^{j_2j_1}G_{a_1r}^{j_1s}\\ &=&\sum_aT_{ia}G^{k+2}_{rap,sjq}\\ &=&(T^\circ G^{k+2})_{rip,sjq} \end{eqnarray*} [[/math]]

As for the right term [math]R[/math], this is given by:

[[math]] \begin{eqnarray*} \lt R_{ij}\xi_{pq},\xi_{rs} \gt &=&\sum_b \lt P_{i_1b_1}\ldots P_{i_lb_l}\xi_{pq},\xi_{rs} \gt T_{bj}\\ &=&\sum_b \lt \xi_{pq},\xi_{i_lb_l} \gt \ldots \lt \xi_{i_1b_1},\xi_{rs} \gt T_{bj}\\ &=&\sum_bG_{pi_l}^{qb_l}G_{i_li_{l-1}}^{b_lb_{l-1}}\ldots G_{i_2i_1}^{b_2b_1}G_{i_1r}^{b_1s}T_{bj}\\ &=&\sum_bG^{l+2}_{rip,sbq}T_{bj}\\ &=&(G^{l+2}T^\circ)_{rip,sjq} \end{eqnarray*} [[/math]]

Thus, we obtain the formula in the statement. See .

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Let us discuss now the computation of the Haar functional for the quantum permutation group [math]G\subset S_N^+[/math] associated to a complex Hadamard matrix [math]H\in M_N(\mathbb C)[/math]. In the general random matrix model context, we have the following formula for the Haar integration functional of the Hopf image, coming from the work of Wang in :

Theorem

Given an inner faithful model [math]\pi:C(G)\to M_N(C(T))[/math], we have

[[math]] \int_G=\lim_{k\to\infty}\frac{1}{k}\sum_{r=1}^k\int_G^r [[/math]]

with the truncated integrals on the right being given by the formula

[[math]] \int_G^r=(\varphi\circ\pi)^{*r} [[/math]]

where [math]\varphi=tr\otimes\int_T[/math] is the random matrix trace on the target algebra.

Show Proof

As a first observation, there is an obvious similarity here with the Woronowicz construction of the Haar measure, explained in chapter 13. In fact, the above result holds for any model [math]\pi:C(G)\to B[/math], with [math]\varphi\in B^*[/math] being a faithful trace, and with this picture in hand, the Woronowicz construction corresponds to the case [math]\pi=id[/math], and the result itself is therefore a generalization of Woronowicz's existence result for the Haar measure. In order to prove now the result, we can proceed as in chapter 13. If we denote by [math]\int_G'[/math] the limit in the statement, we must prove that this limit converges, and that we have:

[[math]] \int_G'=\int_G [[/math]]

It is enough to check this on the coefficients of corepresentations, and if we let [math]v=u^{\otimes k}[/math] be one of the Peter-Weyl corepresentations, we must prove that we have:

[[math]] \left(id\otimes\int_G'\right)v=\left(id\otimes\int_G\right)v [[/math]]

We know from chapter 1 that the matrix on the right is the orthogonal projection onto [math]Fix(v)[/math]. Regarding now the matrix on the left, this is the orthogonal projection onto the [math]1[/math]-eigenspace of [math](id\otimes\varphi\pi)v[/math]. Now observe that, if we set [math]V_{ij}=\pi(v_{ij})[/math], we have:

[[math]] (id\otimes\varphi\pi)v=(id\otimes\varphi)V [[/math]]

Thus, as in chapter 13, we conclude that the [math]1[/math]-eigenspace that we are interested in equals [math]Fix(V)[/math]. But, according to Theorem 14.8, we have:

[[math]] Fix(V)=Fix(v) [[/math]]

Thus, we have proved that we have [math]\int_G'=\int_G[/math], as desired.

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In practice now, we are led to the computation of the truncated integrals [math]\int_G^r[/math] appearing in the above result, and the formula of these truncated integrals is as follows:

Proposition

The truncated integrals in Theorem 14.10, namely

[[math]] \int_G^r=(\varphi\circ\pi)^{*r} [[/math]]

are given by the following formula, in the orthogonal case, where [math]u=\bar{u}[/math],

[[math]] \int_G^ru_{a_1b_1}\ldots u_{a_pb_p}=(T_p^r)_{a_1\ldots a_p,b_1\ldots b_p} [[/math]]

with the matrix on the right being given by the formula

[[math]] (T_p)_{i_1\ldots i_p,j_1\ldots j_p}=\left(tr\otimes\int_T\right)(U_{i_1j_1}\ldots U_{i_pj_p}) [[/math]]

where [math]U_{ij}=\pi(u_{ij})[/math] are the images of the standard coordinates in the model.

Show Proof

This is something straightforward, which comes from the definition of the truncated integrals. Indeed, we have the following computation:

[[math]] \begin{eqnarray*} \int_G^ru_{a_1b_1}\ldots u_{a_pb_p} &=&(\varphi\circ\pi)^{*r}(u_{a_1b_1}\ldots u_{a_pb_p})\\ &=&(\varphi\circ\pi)^{\otimes r}\Delta^{(r)}(u_{a_1b_1}\ldots u_{a_pb_p})\\ &=&(T_p^r)_{a_1\ldots a_p,b_1\ldots b_p} \end{eqnarray*} [[/math]]

In addition to this, let us mention as well that in the general compact quantum group case, where the condition [math]u=\bar{u}[/math] does not necessarily hold, an analogue of the above result holds, by adding exponents [math]e_1,\ldots,e_p\in\{1,*\}[/math] everywhere. See .

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Regarding now the main character, the result here is as follows:

Theorem

In the context of Theorem 14.10, let [math]\mu^r[/math] be the law of the main character [math]\chi=Tr(u)[/math] with respect to the truncated integration:

[[math]] \int_G^r=(\varphi\circ\pi)^{*r} [[/math]]

• The law of the main character is given by the following formula:
  [[math]] \mu=\lim_{k\to\infty}\frac{1}{k}\sum_{r=0}^k\mu^r [[/math]]
• The moments of the truncated measure [math]\mu^r[/math] are the following numbers:
  [[math]] c_p^r=Tr(T_p^r) [[/math]]

Show Proof

These results are both elementary, the proof being as follows:

(1) This follows from the general limiting formula in Theorem 14.10.

(2) This follows from the formula in Proposition 14.11, by summing the integrals computed there over pairs of equal indices, [math]a_i=b_i[/math].

■

In connection with the Hadamard matrices, we can use the above technology in order to compute the law of the main character, and also discuss the behavior of the construction [math]H\to G[/math] with respect to the various operations on the Hadamard matrices, such as the transposition [math]H\to H^t[/math]. Following , we have the following result, at the general level:

Theorem

Consider an inner faithful model, as follows:

[[math]] \pi:C(G)\to M_N(\mathbb C)\quad,\quad u_{ij}\to U_{ij} [[/math]]

• We set [math](U'_{kl})_{ij}=(U_{ij})_{kl}[/math], and we define a model as follows:
  [[math]] \widetilde{\rho}:C(U_N^+)\to M_N(\mathbb C)\quad,\quad v_{kl}\to U_{kl}' [[/math]]
• We perform the Hopf image construction, as to get a model as follows:
  [[math]] \rho:C(G')\to M_N(\mathbb C) [[/math]]

The operation [math]A\to A'[/math] is then a duality, in the sense that we have [math]A''=A[/math], and in the Hadamard matrix case, this duality comes from the operation [math]H\to H^t[/math].

Show Proof

This is something quite technical, the idea being as follows:

(1) First, regarding the statement, the quantum group [math]U_N^+[/math] is Wang's quantum unitary group, whose standard coordinates are subject to the condition [math]u^*=u^{-1},u^t=\bar{u}^{-1}[/math].

(2) Observe that [math]U'[/math] is given by [math]U'=\Sigma U[/math], where [math]\Sigma[/math] is the flip. Thus this matrix is indeed biunitary, and produces a representation [math]\rho[/math] as above.

(3) In what regards now the proof, the fact that [math]A\to A'[/math] is a duality is clear, and the Hadamard matrix assertion can be proved via algebraic methods. See .

■

We denote by [math]D[/math] the dilation operation for probability measures, or for general [math]*[/math]-distributions, given by the formula [math]D_r(law(X))=law(rX)[/math]. Following , we have:

Theorem

Consider the rescaled measure [math]\eta^r=D_{1/N}(\mu^r)[/math].

• The moments [math]\gamma_p^r=c_p^r/N^p[/math] of [math]\eta^r[/math] satisfy the following formula:
  [[math]] \gamma_p^r(G)=\gamma_r^p(G') [[/math]]
• [math]\eta^r[/math] has the same moments as the following matrix:
  [[math]] T_r'=T_r(G') [[/math]]
• In the orthogonal case, where [math]u=\bar{u}[/math], we have:
  [[math]] \eta^r=law(T_r') [[/math]]

Show Proof

All the results follow from Theorem 14.12, as follows:

(1) We have the following computation:

[[math]] \begin{eqnarray*} c_p^r(A) &=&\sum_i(T_p)_{i_1^1\ldots i_p^1,i_1^2\ldots i_p^2}\ldots\ldots(T_p)_{i_1^r\ldots i_p^r,i_1^1\ldots i_p^1}\\ &=&\sum_itr(U_{i_1^1i_1^2}\ldots U_{i_p^1i_p^2})\ldots\ldots tr(U_{i_1^ri_1^1}\ldots U_{i_p^ri_p^1})\\ &=&\frac{1}{N^r}\sum_i\sum_j(U_{i_1^1i_1^2})_{j_1^1j_2^1}\ldots(U_{i_p^1i_p^2})_{j_p^1j_1^1}\ldots\ldots(U_{i_1^ri_1^1})_{j_1^rj_2^r}\ldots(U_{i_p^ri_p^1})_{j_p^rj_1^r} \end{eqnarray*} [[/math]]

In terms of the matrix [math](U'_{kl})_{ij}=(U_{ij})_{kl}[/math], then by permuting the terms in the product on the right, and finally with the changes [math]i_a^b\leftrightarrow i_b^a,j_a^b\leftrightarrow j_b^a[/math], we obtain:

[[math]] \begin{eqnarray*} c_p^r(A) &=&\frac{1}{N^r}\sum_i\sum_j(U'_{j_1^1j_2^1})_{i_1^1i_1^2}\ldots(U'_{j_p^1j_1^1})_{i_p^1i_p^2}\ldots\ldots(U'_{j_1^rj_2^r})_{i_1^ri_1^1}\ldots(U'_{j_p^rj_1^r})_{i_p^ri_p^1}\\ &=&\frac{1}{N^r}\sum_i\sum_j(U'_{j_1^1j_2^1})_{i_1^1i_1^2}\ldots(U'_{j_1^rj_2^r})_{i_1^ri_1^1}\ldots\ldots(U'_{j_p^1j_1^1})_{i_p^1i_p^2}\ldots(U'_{j_p^rj_1^r})_{i_p^ri_p^1}\\ &=&\frac{1}{N^r}\sum_i\sum_j(U'_{j_1^1j_1^2})_{i_1^1i_2^1}\ldots(U'_{j_r^1j_r^2})_{i_r^1i_1^1}\ldots\ldots(U'_{j_1^pj_1^1})_{i_1^pi_2^p}\ldots(U'_{j_r^pj_r^1})_{i_r^pi_1^p} \end{eqnarray*} [[/math]]

On the other hand, if we use again the above formula of [math]c_p^r(A)[/math], but this time for the matrix [math]U'[/math], and with the changes [math]r\leftrightarrow p[/math] and [math]i\leftrightarrow j[/math], we obtain:

[[math]] c_r^p(A')\\ =\frac{1}{N^p}\sum_i\sum_j(U'_{j_1^1j_1^2})_{i_1^1i_2^1}\ldots(U'_{j_r^1j_r^2})_{i_r^1i_1^1}\ldots\ldots(U'_{j_1^pj_1^1})_{i_1^pi_2^p}\ldots(U'_{j_r^pj_r^1})_{i_r^pi_1^p} [[/math]]

Now by comparing this with the previous formula, we obtain:

[[math]] N^rc_p^r(A)=N^pc_r^p(A') [[/math]]

Thus we have the following equalities, which give the result:

[[math]] \frac{c_p^r(A)}{N^p}=\frac{c_r^p(A')}{N^r} [[/math]]

(2) By using (1) and the formula in Theorem 14.12, we obtain:

[[math]] \frac{c_p^r(A)}{N^p} =\frac{c_r^p(A')}{N^r} =\frac{Tr((T'_r)^p)}{N^r} =tr((T'_r)^p) [[/math]]

But this gives the equality of moments in the statement.

(3) This follows from the moment equality in (2), and from the standard fact that for self-adjoint variables, the moments uniquely determine the distribution.

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General references

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