SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS

SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS TEO BANICA AND JULIEN BICHON Abstract. We study the subfactor invariants of the deformed Fourier matrices H = F M Q F N, when the parameter matrix Q M M N (T) is generic. In general, the associated spectral measure decomposes as µ = (1 1 N )δ 0 + 1 N law(a), where A C(T MN, M M (C)) is a certain random matrix (A(q) is the ram matrix of the rows of q). In the case M = 2 we deduce from this a true blowup result, µ = (1 1 N )δ 0 + 1 N Φ ε, where ε is the uniform measure on Z 2 T N, and Φ(e, a) = N + e i a i. We conjecture that such decomposition results should hold as well in the non-generic case. Contents Introduction 1 1. Deformed Fourier matrices 4 2. Representation theory 6 3. Invariants, self-duality 8 4. eometric interpretation 10 5. Toral blowup at M=2 12 References 15 Introduction A complex Hadamard matrix is a matrix H M N (C) whose entries are on the unit circle, H ij = 1, and whose rows are pairwise orthogonal. See [19]. The basic example is the Fourier matrix, F N = (w ij ) with w = e 2πi/N : 1 1 1... 1 F N = 1 w w 2... w N 1............... 1 w N 1 w 2(N 1)... w (N 1)2 Popa discovered in [16] that these matrices parametrize the pairs of orthogonal MASA in the simplest von Neumann algebra, M N (C). As a consequence, each Hadamard matrix 2000 Mathematics Subject Classification. 46L37 (16T05). Key words and phrases. Random walk, Hadamard subfactor. 1

2 TEO BANICA AND JULIEN BICHON H M N (C) produces an irreducible subfactor M R of the Murray-von Neumann hyperfinite factor R, having index [R : M] = N. The computation of the standard invariant of this subfactor is a key problem. As explained by Jones in [13], the associated planar algebra P k = M M k has a direct combinatorial description in terms of H, and the main problem is that of computing the corresponding Poincaré series: f(z) = dim(p k )z k k=0 An alternative approach to these questions is via quantum groups [21], [22]. The idea is that associated to H M N (C) is a certain quantum permutation group S + N, and the problem is to compute the spectral measure µ P(R + ) of the main character χ : C. This is the same problem as above, because f is the Stieltjes transform of µ: 1 f(z) = 1 zχ We refer to [1], [3] for a discussion of these questions, from the quantum permutation group viewpoint, and to [10], [13] for subfactor background on the problem. Let us summarize the above considerations as: Definition. Associated to a complex Hadamard matrix H M N (C) are a planar algebra P = (P k ) and a quantum group S + N. The quantum invariants of H are: (1) The Poincaré series f(z) = k=0 dim(p k)z k. (2) The spectral measure µ P(R + ) of the main character χ : C. These invariants are related by the fact that f is the Stieltjes transform of µ. In our previous paper [3] we studied these invariants for an important class of complex Hadamard matrices, constructed by Diţă in [9]. iven two complex Hadamard matrices H M M (T), K M N (T), the Diţă deformation of their tensor product (H K) ia,jb = H ij K ab, with matrix of parameters Q M M N (T), is by definition H Q K = (Q ib H ij K ab ) ia,jb. In the Fourier matrix case, H = F M, K = F N, Burstein proved in [8] that the corresponding subfactors are of Bisch-Haagerup type [7]. Our main result in [3] is the fact that, for F M Q F N with Q M M N (T) generic, the associated quantum algebra is a crossed coproduct, C() = C (Γ M,N ) C(Z M ), where: Γ M,N = Z (M 1)(N 1) Z N As pointed out in [3], the above result shows that, in the generic case, the quantum invariants of F M Q F N are related to the random walks on Γ M,N. We will investigate here these random walks, by taking some inspiration from the case N = 2, where Γ M,N is related to the half-liberation procedure in [5], [6], to the counting problem for abelian squares [17] and (at M = 3) to the random walks on the honeycomb lattice [20]. We will obtain in this way several concrete results regarding the quantum invariants of F M Q F N, including a purely combinatorial formula for the moments of µ. In the

SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS 3 particular cases M = 2 and N = 2, which are quite special, we will obtain some improved results. Note that these two particular cases fully cover the index 6 case. Our main results, which are rather of geometric nature, are as follows: Theorem. Consider the matrix H = F M Q F N, with Q M M N (T) generic. (1) In general, µ = (1 1 N )δ 0 + 1 N law(a), where A C(TMN, M M (C)) is a certain random matrix (A(q) is the ram matrix of the rows of q). (2) At M = 2 we have µ = (1 1 N )δ 0 + 1 N Φ ε, where ε is the uniform measure on Z 2 T N, and Φ(e, a) = N + e i a i. These formulae are part of the spectral measure blowup philosophy, a potentially powerful subfactor concept, that is still in its pioneering stages, however. More precisely, the origins of writings of this type go back to [4], where it was discovered that the spectral measures of the ADE graphs have very simple formulae, when blown up on the circle T. This finding, while elementary and experimental, was however inspired by some quite heavy results, namely those of Jones regarding the theta series in [12]. Further details regarding the subfactor meaning of this circular blowup phenomenon for ADE graphs came later on, from the work of Evans and Pugh in [11]. Let us also mention that this result was used in [2], for the ADE classification of the quantum subgroups of S + 4. Now back to the above theorem, this is certainly a new, interesting input for this spectral measure blowup theory, because the formula there is no longer a small index one, but takes now place in index MN, with M, N N arbitrary. There are of course many interesting questions here. enerally speaking, the main question is that of deciding if, given a subfactor M R, the associated spectral measure µ, having as moments the numbers c k = dim(p k ), has or not a nice blowup on a suitable manifold. More concretely now, back to the Hadamard matrices, we expect the blowup to appear as well in the non-generic case, our conjecture here being as follows: Conjecture. For H = F M Q F N, with Q M M N (T) arbitrary, the spectral measure blowup should land into a certain arithmetic lattice, canonically associated to Q. This is probably a quite difficult question. The evidence here comes from our previous paper [3], where, using tools from [2], [15], we solved the problem at M = N = 2. More precisely, with Q = ( a c b d ), q = ad/bc and n = ord(q4 ), the formula is as follows, where ε k is the uniform measure on the k-roots of unity, and Φ(q) = (q + q 1 ) 2 : µ = 1 2 (δ 0 + Φ ε 4n ) Finally, we will discuss several questions in relation with the work in [1], [14], [18]. The paper is organized as follows: 1 is a preliminary section, in 2 we discuss representation theory aspects, in 3-4 we work out the combinatorial formula for the invariants, and its geometric interpretation, and in 5 we discuss the case M = 2.

4 TEO BANICA AND JULIEN BICHON 1. Deformed Fourier matrices A complex Hadamard matrix is a matrix H M N (C) whose entries are on the unit circle, H ij = 1, and whose rows are pairwise orthogonal. The basic example is the Fourier matrix, F N = (w ij ) with w = e 2πi/N. For examples and motivations, see [19]. Our starting point is the following observation. Let H 1,..., H N T N be the rows of H, and consider the vectors H i /H j T N. We have then: Hi H j, H i H k = l H il H jl Hkl H il = l H kl H jl =< H k, H j >= δ jk Similarly, < H i H j, H k H j >= δ ik, so the matrix of rank one projections P ij = P roj(h i /H j ) is magic, in the sense that its entries sum up to 1 on each row and column. Recall now that C(S + N ) is the universal C -algebra generated by the entries of a N N magic matrix u, with comultiplication (u ij ) = k u ik u kj, counit ε(u ij ) = δ ij and antipode S(u ij ) = u ji. This algebra satisfies the axioms of Woronowicz in [22], and so S + N is a compact quantum group, called quantum permutation group. See Wang [21]. Definition 1.1. Associated to H M N (C) is the representation π H (u ij ) = P roj(h i /H j ), then is the minimal quantum group S + N producing a factorization of type C(S + N ) π H M N (C) C() and finally is the spectral measure µ P(R + ) of the main character χ C(). Here we have used the fact, due to Woronowicz [22], that any compact quantum group, and in particular the above quantum group, has a Haar functional. Thus we can indeed speak about the spectral measure of χ = i u ii, which is by definition given by: ϕ(x)dµ(x) = ϕ(χ) R The computation of µ is in general a difficult question, one basic result here being the fact that for H = F N we have = Z N, and so µ = (1 1 N )δ 0 + 1 N δ N. We will be interested in what follows in certain affine deformations of F N, constructed by Diţă in [9]. These deformations appear in the tensor product context, as follows: Definition 1.2. iven two Hadamard matrices H M M (T), K M N (T), the Diţă deformation of (H K) ia,jb = H ij K ab, with matrix of parameters Q M M N (T), is: H Q K = (Q ib H ij K ab ) ia,jb We call a dephased matrix Q M M N (T) generic if its entries are as algebrically independent as possible, in the sense that i,j 1 Qr ij ij = 1 with r ij Z implies r ij = 0.

SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS 5 Here, and in what follows, we use indices i {0, 1,..., M 1} and j {0, 1,..., N 1}, usually taken modulo M, N, as is the most convenient in the Fourier matrix setting. We can now recall the main result in our previous paper [3]. Recall that if H Γ is a finite group acting by automorphisms on a discrete group, the corresponding crossed coproduct Hopf algebra is C (Γ) C(H) = C (Γ) C(H), with comultiplication given by (r δ k ) = h H (r δ h) (h 1 r δ h 1 k), for r Γ, k H. Theorem 1.3. When the parameter matrix Q M M N (T) is generic, the quantum group S + MN associated to the matrix H = F M Q F N is given by C() = C (Γ M,N ) C(Z M ) where Γ M,N =< g 0,..., g M 1 g N 0 =... = g N M 1 = 1, [g i 1... g in, g j1... g jn ] = 1 >, and where the action of Z M on Γ M,N is by cyclic permutation of the generators. Proof. The idea is that for any Q M M N (T) the representation u ia,jb P ia,jb associated to F M Q F N factorizes as follows, where w = e 2πi/N : C(S + MN ) C (Γ M,N ) C(Z M ) M MN (C) u ia,jb 1 N c w(b a)c gi c v ij P ia,jb The point now is that when Q is generic this factorization is minimal. See [3]. A precise description of the group Γ M,N goes as follows. iven a group acting on another group, H, as usual H denotes the semi-direct product of by H, i.e. the set H, with multiplication (a, s)(b, t) = (as(b), st). Now given a group, we can make Z N act cyclically on N, and form the product N Z N. Since the elements (g,..., g) are invariant, we can form as well the product ( N /) Z N, and by identifying N / N 1 via (1, g 1,..., g N 1 ) (g 1,..., g N 1 ) we obtain a product N 1 Z N. We have the following result, basically from [3]: Proposition 1.4. The group Γ M,N has the following properties: (1) Γ M,N Z (M 1)(N 1) Z N via g 0 (0, t) and g i (a i0, t) for 1 i M 1, where a ic and t are the standard generators of Z (M 1)(N 1) and Z N. (2) Γ M,N Z (M 1)N Z N via g 0 (0, t) and g i (b i0 b i1, t) for 1 i M 1, where b ic and t are the standard generators of Z (M 1)N and Z N. Proof. Here (1) is a standard result, explained in detail in [3], and (2) follows from it, after some standard manipulations. Now let us go back to Theorem 1.3. At M = 1 we have Γ 1,N = Z N, so = Z N. Also, at N = 1 we have Γ M,1 = {1}, so = Z M. At M = N = 2 now, we have Γ 2,2 = D, and this leads to the quantum group O2 1 from [2]. In fact, we have:

6 TEO BANICA AND JULIEN BICHON Proposition 1.5. The quantum group associated to F 2 Q F 2, with Q = ( a c b d ), is Z 2 Z 2 :: Z 4 if n = 1 and m = 1, 2 :: m = 4 = D2n 1 :: DCn 1 if 1 < n < and m / 4N :: m 4N O2 1 if n = where m = ord(q), n = ord(q 4 ), with q = ad/bc. Proof. As already explained, the n = assertion follows from Theorem 1.3. Regarding now the n < case, the idea here is that the machinery in [2] leads to the quantum groups D2n 1, DCn 1 constructed by Nikshych in [15], as stated. See [3]. As a concrete consequence of Proposition 1.5, we have: Proposition 1.6. For F 2 Q F 2 with Q = ( a c b d ), with q = ad/bc and n = ord(q4 ), µ = 1 2 (δ 0 + Φ ε 4n ) where ε k is the uniform measure on the k-roots of unity, and Φ(q) = (q + q 1 ) 2. Proof. The idea here is that Proposition 1.5 and the ADE tables in [2] show that the associated principal graph is D 2n+2. On the other hand, according to [2], for D m+2 with m {2, 3,..., } we have µ = 1(δ 2 0 + Φ ε 2m ), and this gives the result. See [3]. At n = the corresponding measure ε is of course the uniform measure on T. Finding an analogue of this result for higher M, N will be our main purpose here. 2. Representation theory We discuss in this section the representation theory of the quantum group associated to the matrix H = F M Q F N, in the case where Q M M N (T) is generic. Let us first discuss the case of arbitrary crossed coproducts. If is a quantum group, we denote by Irr() the set of isomorphism classes of irreducible representations of, and by Irr 1 () the group of 1-dimensional representations of. We have: Lemma 2.1. Let H Γ be a finite group acting on a discrete group, and let be the quantum group given by C() = C (Γ) C(H). Then there is a group isomorphism Irr 1 () Γ H Irr 1 (H) where Γ H denotes the subgroup of H-fixed points in Γ. Proof. The elements of Irr 1 () correspond to group-like elements in CΓ C(H). For r H Γ and φ a 1-dimensional representation of H, it is straightforward to check that r φ is a group-like element in CΓ C(H), and that conversely any group-like element arises in that way, and this proves the assertion. When H acts freely on Γ {1}, we have a more precise result, as follows:

SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS 7 Lemma 2.2. Let H Γ and C() = C (Γ) C(H) be as above, and assume that H acts freely on Γ {1}. Then there exists a bijection Irr() Irr(H) (Γ {1})/H where (Γ {1})/H is the orbit space for the action of H on Γ {1}, and the irreducible representations corresponding to elements of (Γ {1})/H have dimension H. Proof. For r H, consider the following space: C r = Span ( h 1 r δ h 1 k, h, k H ) = Span (h r δ k, h, k H) The subspace C r is a subcoalgebra with C 1 = C(H), and by the freeness assumption, for r 1 we have dim(c r ) = H 2. If (r i ) i I denotes a set of representative of the H-orbits in Γ, then we have a direct sum decomposition, as follows: CΓ C(H) = C(H) ( i I C ri ) For r 1, the coalgebra C r is the coefficent coalgebra of the irreducible corepresentation V r = Span(r δ h, h H) (V r is indeed irreducible since dim(c r ) = dim(v r ) 2 ). Thus from the Peter-Weyl decomposition of C(H) we get the Peter-Weyl decomposition of C (Γ) C(H), and this provides the full description of Irr(), as in the statement. We can now state and prove our main result in this section: Theorem 2.3. Let be the quantum group associated to F M Q F N. Then we have a group isomorphism Irr 1 () Z M, and if M is prime, there exists an explicit bijection Irr() Z M (Γ M,N {1})/Z M where the elements of Z M correspond to 1-dimensional representations, and the elements of (Γ M,N {1})/Z M correspond to M-dimensional irreducible representations. Proof. We have to prove that (Γ M,N ) Z M = {1}. By Lemma 2.1 and Lemma 2.2 this will prove indeed the first assertion, as well as the second one (because when M is prime, Z M has no proper subgroup and hence will act freely on Γ M,N {1}). We use a variation on the description of Γ M,N given in Proposition 1.4. We identify Z (M 1)(N 1) with the quotient of Z MN, the free (multiplicative) abelian group on generators a ic, 0 i M 1, 0 c N 1, by the following relations: N 1 c=0 a ic = 1, 0 i M 1, a 0c = 1, 0 c N 1 The resulting quotient is indeed freely generated by the following elements: a ic, 1 i M 1, 0 c N 2

8 TEO BANICA AND JULIEN BICHON We have an action of Z N = t on Z (M 1)(N 1), given by t(a ic ) = a i,c+1. We therefore get a group identification, as follows: Γ M,N Z (M 1)(N 1) Z N g i (a i0, t) Under this identification, the action of Z M = s on the generators g i corresponds to: s(a ic ) = a i+1,c a 1 1c, s(t) = a 10 t We have to show that for w Γ M,N, then s(w) = w w = 1. First assume that w Z (M 1)(N 1), w = M 1 M 2 i=1 c=0 aλ ic ic, λ ic Z. Then: s(w) = M 1 i=1 M 2 c=0 a λ ic i+1,c a λ ic 1c Thus if s(w) = w then λ 1,c = λ 2,c = λ M 1,c = M 1 i=1 λ ic, for any c, so that w = 1. Now if w = ut l with u = M 1 M 2 i=1 c=0 aλ ic ic Γ M,N and 1 l N 1, we have: s(w) = M 1 i=1 M 2 c=0 a λ ic i+1,c a λ ic 1c a 10 a 11 a 1,l 1 t l From this we see that s(w) w, which concludes the proof. 3. Invariants, self-duality We are interested in what follows in the computation of the spectral measure of the main character χ C() of the quantum group in Theorem 1.3 above. We have here: Proposition 3.1. For the matrix F M Q F N, with Q M M N (T) generic, we have χ k = 1 { } MN # i1,..., i k Z M [(i 1, d 1 ), (i 2, d 2 ),..., (i k, d k )] d 1,..., d k Z N = [(i 1, d k ), (i 2, d 1 ),..., (i k, d k 1 )] where the sets between square brackets are by definition sets with repetition. Proof. According to Theorem 1.3, the main character is given by: χ = 1 gi c v ii = ( ) gi c v ii = gi c δ 1 N ic ic iac Now since the Haar functional of C (Γ) C(H) is the tensor product of the Haar functionals of C (Γ), C(H), this gives the following formula, valid for any k 1: ( ) k χ k = 1 M Γ M,N ic g c i

SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS 9 It is convenient to use the notation 1 1 g i χ k = 1 M = c gc i. The above formula becomes: ( ) k 1 1 g i We use now the embedding in Proposition 1.4 (2). With the notations there: 1 = (b i0 b ic, t c ) 1 g i c Now by multiplying two such elements, we obtain: 1 1 = (b i0 b ic + b jc b j,c+d, t c+d ) 1 g i 1 g j cd In general, the product formula for these elements is: 1 1... = b i 1 0 b i1 c 1 + b i2 c 1 b i2,c 1 +c 2 +b i3,c 1 g i1 1 g 1 +c 2 b i3,c 1 +c 2 +c 3 +......, t c 1+...+c k ik c 1...c k...... + b ik,c 1 +...+c k 1 b ik,c 1 +...+c k Γ M,N In terms of the new indices d r = c 1 +... + c r, this formula becomes: 1 1... = b i 1 0 b i1 d 1 + b i2 d 1 b i2 d 2 +b i3 d 1 g i1 1 g 2 b i3 d 3 +......, t d k ik d 1...d k...... + b ik d k 1 b ik d k Now by integrating, we must have d k = 0 on one hand, and on the other hand: [(i 1, 0), (i 2, d 1 ),..., (i k, d k 1 )] = [(i 1, d 1 ), (i 2, d 2 ),..., (i k, d k )] Equivalently, we must have d k = 0 on one hand, and on the other hand: [(i 1, d k ), (i 2, d 1 ),..., (i k, d k 1 )] = [(i 1, d 1 ), (i 2, d 2 ),..., (i k, d k )] Thus, by cyclic invariance with respect to d k, we obtain the following formula: 1 1... = 1 { } Γ M,N 1 g i1 1 g ik N # [(i d 1,..., d k Z 1, d 1 ), (i 2, d 2 ),..., (i k, d k )] N = [(i 1, d k ), (i 2, d 1 ),..., (i k, d k 1 )] It follows that we have the following moment formula: ( ) k 1 = 1 { } 1 g i N # i1,..., i k Z M [(i 1, d 1 ), (i 2, d 2 ),..., (i k, d k )] d 1,..., d k Z N = [(i 1, d k ), (i 2, d 1 ),..., (i k, d k 1 )] Γ M,N i Now by dividing by M, we obtain the formula in the statement. A first interesting consequence of Proposition 3.1 is as follows: i Proposition 3.2. In the generic case, the quantum invariants of the matrices F M Q F N and F N Q t F M coincide.

10 TEO BANICA AND JULIEN BICHON Proof. This follows indeed from Proposition 3.1, because by rotating the i indices, we see that the formula there is symmetric in M, N. Now let us recall the following basic fact: Proposition 3.3. We have an equivalence (H Q K) t K t Q t H t, so in the Fourier matrix case we have an equivalence (F M Q F N ) t (F N Q t F M ). Proof. We have indeed the following computation, where the last manipulation is an Hadamard matrix equivalence, implemented by a certain permutation of the indices: ((H Q K) t ) ia,jb = Q ja H ji K ba = (K t Q t H t ) ai,bj (K t Q t H t ) ia,jb As for the second formula, this follows from the fact that F M, F N are symmetric. Now by getting back to Proposition 3.2, we have: Theorem 3.4. Let H = F M Q F N, with Q M M N (C) generic, and with M N. (1) The quantum groups associated to H, H t have different fusion rules. (2) However, the laws of the main characters of these quantum groups coincide. Proof. Since having the same fusion rules implies having the same Irr 1 groups, (1) follows from Theorem 2.3. As for (2), this follows from Proposition 3.2 and Proposition 3.3. This result is probably quite interesting in connection with the investigations in [1], [14], [18], regarding the duality between the quantum groups associated to H, H t. 4. eometric interpretation In this section we work out a geometric/random matrix interpretation of the formula in Proposition 3.1. First of all, the formula there can be reformulated as follows: Proposition 4.1. For the matrix F M Q F N, with Q M M N (T) generic, we have χ k = 1 q i1d1... q ikdk dq MN q i1 d k... q ik d k 1 T MN i 1...i k d 1...d k where the integral on the right is with respect to the uniform measure on T MN. Proof. We use the fact that the polynomial integrals over a torus T n are given by: q i1... q ik dq = δ [i1,...,i q j1... q k ],[j 1,...,j k ] jk T n Here δ is a Kronecker symbol, and the sets between square brackets at right are as usual sets with repetitions. Now with n = MN, we obtain the following formula: q i1d1... q ikdk dq = δ [i1 d q i1 d k... q 1,...,i k d k ],[i 1 d k,...,i k d k 1 ] ik d k 1 T NM Now since at right we have exactly the contributions to the integral computed in Proposition 3.1 above, this gives the result.

SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS 11 A useful version of Proposition 4.1 comes by dephasing, as follows: Proposition 4.2. For F M Q F N, with Q M M N (T) generic, we have χ k = 1 r i1d1... r ikdk dr MN r i1 d k... r ik d k 1 T (M 1)(N 1) i 1...i k d 1...d k where r = ( 1 1 1 r) is the matrix obtained from r by adding a row and column of 1 entries. Proof. The dephasing is an identification T T MN T M T N T (M 1)(N 1), given by cq ij = a i b j r ij. Now if we denote by Φ the quantity in Proposition 4.1, we have: Φ(q)dq = Φ(cq)d(c, q) T MN T T MN = Φ((a i b j r ij ) ij )d(a, b, r) T M T N T (M 1)(N 1) = Φ( r)dr T (M 1)(N 1) But this gives the formula in the statement, and we are done. Yet another useful reformulation of Proposition 4.1 is as follows: Proposition 4.3. For F M Q F N, with Q M M N (T) generic, we have χ k = 1 q i1i2 d MN 1 d k q i 2i 3 d 2 d k... q i k 1i k d k 1 d k T (M 1)(N 1) d 1...d k i 1...i k where q ij de = q idq je q ie q jd are the variables appearing in the expansion of χ2. Proof. We have indeed the following formula: q i1 d 1 q i2 d 2... q ik d k = q i 1 d 1 q i2 d k qi 2 d 2 q i3 d k... q i q k 1d k 1 i k d k q i1 d k q i2 d 1... q ik d k 1 q i1 d k q i2 d 1 q i2 d k q i3 d 2 q ik 1 d k q ik d k 1 Now by summing over i, d and integrating, this gives the formula in the statement. Observe that Proposition 4.2 and Proposition 4.3 tell us that the moments of χ can be expressed in terms of the second moment variables q ij de = q idq je q ie q jd, and that when defining these variables, we can in addition assume that q = r is a dephased matrix. We will see in the next section that this observation applies particularly well at M = 2. In the general case, Proposition 4.1 is best interpreted as follows: Proposition 4.4. For F M Q F N, with Q M M N (T) generic, we have χ k = 1 T r(a(q) k )dq MN T MN where A(q) ij =< R i, R j >, where R 1,..., R M are the rows of q T MN M M N (T).

12 TEO BANICA AND JULIEN BICHON Proof. We use the formula in Proposition 4.1. We have: ( ) ( χ k 1 q i1d1 = MN q i2 d 1 = T MN i 1...i k d 1 d 2 q i2d2 q i3 d 2 )... ( d k q ikdk q i1 d k 1 < R i1, R i2 >< R i2, R i3 >... < R ik, R i1 > MN T MN 1 MN = A(q) i1 i 2 A(q) i2 i 3... A(q) ik i 1 T MN But this gives the formula in the statement, and we are done. We can formulate now our first spectral measure blowup result, as follows: Theorem 4.5. For F M Q F N, with Q M M N (T) generic, we have ( law(χ) = 1 1 ) δ 0 + 1 N N law(a) ) dq where A C(T MN, M M (C)) is given by A(q) = ram matrix of the rows of q. Proof. This follows from Proposition 4.4. Indeed, in terms of tr = 1 T r, we have: M χ k = 1 tr(a(q) k )dq N T MN But this tells us precisely that, up to a rescaling as in the statement, the random variable χ follows the same law as the random variable A. In general, the moments of the ram matrix A are given by a quite complicated formula, and we cannot expect to have a refinement of Theorem 4.5, with A replaced by a plain (non-matricial) random variable, say over a compact abelian group. However, this kind of simplification does appear at M = 2, as we will see in the next section. 5. Toral blowup at M=2 In this section we discuss in detail the cases M = 2 and N = 2, with the idea in mind of generalizing Proposition 1.6 at n =. As a first remark, at N = 2 we have: Proposition 5.1. The groups Γ M,2 have the following properties: (1) Γ 2,2 D. (2) Γ M,2 = Z M 2 / < abc = cba >. (3) Γ 3,2 has the honeycomb lattice as Cayley graph. Proof. All these results are quite well-known, the proof being as follows: (1) At M = 2 the commutation relations [ab, cd] = 1 are automatic, and so we obtain Γ 2,2 = Z 2 Z 2 D, the last isomorphism being well-known.

SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS 13 (2) We have to prove that the commutation relations [ab, cd] = 1 are equivalent to the half-commutation relations abc = cba from [5]. Indeed, assuming that [ab, cd] = 1 holds, we have ab ac = ac ab, and so bac = cab. Conversely, assuming that abc = cba holds, we have ab cd = a bcd = a dcb = adc b = cda b = cd ab. (3) At M = 3 our group is Γ 3,2 = Z 3 2 / < abc = cba >, whose Cayley graph is the honeycomb lattice (one basic hexagon being 1 a ac acb bc b). We should mention that the above result was extremely useful at some early stages of the present work, when we were exploring various small M, N situations. Let us compute now the associated measures, at M = 2. We first have: Lemma 5.2. For F 2 Q F N, with Q M 2 N (T) generic, we have ( χ ) k TN ( ) k 2p a 1 +... + a N N = N 2p N da where the integral on the right is with respect to the uniform measure on T N. p 0 Proof. Consider the quantity that appears on the right in Proposition 4.1, namely: Φ(q) = q i1d1... q ikdk q i1 d k... q ik d k 1 i 1...i k d 1...d k As explained in Proposition 4.2 and its proof, we can dephase the matrix q M 2 N (T) if we want, so in particular we can half-dephase it, as follows: ( ) 1... 1 q = a 1... a N We have to compute the above quantity Φ(q) in terms of the numbers a 1,..., a N. Our claim is that we have the following formula: Φ(q) = 2 ( k N 2p) k 2p 2p a i p 0 i Indeed, the 2N p contribution will come from i = (1... 1) and i = (2... 2), then we will have a k(k 1)N p 2 i a i 2 contribution coming from indices of type i = (2... 21... 1), up to cyclic permutations, then a 2 ( k 4) N k 4 i a i 4 contribution coming from indices of type i = (2... 21... 12... 21... 1), and so on. More precisely, in order to find the N k 2p i a i 2p contribution, we have to count the circular configurations consisting of k numbers 1, 2, such that the 1 values are arranged into p non-empty intervals, and the 2 values are arranged into p non-empty intervals as well. Now by looking at the endpoints of these 2p intervals, we have 2 ( k 2p) choices, and this gives the above formula. Now by integrating, this gives the formula in the statement.

14 TEO BANICA AND JULIEN BICHON Observe that the integrals in Lemma 5.2 can be computed as follows: a 1 +... + a N 2p a i1... a ip da = da T N T a N i 1...i p j 1...j j1... a jp p { } [i1 = # i 1... i p, j 1... j p,..., i p ] = [j 1,..., j p ] = ( ) 2 p r 1,..., r N p=σr i Regarding the combinatorics of these latter numbers, see [17], [20]. Let us go back now to the blowup problematics. First, we have: Proposition 5.3. For F 2 Q F N, with Q M 2 N (T) generic, we have ( µ = 1 1 ) δ 0 + 1 ( Ψ + N 2N ε + Ψ ε ) where ε is the uniform measure on T N, and Ψ ± (a) = N ± i a i. Proof. We use the formula in Lemma 5.2, along with the following identity: ( k )x 2p = (1 + x)k + (1 x) k 2p 2 p 0 By using this identity, Lemma 5.2 reformulates as follows: ( χ ) k ( 1 N = 1 + i a i k ( N 2 T N ) + 1 i a i k N ) da N Now by multiplying by N k 1, we obtain the following formula: χ k = 1 ( ) k ( ) k N + a 2N i + N a i da T N But this gives the formula in the statement, and we are done. i As an example, at N = 1 the δ 0 term dissapears, ε is the uniform measure on T, and Ψ = 0, Ψ + = 2. Thus we obtain indeed the spectral measure of Z 2, µ = 1(δ 2 0 + δ 2 ). At N = 2 now, µ = 1δ 2 0 + 1 4 (Ψ+ ε + Ψ ε). By dephasing, a = (1, q), we may assume that ε is the uniform measure on T, and our blowup maps become Ψ ± (q) = 2 ± 1 + q. Now since with q = t 4 we have {Ψ + (q), Ψ (q)} = {(t + t 1 ) 2, (t t 1 ) 2 }, and since with t it we have (t t 1 ) 2 (t + t 1 ) 2, we conclude that averaging the measures Ψ ± ε on [0, 2] and [2, 4] produces the measure on [0, 4] obtained via Φ(t) = (t + t 1 ) 2. Thus we have recovered Proposition 1.6, at n =. These N = 2 manipulations, however, are quite special and don t have a higher dimensional analogue, because at N 3 the quantity i a i 2 = ij does not have a simple writing as a square. a i a j i

SPECTRAL MEASURE BLOWUP FOR BASIC HADAMARD SUBFACTORS 15 However, we can reduce the maps Ψ ± to a single one, as follows: Theorem 5.4. For F 2 Q F N, with Q M 2 N (T) generic, we have ( µ = 1 1 ) δ 0 + 1 N N Φ ε where ε is the uniform measure on Z 2 T N, and Φ(e, a) = N + e i a i. Proof. This is clear indeed from Proposition 5.3 above. As explained in the introduction, this result is part of an emerging subfactor machinery, related to [2], [4], [11], [12]. We have as well the following conjecture: Conjecture 5.5. In the non-generic case, the spectral measure blowup should land into a certain arithmetic lattice, canonically associated to Q M M N (T). Observe that this conjecture is supported by Proposition 1.6. In order to further comment here, recall from [3] that the parameter q = ad appears by dephasing: bc ( ) ( ) ( ) a b 1 1 1 1 Q = c d c d 1 ad a b bc On the other hand, the dephasing is given by the action T (A,B) (H) = AHB of the group (K 2 K 2 )/T on M 4 (T), where K 2 = T S 2. Thus the group < q > T is more or less the orbit of the group < Q > M 4 (T) under this action. The problem, however, is that Proposition 1.6 involves the group < q 4 > T n, instead of the group < q > itself. References [1] T. Banica, Compact Kac algebras and commuting squares, J. Funct. Anal. 176 (2000), 80 99. [2] T. Banica and J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009), 74 114. [3] T. Banica and J. Bichon, Quantum invariants of deformed Fourier matrices, arxiv:1310.6278. [4] T. Banica and D. Bisch, Spectral measures of small index principal graphs, Comm. Math. Phys. 269 (2007), 259 281. [5] T. Banica and R. Vergnioux, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier 60 (2010), 2137 2164. [6] J. Bichon and M. Dubois-Violette, Half-commutative orthogonal Hopf algebras, Pacific J. Math. 263 (2013), 13 28. [7] D. Bisch and U. Haagerup, Composition of subfactors: new examples of infinite depth subfactors, Ann. Sci. Ecole Norm. Sup. 29 (1996), 329 383. [8] R.D. Burstein, roup-type subfactors and Hadamard matrices, arxiv:0811.1265. [9] P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (2004), 5355 5374. [10] D.E. Evans and Y. Kawahigashi, Quantum symmetries on operator algebras, Oxford University Press (1998). [11] D.E. Evans and M. Pugh, Spectral measures and generating series for nimrep graphs in subfactor theory, Comm. Math. Phys. 295 (2010), 363 413.

16 TEO BANICA AND JULIEN BICHON [12] V.F.R. Jones, The annular structure of subfactors, Monogr. Enseign. Math. 38 (2001), 401 463. [13] V.F.R. Jones, Planar algebras I, math.qa/9909027. [14] Y. Kawahigashi, Quantum alois correspondence for subfactors, J. Funct. Anal. 167 (1999), 481 497. [15] D. Nikshych, K 0 -rings and twisting of finite-dimensional semisimple Hopf algebras, Comm. Algebra 26 (1998), 321 342. [16] S. Popa, Orthogonal pairs of -subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), 253 268. [17] L.B. Richmond and J. Shallit, Counting abelian squares, Electron. J. Combin. 16 (2009), 1 9. [18] N. Sato, Two subfactors arising from a non-degenerate commuting square. An answer to a question raised by V.F.R. Jones, Pacific J. Math. 180 (1997), 369 376. [19] W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006), 133 177. [20] B. Vidakovic, All roads lead to Rome even in the honeycomb world, Amer. Statist. 48 (1994), 234 236. [21] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195 211. [22] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613 665. T.B.: Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France. teo.banica@gmail.com J.B.: Department of Mathematics, Clermont-Ferrand University, F-63177 Aubiere, France. bichon@math.univ-bpclermont.fr