E Falbel

I Pasquinelli

A Ucan-Puc

Representations of Deligne-Mostow lattices into PGL(3, C)

We classify representations of a class of Deligne-Mostow lattices into PGL(3, C). In particular, we show local rigidity for the representations (of Deligne-Mostow lattices with 3-fold symmetry and of type one) where the generators we chose are in the same conjugacy class as the generators of Deligne-Mostow lattices. We use formal computations in SAGE to obtain the results. The code files are available on GitHub ([FPUP21]).

Introduction

Lattices in semi-simple Lie groups have strong rigidity properties. When the Lie group is a connected component of the isometry group G of an irreducible hermitian space, Mostow's rigidity theorem states that, in dimension greater than two, any cocompact (or finite covolume by Prasad) lattice Γ ⊂ G is rigid in the sense that an isomorphism between lattices extends to a homomorphism between isometry groups. For hermitian symmetric spaces of higher rank, Margulis' super-rigidity theorem implies that any homomorphism φ : Γ → G (with G , connected with trivial centre and no compact factors) with Zariski dense image extends to a homomorphism φ : G → G .

In rank one, it is well known that super rigidity fails. In particular, there might exist representations of a given lattice in PU(2, 1) into other Lie groups which do not extend to homomorphisms of the group PU(2, 1). A very active study of representations of lattices into hermitian Lie groups originated with Toledo's ( [START_REF] Toledo | Representations of surface groups in complex hyperbolic space[END_REF]) description of maximal representations of lattices in PU(1, 1) into PU(2, 1) (see [START_REF] Beatrice | Maximal representations of complex hyperbolic lattices into SU(M, N )[END_REF] for a recent overview).

In this paper we are interested in a target Lie group which is not hermitian. We classify representations (up to conjugation) of a class of Deligne-Mostow lattices into PGL(3, C).

An important motivation for our results is the study of complex projective structures. Indeed, given a complex projective structure on a complex surface M , one can define a developing map D : M → CP 2 and a holonomy representation ρ : π 1 (M ) → PGL(3, C). Complex hyperbolic manifolds are complete complex projective structures. It is an open question whether a complex hyperbolic manifold has a different complex projective structure. On the other hand, fixing the complex structure of a finite volume complex hyperbolic surface, its complex projective structure is unique by a theorem of Mok-Yeung [START_REF] Mok | Geometric realizations of uniformization of conjugates of Hermitian locally symmetric manifolds[END_REF] (see also [START_REF] Klingler | Structures affines et projectives sur les surfaces complexes[END_REF]). It is possible that some of the representations in our list correspond (taking torsion free subgroups) to complex projective structures on complex manifolds diffeomorphic but not biholomorphic to complex hyperbolic manifolds. One should be aware that in the compact case, Siu's rigidity theorem (valid for any compact locally hermitian symmetric space) implies that any other Kähler complex structure on the manifold is biholomorphic or conjugate-biholomorphic to the original complex hyperbolic structure.

In section 2 we give the classification of all representations where the generators we chose are of the same type as the generators of the Deligne-Mostow lattice.

In Sections 3 and 4 we review rigidity results and the definition of the Deligne-Mostow lattices in P U (2, 1). The main result which allows computations is the explicit presentation of the Deligne-Mostow lattices. It was obtained in several steps culminating in [START_REF] Pasquinelli | Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere[END_REF], where a unified treatment of all lattices is given. We will use a presentation of each Deligne-Mostow group using two generators easily obtained from the presentations in [START_REF] Pasquinelli | Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere[END_REF].

In section 5 we give the detailed proofs for one case. The other cases are calculated in the same way. The main strategy is to use Gröbner basis methods to solve the equations imposed by the relators in the presentation. The computations show that the representations obtained are all locally rigid. With the exception of the Deligne-Mostow lattice itself (because of the local rigidity theorem for complex hyperbolic lattices), we do not know if this local rigidity phenomenon is specific to the lattices we studied or has a more general scope. With an appropriate conjugation, the representations have values in an algebraic field extension of Q and we studied the orbits by the Galois group.

We thank Sorin Dumitrescu for useful discussions and for pointing to us the local rigidity result in [START_REF] Mok | Geometric realizations of uniformization of conjugates of Hermitian locally symmetric manifolds[END_REF]. The second author also acknowledges the support of the FSMP and of the EPSRC grant EP/T015926/1.

The results

We refer to section 4 for the definition of Deligne-Mostow lattices. For the statement of the results we simply recall that 3-fold Deligne-Mostow lattices of type one are parametrised by a couple of integers (p, k) and are generated by two elements J and R 1 which are, respectively, a regular elliptic element and a complex reflection.

p, k) Total Number Galois Orbits Irreducible Q-extension (3,4) 18 2 2 Q(1 -x 6 + x 12 ) (3,5) 24 1 1 Q(1 -x 3 + x 9 -x 12 + x 15 -x 21 + x 24 ) (4,3) 22 5 3 Q(1 -x 6 + x 12 ) (5,2) 12 2 2 Q(1 -x + x 3 -x 4 + x 5 -x 7 + x 8 ) (5,3) 24 1 1 Q(1 + x 3 -x 9 -x 12 -x 15 + x 21 + x 24 )
Q(x 4 -x 2 +1) 1 0 (2,1) 0 1 0 (3,0) 1 0 (2,2) Degenerate 4 Q(x 2 -x + 1) 1 0 (3,0) 1 Q(x 2 +1) 1 0 (2,1) 0 1 0 (3,0) 1 0 (2,0) Degenerate 2 Q 1 0 (3,0) 1 (6,2) 12 4 Q(x 6 -x 3 +1) 1 0 (2,1) 0 1 0 (3,0) 1 Q(x 2 -x + 1) 1 0 (3,0) 1 Q 1 0 (3,0) 1 (6,3) 30 13 Q(x 6 + x 3 + 1) 1 0 (2,1) 0 2 0 (3,0) 2 Q(x 2 + x + 1) 0 (6,4) Degenerate 9
Table 2: Non-compact 3-fold Deligne-Mostow Lattices

In Table 1 are the information for each compact 3-fold type one Deligne-Mostow lattice. The representations are defined with coefficients in a cyclotomic extension of Q, which depends on (p, k) and for which we give the corresponding minimal polynomial. We also compute the orbits of the Galois group action on the space of representations. For the compact Deligne-Mostow lattices, there exits an invariant Hermitian form which is always non-degenerate.

In Table 2 are the information on non-compact 3-fold type one lattices. Each representation has coefficients in a subfield of a Q-extension which depends on (p, k). For each representation there exists an invariant Hermitian form that can be non-degenerate with signatures (2,1) or (3,0), or degenerate. While the irreducible representations can only come from non-degenerate configurations, the reducible ones can come from either degenerate or non-degenerate configurations (see Section 5). In the Reducible column the numbers (m, n) mean that m reducible representations come from non-degenerate configurations and n come from degenerate ones. The last column corresponds to the number of representations which contain in their kernel an element of the centraliser of the cusp group. One may interpret those representations as representations which factor through the orbifold fundamental group of a compactification of the complex hyperbolic orbifold defined by the Deligne-Mostow lattice.

We also decide which representations may be lifted to GL(3, C) representations (see [START_REF] Falbel | Code Representations of Deligne-Mostow lattices into P GL(3, C)[END_REF]). The representations with generators ρ(J) and ρ(R 1 ) of other types are not tabulated. The computations show that they are all locally rigid except for one dimensional branches for Deligne-Mostow lattices (4, 4), (6, 2) and (4, 3).

In the following sections, we give details for the calculations in the case of the Deligne-Mostow lattice (3, 6). We also compute the reducible representations of this lattice when the generators are not of the same type as the ones of the Deligne-Mostow lattice. The same calculations for the other Deligne-Mostow lattices in the table can be found on GitHub ( [START_REF] Falbel | Code Representations of Deligne-Mostow lattices into P GL(3, C)[END_REF]).

Rigidity of lattices and projective structures

In this section we recall some rigidity theorems in the two dimensional complex hyperbolic context. The first one is Mostow's (Prasad's in the non-compact quotient case) rigidity for lattices in P U (2, 1):

Theorem 3.1. If ρ : Γ 1 → Γ 2 is
an isomorphism between two lattices in P U (2, 1) then it extends to an automorphism of P U (2, 1).

Therefore Γ 2 is conjugated to Γ 1 or to its complex conjugate by an element of P U (2, 1). The representations we find in this paper have images in PGL(3, C) and are counterexamples to Margulis' super-rigidity theorem in rank one, which states as follows. Let G and G be connected Lie groups, with trivial centre and no compact factors. Suppose that G has rank higher than one and that it is not isogenous to a group of the form SO(1, n) × K or SU (n, 1) × K where K is compact. Let Γ ⊂ G be a lattice and ρ : Γ → G a homomorphism with Zariski dense image. Margulis' super-rigidity theorem states that ρ extends to a continuous homomorphism of G into G .

In this paper we construct a large class of homomorphisms of Deligne-Mostow lattices in P U (2, 1) into P GL(3, C). One could think that these representations (taking torsion free subgroups of the lattice) might be holonomy representations of different complex projective structures on the quotient of complex hyperbolic space by the lattice. But that is not the case, at least if the complex structure of the quotient is fixed:

Theorem 3.2 (Mok-Yeung [MY93]). Let Γ\H 2
C be a complex hyperbolic manifold of finite volume. Then its complex projective structure induced by the embedding H 2 C ⊂ CP 2 is the unique projective structure compatible with its complex structure.

The theorem is also true for complex hyperbolic manifolds of dimension greater than two (see [START_REF] Mok | Geometric realizations of uniformization of conjugates of Hermitian locally symmetric manifolds[END_REF]) and for higher rank hermitian symmetric domains (see [START_REF] Klingler | Structures affines et projectives sur les surfaces complexes[END_REF]). On the other hand, the existence of different complex structures on the quotient Γ\H 2 C (for a torsion free discrete group Γ) is an open problem except for the local rigidity result of ). This states that, in the two dimensional complex hyperbolic setting, the complex structure of Γ\H 2 C is locally rigid. One should also observe that local rigidity (up to conjugation) in PGL(3, C) of the holonomy representation of the complex hyperbolic structure, that is H 1 (Γ, sl(3, C)) = 0, follows easily from the local rigidity of hyperbolic structures (see [START_REF] Klingler | Structures affines et projectives sur les surfaces complexes[END_REF] prop. 3.2).

Deligne-Mostow lattices

The Deligne-Mostow lattices were introduced and studied by Mostow and by Deligne and Mostow in various works, including [START_REF] Mostow | On a remarkable class of polyhedra in complex hyperbolic space[END_REF] and [START_REF] Deligne | Monodromy of hypergeometric functions and nonlattice integral monodromy[END_REF]. They have a long history dating back to by Picard, Le Vavasseur and Lauricella. Deligne and Mostow start with a ball N-tuple µ = (µ 1 , . . . , µ N ) which is a set of N real numbers in (0, 1) whose sum is 2. Then they look at a hypergeometric function defined using these parameters. Finally, they use the monodromy action to build some lattices in P U (N -3, 1). This list of lattices contains the first known examples on non-arithmetic complex hyperbolic lattices in dimensions 2 and 3.

Here we will concentrate on a very special class of the Deligne-Mostow lattices in P U (2, 1), called the lattices with 3-fold symmetry and of type one. More precisely, first we restrict to the case of dimension 2. This means that we are looking at the group P U (2, 1) acting on 2-dimensional complex hyperbolic space and hence that we are looking at ball 5tuples. In this dimension, the finite list of lattices that Deligne and Mostow found contains only lattices with some special symmetry. These are called the 2-fold and 3-fold symmetry lattices. A lattice has m-fold symmetry is m of the N elements of the ball N -tuple are equal.

In this work we will only look at some of the lattices with 3-fold symmetry. For all of those, one can find a construction for a fundamental domain and a presentation in [START_REF] Pasquinelli | Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere[END_REF]. In principle the 2-dimensional Deligne-Mostow lattices depend on the 5 elements of the ball 5-tuple. The condition on the sum of the elements of the ball 5-tuple reduces the parameters to 4. Of these 4, 3 are equal, so the lattices only depend on 2 parameters. One common choice for the parameters is to choose the orders of some of the generators. We will hence identify the lattices using a pair (p, k) which corresponds to the ball 5-tuple (1/2 -1/p, 1/2 -1/p, 1/2 -1/p, 1/2 + 1/p -1/k, 2/p + 1/k).

The Deligne-Mostow lattices in P U (2, 1) with 3-fold symmetry are divided in 4 types according to the ranges of p and k. The type determines the presentation, the volume formula and the combinatorics of a fundamental domain. Here we will look at the lattices of type one, which were first looked at in [START_REF] Boadi | Mostow's lattices and cone metrics on the sphere[END_REF]. We are choosing these because they have minimal complexity (minimum number of facets in a fundamental domain, shortest presentation). In terms of p and k they are characterised by

0 < p ≤ 6, k ≤ 2p p -2 .
Then a presentation for them is given by

Γ = J, P, R 1 , R 2 : J 3 = R p 1 = R p 2 = (P -1 J) k = I, R 2 = P R 1 P -1 = JR 1 J -1 , P = R 1 R 2 , (1)
We can rewrite this presentation in terms of the two generators J and R 1 in the following way:

Γ = J, R 1 : J 3 = R p
5 The group (3,6)

In this section we look in details at the case (p, k) = (3, 6). The presentation for this case is:

Γ = J, R 1 : J 3 = R 3 1 = (R 1 J) 12 = R 1 JR 1 J 2 R 1 JR 2 1 J 2 R 2 1 JR 2 1 J 2 = Id , (3)
We first note that for the Deligne-Mostow lattices, one has that J is a regular elliptic element, while R 1 is a complex reflection. In general, while the presentation guarantees that ρ(J) and ρ(R 1 ) are both elliptic, we cannot say more about their type. We will hence look at all possible combinations of types:

• when ρ(J) is regular elliptic and ρ(R 1 ) is a complex reflection,

• when ρ(J) is a complex reflection and ρ(R 1 ) is regular elliptic,

• when ρ(J) and ρ(R 1 ) are both regular elliptic,

• when ρ(J) and ρ(R 1 ) are both complex reflections.

Note that if ρ(J) and ρ(R 1 ) are both complex reflections, then the group preserves the intersection of two complex lines in CP 2 . The group is therefore reducible and we will hence ignore this last case.

ρ(J) is regular elliptic and ρ(R 1 ) is a complex reflection

First, let us assume that the fixed points of ρ(J) do not intersect the fixed points of ρ(R 1 ). We will call these the non-degenerate configurations. We remark that this case might give rise to reducible representations when there exists a degenerate Hermitian form preserved by the generators. We note in the following ω = -1 2 + √ 3 2 i and ζ 9 a primitive 9th-root of unity.

Proposition 5.1. A representation ρ : Γ → P GL(3, C) such that ρ(R 1 ) is a complex reflection and ρ(J) has three distinct eigenvectors which do not intersect the fixed line of ρ(R 1 ) are given, up to conjugation, by the matrices

J =   0 0 1 -1 0 0 0 1 0   (4) and ρ(R 1 ) =   1 -r 1 r 1 0 1 -r 2 r 2 0 1 -r 2 -ω r 2 + ω   , (5)
with (r 1 , r 2 ) in the following list appear from a simple factorisation of the equations. The others come from the following system:

(8r

3 2 + 6(i √ 3 -3)r 2 2 -12(i √ 3 -1)r 2 + 6i √ 3 -1)(8r 3 2 + 6(i √ 3 -3)r 2 2 -12(i √ 3 -1)r 2 + 6i √ 3 + 1) = 0, ( 32ir
4 2 √ 3 -96(i + 1)r 3 2 √ 3 + (92i + 276) √ 3r 2 2 -25i √ 3 -12r 1 -264r 2 + 75)(8r 3 2 + 6(i √ 3 -3)r 2 2 -12(i √ 3 -1)r 2 + 6i √ 3 -1) = 0, (3 -i √ 3)r 1 + (3 -i √ 3)r 2 + 2r 2 1 -2r 2 2 -1 + √ 3 = 0.
Writing

p 1 (r 2 ) = 8r 3 2 + 6(i √ 3 -3)r 2 2 -12(i √ 3 -1)r 2 + 6i √ 3 -1, p 2 (r 2 ) = 8r 3 2 + 6(i √ 3 -3)r 2 2 -12(i √ 3 -1)r 2 + 6i √ 3 + 1,
both polynomials factor over Q(ζ 9 ). For each root of p 1 , the corresponding values of r 1 satisfy a quadratic polynomial equation which factors over Q(ζ 9 ). In a similar way, for each root of p 2 , the corresponding values of r 1 satisfy a linear polynomial equation whose root belong to Q(ζ 9 ). Therefore the whole set of solutions belongs to the field Q(ζ 9 ). The solutions are then written in the basis of Q(ζ 9 ).

We now look at the degenerate configurations, where one eigenvector of ρ(J) intersects the fixed complex line of ρ(R 1 ). This gives rise to reducible representations: Proposition 5.2. The representation ρ : Γ → P GL(3, C) such that ρ(R 1 ) is a complex reflection and ρ(J) has three distinct eigenvectors with one intersecting the fixed line of ρ(R 1 ) are given, up to conjugation, by the matrices

ρ(J) =   1 0 0 0 e 2iπ/3 0 0 0 e 4iπ/3   (7)
and

ρ(R 1 ) =   1 -1 1 0 1 -r 2 r 2 0 1 -r 2 -x r 2 + x   , (8)
where x 2 + x + 1 = 0 and with r 2 determined as follows:

1. for x = e 2iπ/3 , r 2 = 1 - √ 3i/3 or r 2 = 1/2 - √ 3i/6, 2. for x = e 4iπ/3 , r 2 = 1 + √ 3i/3 or r 2 = 1/2 + √ 3i/6.
Proof. We choose a diagonal form for ρ(J) and impose that ρ(R 1 ) has the same eigenvector e 1 . One can then impose that the fixed line in projective space passes through [1, 0, 0] and [1, 1, 1]. By conjugating by a diagonal matrix, one can suppose that ρ(R 1 ) is as above with r 2 to be determined. As in the previous proposition a computation using Gröbner basis package in SAGE gives the result. Note that the two sets of solutions are complex conjugates.

5.0.2 ρ(J) is a complex reflection and ρ(R 1 ) is regular elliptic

Interchanging the type of ρ(J) and ρ(R 1 ), a computation with Gröbner basis proves the following:

Proposition 5.3. There are no representations ρ : Γ → P GL(3, C) such that ρ(J) is a complex reflection and ρ(R 1 ) has three distinct eigenvectors which do not intersect the fixed line of ρ(R 1 ).

5.0.3 ρ(J) and ρ(R 1 ) are regular elliptic

The final case to consider occurs when both ρ(J) and ρ(R 1 ) are regular elliptic. A simple computation shows that the two regular elliptic elements cannot have the same eigenspaces.

Proposition 5.4. The representations ρ : Γ → P GL(3, C) such that ρ(R 1 ) and ρ(J) are regular elliptic with at least one distinct eigenspace are given, up to conjugation, by the matrices

J =   0 0 1 -1 0 0 0 1 0   (9)
and

ρ(R 1 ) =   1 s 1 r 1 0 s 2 r 2 0 s 3 -1 -s 2   , (10)
with r 1 , r 2 , s 1 , s 2 , s 3 in the following list (up to complex conjugation) 11. r 1 = -5/7 + 1/7i √ 3, r 2 = -5/7 + 1/7i √ 3, s 1 = 9/7 + 1/7i (3), s 2 = -3/14 -5/14i √ 3, s 3 = 5/7 -1/7i √ 3,

Proof. We can again fix J to be of the form above and one eigenvector for R 1 to be the first column vector. Observe that a regular elliptic element of order three has null trace and, therefore, we obtain the form of the matrix for R 1 . We obtain the result finding a Gröbner basis for the system of equations as in the previous proposition.

The Galois Conjugation

Consider the Q-extension by the cyclotomic polynomial x 6 + x 3 + 1, that we will denote Q(ζ 9 ) where ζ 9 is again the 9-th root of unity. This extension has a subfield isomorphic to the Q-extension by x 2 + x + 1, that we will denote by Q(ω) where we recall that ω = ζ 3 9 . Recall also that the Galois group of Q(ζ 9 ), Gal(Q(ζ 9 )/Q), is isomorphic Z 6 .

Remark 5.5. For the first six solutions in Proposition 5.1, the group image belongs to PGL(3, O 3 ) up to a projective representative, therefore the corresponding groups have discrete image.

The group image for the other solutions, up to projective representative, belongs to PGL(3, O 9 ) where O 9 denotes the ring of integers of Q(ζ 9 ). We do not know which of these representations are discrete.

The action of the Galois group Gal(Q(ζ 9 )/Q) over the field Q(ζ 9 ) induces an action over the set of solutions, therefore on the set of representations. Recall that Gal(Q(ζ 9 )/Q) is a cyclic group, and we will denote its generator by g. In particular, g(ζ 9 ) = ζ 2 9 . Therefore the element g 2 sends ζ 9 to ζ 4 9 , and one can verify that

g 2 (ω) = g 2 (ζ 3 9 ) = ζ 3 9 ,
and so the group generated by g 2 is the unique subgroup of the Galois group that fixes ω. Therefore, the subgroup g 2 fixes the subfield Q(ω) and in particular, all solutions {α j } 6 j=1 are fixed points under the Galois group action. By a direct computation, one can check that the action of g 2 on the solutions {α j } 1 j=7 5 has three closed orbits, namely {α 7 , α 8 , α 9 }, {α 10 , α 11 , α 12 } and {α 13 , α 14 , α 15 }, and these are the only closed orbits in the set of solutions under the Galois group action.

The Hermitian form

In what follows, we will find the Hermitian form that is preserved by each representation and it will sometimes be degenerate. Assume that H is an Hermitian form. Now if Γ preserve H, then the generators in the presentation (3) satisfy Once we solved the equation, we have that for the Galois orbits {α 7 , α 8 , α 9 } and {α 10 , α 11 , α 12 }, the Hermitian form is non-degenerate and of signature (3,0). For the orbit {α 13 , α 14 , α 15 } the Hermitian form is non-degenerate of signature (2,1). In the case of the Galois fixed points {α 1 , α 2 , α 3 , α 4 , α 5 , α 6 }, all Hermitian forms are degenerate with two zero eigenvalues.

J * HJ = H, R * 1 HR 1 = H.

The cusp group

For a non-compact lattice, one defines a cusp holonomy as a conjugacy class of maximal subgroups (not containing loxodromic elements) fixing a point in the boundary of complex hyperbolic space. The following proposition was proved using a fundamental domain for the lattice (see [START_REF] Boadi | Mostow's lattices and cone metrics on the sphere[END_REF] and [START_REF] Pasquinelli | Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere[END_REF]).

Proposition 5.6. The Deligne-Mostow (3,6)-lattice Γ (3,6) ⊂ PU(2, 1) has only one cusp. The cusp holonomy is the class determined by

Γ cusp = R 2 = JR 1 J 2 , A 1 = JR 2 1 J 2 R 2 1 J .
The cusp holonomy might contain elliptic elements. The purely parabolic cusp holonomy is the maximal subgroup of the cusp holonomy with no elliptics.

Proposition 5.7. The purely parabolic cusp holonomy of

Γ (3,6) is [A 1 , R 2 ], [A 1 , R 2
2 ], (R 2 A 1 ) 2 . The centre of this group is the cyclic group generated by (R 2 A 1 ) 2 .

A straightforward computation with the list of representations gives the following description of cusp groups. The representations for which the generator of the centraliser is elliptic, may be factorised (up to a finite index subgroup) through a representation of the fundamental group of the Satake-Baily-Borel compactification. Indeed, in the compactification, the centraliser of the cusp holonomy disappears.

Proposition 5.8. The generator of the centraliser, (R 2 A 1 ) 2 , is elliptic of order at most three for all representations except for three (up to conjugation) where it can be chosen to be unipotent of the form

ρ((R 2 A 1 ) 2 ) =   1 ξ 0 0 1 0 0 0 1   .
(11)

Theorem 2. 1 .

 1 Representations of 3-fold type one Deligne-Mostow lattices into PGL(3, C) (with image of generators conjugated to those of the lattice) are locally rigid and are classified in the following tables.

(

  From the first equation, we can assure that H is of the form   a -c c -c a c c c a   , where a ∈ R, and we can assume that c ∈ Q(ζ 9 ). Let c = b 0 +Σ 5 j=1 b j ζ j 9 . Then the second equation provides a polynomial system on Q(ζ 9 )[a, b 0 , b 1 , b 2 , b 3 , b 4 , b 5 ]. For each solution α i , we use Gröbner basis to reduce the system into two linear equations.

Table 1

 1

	: Compact 3-fold Deligne-Mostow Lattices

= (R 1 J)

2k = R 1 JR 1 J 2 R 1 JR p-1 1 J 2 R p-1 1 JR p-1 1 J 2 = Id ,(2)One can go back to the previous presentation writing R 2 = JR 1 J -1 and P = R 1 R 2 .

α 1 = (0, 1), α 2 = (ω -1, 1),

α 5 = (0, -ω),

It is interesting to note that only the first six representations can be lifted to GL(3, C) representations (see [START_REF] Falbel | Code Representations of Deligne-Mostow lattices into P GL(3, C)[END_REF]).

Proof. The projective group acts transitively on ordered sets of three distinct points and a line not containing them in CP 2 . One observes that the fixed points of J are [1,1,[ω,ω,1],[ω,ω, 1] and the line fixed by R 1 (generated by [1, 0, 0] and [1, 1, 1]) does not meet the fixed points.

Then the matrix of ρ(J) may be fixed to be the one in the statement of the lemma and the matrix of ρ(R 1 ) will be of the form

By a computation, ρ(R 1 ) 3 = Id if and only if r 3 = r 2 + x with x 2 + x + 1 = 0. The proof now is a computation using Gröbner basis package in SAGE. In fact, the first six solutions


Qi Xue

Eric Larose
email: eric.larose@univ-grenoble-alpes.fr

Ludovic Moreau

Romain Thery

Odile Abraham

Jean-Marie Henault

Keywords: Coda wave interferometry, diffuse ultrasound, cracks in concrete, velocity change, stretching and correlation

To evaluate the stress level and damage of a reinforced concrete containment wall (similar to those used in nuclear power plants) and its reaction to pressure variations, we implemented successive ultrasonic experiments on the exterior surface of the containment wall in the gusset area for three consecutive years (2017, 2018 and 2019). During each experiment, the pressure inside the containment wall increased gradually from 0 MPa to 0.43 MPa and then decreased back to 0 Mpa. From the analysis of the ultrasonic coda waves obtained in the multiple scattering regime (80-220 kHz), we performed Coda Wave Interferometry to calculate the apparent velocity changes in the structure (denoted by dV /V a ) and Coda Wave Decorrelation (DC) measurements to produce 3D cartographies of stress and crack distribution. From three source-receiver pairs, located at the top, middle and bottom of the experimental region, we observe that coda waves dilate, shrink and remain almost unchanged, respectively. This corresponds to the decreasing, increasing and invariant pressure inside the concrete. The comparison of three years' results demonstrates that the variation of dV /V a and DC under the same pressure test increases through the years, which indicates the progressive deterioration and aging of the concrete. From a large collection of source-receiver pairs at different times, the spatial-temporal variations of dV /V a and DC are then used to produce a map of the structural velocity and scattering changes, respectively. We observe a decreasing velocity on the top part and an increasing in the middle one, which is in line with the dV /V a analysis. The reconstructed scattering changes (or structural changes) highlight the active region during the inflationdeflation procedure, corresponding to the opening and closing (and sometimes the development) of cracks. The larger magnitude in 2019 than in 2017 indicates the increasing damage in the concrete.

Introduction

Concrete containments are generally used to protect the nuclear reactor of a nuclear power plant in case of any leakage of the nuclear fuel or waste, or from possible external aggression. The humidity, temperature and/or pressure changes induce the deformation of the concrete shell and, together with additional chemical processes, progressively alter the mechanical integrity of the structure. Small cracks (millimeter scale) are generated and progressively develop during the lifetime of the structure. The assessment of the state of damage and the characterization of cracks are vital issues for the security of the nuclear power plant in case of emergency, and for the estimation of the residual life time. In France, a leak test of each concrete containment is enforced by the law every ten years 1 , during which the pressure inside the containment is increased from 0 MPa to about 0.43 MPa and then decreased back to 0 MPa.

In order to study the aging of the concrete wall and follow the evolution of the leak rate owning to drying creep and loss of pre-stress in a realistic environment, Électricité de France (EDF) launched a research program on a 1/3 scale mock-up, called VeRCoRs [START_REF] Masson | Objectives and design of the new experimental program vercors based on a 1/3 scaled PWR containment building[END_REF][START_REF] Mathieu | Temperature and humidity-driven ageing of the vercors mock-up[END_REF][START_REF] Henault | How to characterize the airtightness of containment structure; overview of monitoring techniques tested on VeRCoRs mock up[END_REF] . This mockup is a double walled concrete containment building, as a 1/3 scale specimen representative for 24 (about 40%) of the French reactors. The present work explores the applicability and performance of a method based on diffuse ultrasound analysis to image and quantify stress changes and cracks within the structure at VeRCoRs during a standard internal pressure test.

To this end, we take advantage of the high sensitivity of multiply scattered waves at the ultrasonic scale, for frequencies of few hundreds of kHz. The high heterogeneity of the concrete results in the loss of ballistic and coherent properties of the waveforms after a propagation over a few wavelengths. The waveforms of the late-arriving ultrasonic signals, also known as coda waves, EXPERIMENTAL SETUP are known to be highly reproducible for a fixed source and receiver, while being very sensitive to small perturbation of the medium. At the seismic scale, the sensitivity of the coda wave has been extensively exploited to detect small changes in the Earth's crust, e.g., fault zones [START_REF] Wegler | Fault zone monitoring with passive image interferometry[END_REF][START_REF] Brenguier | Postseismic relaxation along the san andreas fault at parkfield from continuous seismological observations[END_REF] or volcanic areas [START_REF] Grêt | Monitoring rapid temporal change in a volcano with coda wave interferometry[END_REF][START_REF] Sens-Schönfelder | Passive image interferometry and seasonal variations of seismic velocities at merapi volcano, indonesia[END_REF][START_REF] Brenguier | Towards forecasting volcanic eruptions using seismic noise[END_REF] .

Coda wave interferometry (CWI), as introduced in [START_REF] Poupinet | Monitoring velocity variations in the crust using earthquake doublets: An application to the calaveras fault, california[END_REF] and further developed in [START_REF] Snieder | Coda wave interferometry for estimating nonlinear behavior in seismic velocity[END_REF][START_REF] Snieder | The theory of coda wave interferometry[END_REF] , is a technique to detect slight velocity perturbations in a highly heterogeneous medium, by comparing coda waves recorded before and after a perturbation. A quantitative way to compare two waveforms is to compute their correlation coefficient (CC). One way of applying CWI consists in stretching or shrinking the waveform to match two coda signals, i.e., to maximize the CC in a given time window, with a dilation coefficient. This coefficient, associated to the local velocity change [START_REF] Lobkis | Coda-wave interferometry in finite solids: Recovery of P-to-S conversion rates in an elastodynamic billiard[END_REF] , is generally referred to as the apparent relative velocity change dV /V a . Alternatively, it can be useful to calculate the DC instead of the CC, because the DC can be linked to small structural changes [START_REF] Planès | Imaging multiple local changes in heterogeneous media with diffuse waves[END_REF] . The dV /V a and DC variables are now widely used to detect ultrasonic velocity changes in a concrete structure induced by thermal or stress variations [START_REF] Larose | Observation of multiple scattering of khz vibrations in a concrete structure and application to monitoring weak changes[END_REF][START_REF] Larose | Monitoring stress related velocity variation in concrete with a 2× 10-5 relative resolution using diffuse ultrasound[END_REF][START_REF] Schurr | Damage detection in concrete using coda wave interferometry[END_REF] , and/or to detect micrometer scale cracks [START_REF] Hilloulin | Small crack detection in cementitious materials using nonlinear coda wave modulation[END_REF] . The reader can refer to [START_REF] Planès | A review of ultrasonic coda wave interferometry in concrete[END_REF] for a review.

The relation between the dV /V a and the local velocity change is proposed in [START_REF] Pacheco | Time-lapse travel time change of multiply scattered acoustic waves[END_REF] by comparing wave propagation with random walk. The relation between DC and the local scattering change (for example, induced by cracks) follows a similar formula proposed later [START_REF] Larose | Locating a small change in a multiple scattering environment[END_REF] . The reader may refer to [START_REF] Planès | Decorrelation and phase-shift of coda waves induced by local changes: multiple scattering approach and numerical validation[END_REF] for a detailed explanation and proof. These two formulas have the same form: an integral equation with the same sensitivity kernel. The sensitivity kernel is a function of the source, receiver, space and time, that statistically estimates the density of time spent by the diffuse wave at a given region of the medium. It can be estimated either from i) the explicit solution of diffusive or radiativetransfer approximation of ultrasonic waves for simple geometries (e.g., infinite plate or cuboid) [START_REF] Planès | Decorrelation and phase-shift of coda waves induced by local changes: multiple scattering approach and numerical validation[END_REF] , ii) numerical methods (e.g., finite difference or finite element) for any shape [START_REF] Xue | Locating structural changes in a multiple scattering domain with an irregular shape[END_REF] or iii) directly solving the wave equation [START_REF] Kanu | Numerical computation of the sensitivity kernel for monitoring weak changes with multiply scattered acoustic waves[END_REF] . By combining information from several sources and receivers, we are able to obtain thousands of dV /V a and DC values between different source-receiver pairs in different times, which can be used to image the velocity and scattering changes through solving the integral equation. Please refer to [START_REF] Planès | Imaging multiple local changes in heterogeneous media with diffuse waves[END_REF] for a numerical simulation, [START_REF] Zhang | Diffuse ultrasound monitoring of stress and damage development on a 15-ton concrete beam[END_REF] and [START_REF] Zhan | Three-dimensional images generated from diffuse ultrasound wave: detections of multiple cracks in concrete structures[END_REF] for laboratory experiments and [START_REF] Zhang | Three-dimensional insitu imaging of cracks in concrete using diffuse ultrasound[END_REF] for a field experiment.

We implemented similar diffuse ultrasonic experiments on the outer surface of the gusset area of VeRCoRs (see Fig. 1(a)) during the internal pressure test over three consecutive years, in order to detect cracks and the effect of damage on stress changes during each test. From the analysis of the dV /V a and DC for three different sourcereceiver pairs between the initial state and other states during the pressure test, we aim at observing that different parts of the wall are expected to have different responses to the same inflation-deflation procedure. We also aim at quantifying the evolution of dV /V a and DC versus aging and damage over the years. An imaging inversion procedure is then applied to obtain 3D cartographies of relative velocity changes and density of probability of cracks.

The comparison of different years' results may quantify the deterioration of the concrete: the sensitivity of dV /V a to internal stress is expected to increase with time (thus with damage) as observed elsewhere [START_REF] Planès | A review of ultrasonic coda wave interferometry in concrete[END_REF] and the decorrelation is expected to show some non-reversible (thus cumulative) behavior.

Experimental setup

VeRCoRs is a 1/3 scale experimental mock-up of a nuclear reactor containment (see Fig. 1(a)) built by EDF near Paris. Construction of the mock-up began in 2013 and was completed in March 2016. The construction procedure is layer by layer. The horizontal dashed lines in Fig. 1(b) and(c) indicate positions of the joint in our experimental region. After the construction, two auxiliary air-conditioning units started to impose relative realistic humidity and temperature inside and outside the containment to simulate regular operations of a nuclear reactor. The 1/3 scale was chosen to accelerate aging, ten times faster than in a realistic environment hopefully [START_REF] Charpin | Ageing and air leakage assessment of a nuclear reactor containment mockup: VERCORS 2nd benchmark[END_REF] .

To monitor the properties of the concrete, the pressure inside the structure increases gradually from 0 MPa to 0.43 MPa, which is followed by a static period, during which the pressure decreases a little bit due to small leakage from cracks, and then the whole structure is deflated to its initial state (see Fig. 2(a)). This test was repeated several times, twice in 2017, and once in 2018 and 2019, which corresponds to the standard leak test for real nuclear power plant containment every ten years. We foresee a large change of the concrete in the first test in 2017, because it was the first pressure test and a lot of micro and macro-cracks were created. Therefore, in this paper we only present the second test in 2017, to take into account the so-called Kaiser effect that has been observed and commented previously in concrete [START_REF] Zhang | Study of stressinduced velocity variation in concrete under direct tensile force and monitoring of the damage level by using thermallycompensated coda wave interferometry[END_REF] . Modifications observed in the following are thus not related to the fact that the pressure level exceeds the first one. Our experiment was implemented on the outer surface of the gusset region (a surface of about 2m×2m). Fig. 1(a) provides a sketch of the structure and the experimental region.

Several transducers are glued directly to the concrete to generate and receive ultrasonic signals. During the experiment, a chirp signal with frequency ranging from 80 to 220 kHz is generated, amplified and sent into the concrete from each source one after another, and then recorded by all the receivers in order to construct all the source-receiver impulse responses, through deconvolution of recorded waveforms with the chirp source signal. The whole acquisition process takes about 3 minutes, within which the change of concrete can be neglected. The same process is repeated every hour to obtain measurements at different loading states. The impulse response lasts 4 ms and consists 10000 discrete points. Because of the high diffusion and dissipation of ultrasound in the concrete, we observe that receivers more than one meter away from the source can only record background noise. We check all the signals and choose the interval 1.8 ms -2.2 ms to estimate the average amplitude of the background noise level.

Apparent velocity change and decorrelation analysis

Let us consider ultrasounds generated and recorded by the same source-receiver pair in two different states: φ 1 (t ) and φ 2 (t ). The apparent velocity change dV /V a on the time interval [t 1 , t 2 ] is defined by

dV /V a = arg max α t 2 t 1 φ 1 ((1 + α)t )φ 2 (t )d t t 2 t 1 φ 2 1 ((1 + α)t )d t t 2 t 1 φ 2 2 (t )d t ,
i.e., the amount of stretching that maximizes CC between the signals, and the DC is one minus CC, i.e.,

DC = 1 -max α t 2 t 1 φ 1 ((1 + α)t )φ 2 (t )d t t 2 t 1 φ 2 1 ((1 + α)t )d t t 2 t 1 φ 2 2 (t )d t .
As an example, if the velocity decreases in the second state and if we focus on a peak at t 0 in the wave field φ 1 , then the same peak should arrive after t 0 in φ 2 . Therefore, to match the peak in these two waveforms, α should be negative. That is to say, a dilation of the time axis (a negative dV /V a ) indicates a decreasing velocity.

To quantitatively study the response of concrete to pressure changes in different regions and to compare the state of the concrete over different years, we compute dV /V a and DC for three source-receiver pairs between the original state and other states during the inflationdeflation procedure: the top pair (S2-R3 in Fig. 1), the middle pair (S6-R6) and the bottom pair (S10-R10). The time window is chosen to be [0.16, 0.48] ms in this section, which is much larger than those chosen in the following section. We intentionally choose a large interval and the early part of the waveform to stabilize the computation because late-arrival waves vary too much for ultrasound to handle. The apparent velocity change is obtained by interpolating φ 1 on a fine mesh of α and finding the one maximizing the inner product of φ 1 ((1 + α)t ) and φ 2 (t ). Fig. 2(c) shows the value of dV /V a for source-receiver pairs located at the top (in blue), middle (in green) and bottom (in red) zones of the structure, during inflation cycles from 2017 to 2019. When the the structure is inflated, stress builds up in concrete, as shown in Fig. B1 and B2 in appendix B, where we plot tangential stress distribution calculated with linear elastic simulations. The stress distribution shows traction in the upper part of the structure. This is consistent with what is observed in Fig. 2(c), which exhibits a decrease of velocity for the source-receiver pair located at the top region. However, the apparent velocity change remains small in the middle and marginal at the bottom, where the simulation shows a constant stress or compression.

We have also checked the uncertainties of the apparent velocity changes. Appendix A describes how the uncertainties are calculated, following the formula introduced by [START_REF] Weaver | On the precision of noise correlation interferometry[END_REF] . These uncertainties are shown as colored areas in Fig. A1, for the source-receiver pair located at the upper region of the structure. One may see that they remain small despite strong apparent velocity changes.

From 2017 to 2019 one notices an increase in the slope of dV /V a , which suggests that the sensitivity of concrete to the same pressure change becomes larger. That is to say, the deterioration of the concrete is more pronounced [START_REF] Planès | A review of ultrasonic coda wave interferometry in concrete[END_REF] .Another interesting phenomenon is that in 2017 the line during inflation almost overlaps with the line during deflation which means that expansion and contraction of the concrete are fully reversible, because the concrete properties have not evolved significantly between the first ever pressure test in 2017 and the one performed soon after presented here, which is no more the case in 2019. This might be a good criterion to quantify the damaging impact of the pressure tests, and to test the quality of the concrete. Similar conclusions can be drawn for the middle and bottom pairs, though a much lower resolution: the middle region experiences a contraction and the pressure in the bottom region keeps almost unchanged. We leave the precision study of the dV /V a due to changes in wave speed in the Appendix.

In 2017, the concrete retrieves its initial state after the whole inflation-deflation procedure, as indicated by DC (solid lines) in Fig. 2(b), which is not the case in 2018 and 2019. We conclude from lines with the same color that the change of DC becomes more irreversible as time goes by, which is caused by the cumulative deterioration of the concrete (including micro-cracks and cracks). The large value of DC between the initial and final states implies that the damage induced by pressure changes is permanent: some cracks open and close reversibly while others develop further and the crack surface is altered such that they do not close exactly back to their initial state. What's also interesting is the bottom pair: we do not observe significant pressure changes in the bottom region from dV /V a , but they do suffer dramatic scattering (or structural) changes (see Fig. 2(b)).

Starting from the 30th hour, there is an increasing of DC for S10-R10, which is due to the following action on the whole structure. In order to estimate the amount of leakage, visually detect the cracks and perform other experiments, during the pressure-steady stage (from around the 30th to 38th hour), soap water is poured from the top of the structure. From our data we conclude that the impact of the water is more significant for the bottom region of our experimental domain.

3D cartography of velocity and structural changes

The relation between the relative velocity change dV V (x) and dV /V a is given by

dV /V a t = dV V (x)K (s, r, x, t )d x,
and the formula connecting the scattering change and DC has a similar form

DC = V σ(x) 2 K (s, r, x, t )d x,
where V is the average velocity of the ultrasound and σ is the local scattering cross section density. The reader may refer to [START_REF] Planès | Decorrelation and phase-shift of coda waves induced by local changes: multiple scattering approach and numerical validation[END_REF] for more details. These two formula share a common sensitivity kernel K (s, r, x, t ), which is computed from the diffusive approximation of the coda wave. The sensitivity kernel reflects the probability that a wave emitted at s at time 0 and collected at r at time t has passed through the position x. Here t is in practice the center of the time window where we evaluate dV /V a and DC experimentally.

In order to solve the integral equations, the medium is divided into equally sized cuboids. The value of the unknown variable inside one cuboid, dV V or σ, is assumed to be constant and approximated by the value at the center of each cuboid. In our experiment, the size of the cuboids along x, y and z axes is 10 cm, 8 cm and 10 cm, respectively. We choose 13 consecutive time intervals starting from 0.2 ms. Each of them has a length of 0.08 ms. We remove intervals whose average waveform amplitude is no larger than 2.5 times the noise amplitude. The calculating of the sensitivity kernel and integral equation is implemented in 3D. Refer to [START_REF] Xue | Locating structural changes in a multiple scattering domain with an irregular shape[END_REF] for a detailed discussion about computing the sensitivity kernel and solving the integral equations. We compute the velocity change between every two consecutive states and take the average value from all consecutive pairs in the second inflation stage (from 0.21 MPa to 0.41 MPa) and the first deflation stage (from 0.40 Mpa to 0.21 Mpa) to generate Fig. 3. Each step has, in average, a change of + or -0.02 MPa. The decreasing (increasing, respectively) velocity is observed in the inflation stage in the top region (in the lower middle region, respectively), and the opposite phenomenon occurs during the deflation stage. The magnitude in 2019 is much larger than in 2017, which indicates again the cumulative damage of the concrete over years. We only FUNDING present velocity changes of the top layer (at a depth of 5 cm from the surface). We obtain similar results for other depth, except a decrease of magnitude and resolution for deeper layers.

Fig. 4 shows the average structural changes, or more precisely the scattering cross section density σ(x, y) of the changes, between two consecutive states during the second inflation and first deflation stages. The units (m 2 /m 3 ) can be understood as a density of micro-cracks per volume unit. We observe an increase of scattering change from 2017 to 2019. Our result shows that the most active region is along the top construction joint. We also observe the developing of the crack near dotted lines from 2017 to 2019.

Conclusion and discussion

In this work, we use diffuse ultrasound to study the pressure and structural changes in the gusset area of a 1/3 replica of a nuclear power plant containment wall. Changes are induced by the increase and decrease of the inner pressure. We list the main results below:

1. We are able to image the ultrasonic relative velocity change with a magnitude as small as 10 -4 thanks to the high sensitivity of diffuse ultrasound. 2. The stretching of diffuse ultrasound and reconstructed velocity change indicate that the concrete near the base is in compression and in traction in the upper region in the inflation process, in agreement with mechanical models. 3. The increase of the slope of dV /V a versus internal pressure change from 2017 to 2019, in other words the sensitivity of dV /V a to internal pressure, suggests that the concrete is progressively damaged and weakened. That is to say, the concrete deteriorates over time. 4. Although we do not detect significant pressure change in the base (bottom) region, it could actually also suffer severe cumulative changes. The large DC value between the initial and final states suggests that the concrete does not restore to its initial state during the test, i.e., the damage is permanent and irreversible, at least on the time scale of the test. 5. The reconstructed structural changes (scattering cross section) shows active regions during the inflation-deflation procedure that could correspond to cracking inside the concrete.

  Fig. 1(b) shows the positions of sources and receivers in 2017 and 2018, and Fig. 1(c) shows positions in 2019. Since the base of the concrete structure (base slab) is extremely thick, the signals recorded by the bottom source-receiver pair (S12-R12) remain almost unchanged during the experiment in 2017 and 2018. We decided to remove them and to add several new transducers to improve the resolution of the upper part in 2019. In addition, we did not receive reasonable signals from R1 in 2018 which may be caused by the failure or detachment of the transducer.

Figure 1 .

 1 Figure 1. Configuration of the experiment. (a) Sketch of the concrete structure and the experimental region (green square). We focus on the lower corner (the gusset) of the exterior surface of VeRCoRs, a 1/3 replica of a nuclear containment wall. (b) Source and receiver positions in 2017 and 2018. (c) Source and receiver positions in 2019 with transducer positions in green on the top of the experimental region (S12 to S15 and R12 to R15). All transducers are glued on the outer surface (z = 0) of the structure. The horizontal solid lines represent the change of thickness of the wall. The thickness of the upper part is 40 cm and the middle part (gusset) is 60 cm. The bottom layer is the base slab. The dashed horizontal lines are positions of the construction joint. Dotted lines are positions of cracks on the surface of the wall observed in 2018.

Figure 2 .

 2 Figure 2. (a) pressure variation during the inflation-deflation procedure. We took the measurement every an hour, during which the pressure has a change of + or -0.02 MPa except in steady states. (b) DC between the original state and other states; (c) dV /V a variation with respect to pressure. Blue, green and red lines in (b) and (c) correspond to the top, middle and bottom source-receiver pairs (S2-R3, S6-R6 and S10-R10 in Fig. 1(b)), respectively. Solid, dashed and dotted lines are results from 2017, 2018 and 2019, respectively.

Figure 3 .Figure 4 .

 34 Figure 3. (a)-(c) average relative velocity change between consecutive measurements (one hour apart) at a depth of 5 cm during the second inflation stage in 2017, 2018 and 2019, respectively; (d)-(f) results during the first deflation stage. The white crosses and squares indicate positions of sources and receivers, respectively. Black lines are the same as those in Fig. 1.

Figure B1 .Figure B2 .

 B1B2 Figure B1. Numerical simulation of stress around the gusset (vertical cut). We represent here the difference in tangential stress from the initial state (no internal pressure) to the plateau at 0.4 MPa. The colored areas indicate traction (form 0 to 0.3 MPa approximately). Black color indicates compression or neutral areas. The yellow vertical dashed line is the position where we plot Fig. B2.

Funding

This work was partially funded by the ANR ENDE (PIA 11-RSNR-0009 ) project, and by the IDEX Université Grenoble Alpes innovation Grant (CND-US).

Appendix A

In [START_REF] Weaver | On the precision of noise correlation interferometry[END_REF] , authors proposed to evaluate the uncertainties of dV /V a due to the change in wave speed through the computation of its root mean square rms(dV /V a ) =

For our experiment, the parameters are as follows:

Appendix B

Here we evaluate numerically the state of stress inside the concrete around the gusset. We perform elastic linear numerical simulations using the ASTER code [START_REF]Finite element code_aster, structures and thermomechanics analysis for studies and research[END_REF] , without taking into account the possible role of cracks. For clarity, we only represent the difference of stress between the initial state (additional internal pressure 0 MPa) and the final state (internal pressure 0.43 MPa). Fig. B1 and B2 represent a 2D vertical cut and a vertical profile of this tangential stress change. Positive values correspond to traction, where cracks are expected to develop and open, and where dV /V a is expected to be negative. Negative values of stress correspond to compression areas, where dV /V a are expected to be positive. Experimental results of dV /V a are perfectly in agreement with this mechanical simulations, demonstrating the ability of dV /V a obtained from ultrasonic coda waves to cartography the state of stress inside the structure.