Paolo Bellingeri
email: paolo.bellingeri@unicaen.fr

Arthur Soulié
email: arthur.soulie@glasgow.ac.uk

A note on representations of welded braid groups

Keywords: 2010 Mathematics Subject Classification: 20C07, 20F36, 57M07 Welded braid groups, Burau representation, Long-Moody construction

Introduction

The theory of linear representations of the braid group B n is a very large topic. One of the most famous representations is the Burau representation [START_REF] Burau | Über zopfgruppen und gleichsinnig verdrillte verkettungen[END_REF], which is non faithful for n ≥ 5. For a long period, it was an open problem whether B n was linear until the independent works of Bigelow [START_REF] Stephen | Braid groups are linear[END_REF], Krammer [START_REF] Krammer | Braid groups are linear[END_REF] and Lawrence [START_REF] Lawrence | Homological representations of the Hecke algebra[END_REF] showing a faithful representation. Since the braid group B n is an ubiquitous object in mathematics it is natural to ask whether other generalizations are linear too, but except some cases (for instance Artin groups of type B and D, braid groups of the sphere and of the projective plane) this question remains widely open.

In this work note we focus on welded braid groups, which, as braid groups, admit several different definitions, for instance in terms of configuration spaces of (euclidean) circles, as automorphisms of free groups, or as tubes in R 4 .

The representation theory for these groups is just at the beginning: Burau representation extends, in terms of Fox derivatives and Magnus expansion, to welded braid groups [START_REF] Valeriy | The structure of the group of conjugating automorphisms and the linear representation of the braid groups of some manifolds[END_REF] and few other results are known on representations arising from braided vector spaces [START_REF] Kádár | Local representations of the loop braid group[END_REF] and on extensions of representations of B n to particular subgroups of welded braid groups [START_REF] Bullivant | Representations of the necklace braid group: Topological and combinatorial approaches[END_REF].

The main idea of this work is to extend Long-Moody procedure [START_REF] Birman | Finite-dimensional representations of Artin's braid group[END_REF][START_REF] Long | Constructing representations of braid groups[END_REF][START_REF] Long | On the linear representation of braid groups[END_REF][START_REF] Long | On the linear representation of braid groups[END_REF][START_REF] Soulié | The Long-Moody construction and polynomial functors[END_REF][START_REF] Soulié | Generalized Long-Moody functors[END_REF] to wB n : in §1 we recall briefly the interpretation of welded braid groups in terms of fundamental groups of configuration spaces of circles and as automorphisms of free groups through Artin homomophism; this latter interpretation will be extended to other possible representations, extending Wada representations of the classical braid group B n . In §1. [START_REF] Valeriy | The structure of the group of conjugating automorphisms and the linear representation of the braid groups of some manifolds[END_REF] we recall and compare different Burau representations for wB n , the reducible one (Proposition 1.1), the reduced one (Proposition 1.2), and the dual version (Proposition 1.4). Then we introduce Tong-Yang-Ma representations of wB n (Proposition 1.5) and extending the heuristic approach proposed in [START_REF] Tong | A new class of representations of braid groups[END_REF] we show that Burau and Tong-Yang-Ma representations are the only representations allowing a certain diagram to commute (Proposition 1.6). The main results are given in §2, where we adapt to welded braid groups the Long-Moody procedure for obtaining iterated linear representations. At the first step we obtain the Burau representation (Theorem 2.4) as in the case of B n . Surprisingly at the second iteration we do not obtain any new information, since we get the tensor product of two Burau representations (Theorem 2.6), while in the case of B n we recover this way Lawrence-Krammer representation (see [START_REF] Soulié | The Long-Moody construction and polynomial functors[END_REF]Section 2.3.1]). This result can be also compared with the fact that the "trivial" extension of Bigelow representation to wB n (associating to "braid" generators corresponding Bigelow matrices and to "permutation" generators the corresponding permutations matrices) is not well defined (see [START_REF] Valeriy | The structure of the group of conjugating automorphisms and the linear representation of the braid groups of some manifolds[END_REF][START_REF] Kádár | Local representations of the loop braid group[END_REF]). Finally, once more contrarily to the case of B n , we show in Theorem 2.7 that it is not possible to recover the Tong-Yang-Ma representation for wB n by a Long-Moody construction and we conclude with some further possible directions on the study of linear representations for wB n . In a further paper, we intend to study the Long-Moody iteration on the Tong-Yang-Ma representation, but here we focus on the two above interesting facts. was also partially supported by the ANR Project ChroK (ANR-16-CE40-0003). The authors are deeply grateful to Emmanuel Wagner for his suggestions and ideas all along the writing of this paper. They also thank Markus Szymik for helpful discussions on that topic.

Notations and conventions.

Throughout this work, for a group defined by a (finite) presentation, we take the convention to read from the right to the left for the group operation.

State of the art of welded braid groups representations 1.Recollections on welded braid groups

We refer the reader to [START_REF] Damiani | A journey through loop braid groups[END_REF] for a complete and unified presentation of the various definitions of welded braid groups, which correspond to (unextended) loop braid groups in this reference.

In the following we will use essentially the interpretation of welded braid groups in terms of automorphisms of free groups and fundamental groups of configuration spaces of circles.

In this work, we focus on a 3-dimensional analogue of B n : it is the fundamental group of all configurations of n unlinked Euclidean circles lying on planes that are parallel to a fixed one (called untwisted rings in [START_REF] Brendle | Configuration spaces of rings and wickets[END_REF]). According to [START_REF] Brendle | Configuration spaces of rings and wickets[END_REF] we will denote by UR n the space of configurations of n unlinked Euclidean circles being all parallel to a fixed plane and by U R n its fundamental group (called group of rings in [START_REF] Brendle | Configuration spaces of rings and wickets[END_REF]). The group U R n is generated by 2 types of moves (see Figure 1.1).

i i + 1 i i + 1 Figure 1.1 The moves τ i and σ i .
The move τ i is the path permuting the i-th and the i + 1-th circles by passing over (or around) while σ i permutes them by passing the i-th circle through the i + 1-th (let us remark that our notation is different from the one of [START_REF] Brendle | Configuration spaces of rings and wickets[END_REF], where τ i was denoted by σ i and σ i by ρ i ; here we change the notation because σ i 's generate a subgroup isomorphic to B n ).

The fundamental group U R n is here denoted by wB n , and it is called welded braid group. Note that the convention of reading from the right to the left for the group operation is coherent with the interpretation of wB n as the fundamental group of the space of configurations of n unlinked Euclidean circles being all parallel to a fixed plane: this convention corresponds to the composition of morphisms. We abuse the notation throughout this work, identifying λ•λ ′ = λλ ′ for all elements λ and λ ′ of wB n .

In [START_REF] Brendle | Configuration spaces of rings and wickets[END_REF] was proven that the welded braid group on n generators wB n admits a presentation with generators {σ i , τ i | i ∈ {1, . . . , n -1}} together with relations:

                             σ i σ k = σ k σ i if | i -k | 2, σ i σ i+1 σ i = σ i+1 σ i σ i+1 if i ∈ {1, . . . , n -2}, τ i τ k = τ k τ i if | i -k | 2, τ i τ i+1 τ i = τ i+1 τ i τ i+1 if i ∈ {1, . . . , n -2}, τ 2 i = 1 if i ∈ {1, . . . , n -1}, σ i τ k = τ k σ i if | i -k | 2, τ i σ i+1 σ i = σ i+1 σ i τ i+1 if i ∈ {1, . . . , n -2}, σ i τ i+1 τ i = τ i+1 τ i σ i+1 if i ∈ {1, . . . , n -2}.
(1.1)

The map UR n → UR n+1 which adds a circle on the left induces an (injective) homomorphism from wB n to wB n+1 .

Let PUR n be the space of configurations of n ordered unlinked Euclidean circles being all parallel to a fixed plane; the fundamental group of PUR n is usually denoted by wP n and called welded pure braid group.

The map PUR n+1 → PUR n which forgets the first circle is a fibration and the long exact sequence in homotopy provides the following (splitting) sequence:

1 / / D n / / wP n+1 / / wP n / / 1,
where D n consists of configurations with n circles in a fixed position and the first circle varying [START_REF] Brendle | Configuration spaces of rings and wickets[END_REF].

Let now UR 1,n be the orbit space PUR n+1 /S n where the symmetric group S n acts by permutation on last n circles. We will denote by wB 1,n its fundamental group. As in previous case we have a splitting sequence:

1 / / D n / / wB 1,n / / wB n / / 1.

Artin homomorphism

In the following we will use another interpretation of wB n , this time in terms of automorphisms of free groups. Let F n be the free group on n generators x 1 , . . . , x n . We call Artin homomorphism the map a n : wB n → Aut (F n ) defined as follows:

σ i -→      x i -→ x i+1 x i+1 -→ x -1 i+1 x i x i+1 x j -→ x j j / ∈ {i, i + 1} τ i -→      x i -→ x i+1 x i+1 -→ x i x j -→ x j j / ∈ {i, i + 1}
This map is well defined: the relations involving only generators {σ i | i ∈ {1, . . . , n -1}} are verified since it is the usual Artin representation of B n in Aut (F n ). The same remark holds for relations involving only generators {τ i | i ∈ {1, . . . , n -1}} (permutation automorphisms), therefore the only relations that we have to verify are relations involving generators of both type σ i and τ i . We check here relations τ i σ i+1 σ i = σ i+1 σ i τ i+1 and τ i+1 τ i σ i+1 = σ i τ i+1 τ i ; commutation relations are evidently verified (see also [START_REF] Fenn | The braid-permutation group[END_REF]).

a n (σ i+1 σ i τ i+1 ) =          x i -→ x i+2 x i+1 -→ x -1 i+2 x i+1 x i+2 x i+2 -→ x -1 i+1 x i x i+1 x j -→ x j j / ∈ {i, i + 1, i + 2} = a n (τ i σ i+1 σ i ) a n (σ i τ i+1 τ i ) =          x i -→ x i+2 x i+1 -→ x i+1 x i+2 -→ x -1 i+1 x i x i+1 x j -→ x j j / ∈ {i, i + 1, i + 2} = a n (τ i+1 τ i σ i+1 )
Geometrically, we are associating to any generator of wB n the corresponding action on the fundamental group of the ball B 3 less n trivial circles, which is a free group on n generators. We refer to [START_REF] Damiani | A journey through loop braid groups[END_REF] for a rigorous proof of this construction and of the fact that Artin homomorphism is injective. Moreover, the image of the group wB n in Aut (F n ) is the subgroup of those automorphisms of F n that send each generator of F n to a conjugate of some generator [START_REF] Fenn | The braid-permutation group[END_REF].

Notice also that the homomorphism from wB n to wB n+1 previously defined ("add a circle on the left") becomes here the restriction of the map id 1 * -: Aut (F n ) ֒→ Aut (F n+1 ). In other words,

id 1 * σ i = σ i+1 and id 1 * τ i = τ i+1 .
The group D n previously defined is isomorphic to the group generated by automorphisms

{ǫ i,1 , ǫ 1,i | i ∈ {2, . . . , n + 1}} [1],
where

ǫ 1,i -→ x i -→ x -1 1 x i x 1 x j -→ x j j = i ǫ i,1 -→ x 1 -→ x -1 i x 1 x i x j -→ x j j = 1
Note that {ǫ i,1 } generate a free group of order n and {ǫ 1,i } a free abelian group of rank n [START_REF] Valeriy | The structure of a group of conjugating automorphisms[END_REF], but D n is not finitely presented when n ≥ 3 [START_REF] Pettet | Finiteness properties for a subgroup of the pure symmetric automorphism group[END_REF].

Alternative to Artin homomorphism. Wada [27] found several local representations of B n in Aut (F n ) of the following form: any generator (and therefore its inverse) of B n acts trivially on generators of F n except a pair of generators:

σ i • x i = u(x i , x i+1 ) , σ i • x i+1 = v(x i , x i+1 ) , σ i • x j = x j j = i, i + 1 ,
where u and v are now words in the generators x i , x i+1 , with x i , x i+1 ≃ F 2 . Wada found seven families of representations of local type (see Section 4 of [START_REF] Valeriy | On representations of braids as automorphisms of free groups and corresponding linear representations[END_REF] for a short survey on these representations), up to the dual equivalence (corresponding to the involution of F n given by simultaneously replacing all x i with x -1 i ) and inverse equivalence (derived from the involution of B n defined by sending

σ i to σ -1 i ): 1 • Type 1, ψ 1 : u(x i , x i+1 ) = x i and v(x i , x i+1 ) = x i+1 ; • Type 2, ψ 2 : u(x i , x i+1 ) = x -1 i+1 and v(x i , x i+1 ) = x i ; • Type 3, ψ 3 : u(x i , x i+1 ) = x -1 i+1 and v(x i , x i+1 ) = x -1 i ; • Type 4, ψ 4,h : u(x i , x i+1 ) = x i+1 and v(x i , x i+1 ) = x -h i+1 x i x h i+1 ; • Type 5, ψ 5 : u(x i , x i+1 ) = x i+1 and v(x i , x i+1 ) = x i+1 x -1 i x i+1 ; • Type 6, ψ 6 : u(x i , x i+1 ) = x -1 i+1 and v(x i , x i+1 ) = x i+1 x i x i+1 ; • Type 7, ψ 7 : u(x i , x i+1 ) = x i x -1 i+1 x -1 i and v(x i , x i+1 ) = x i x 2 i+1 .
We can try to extend Wada representations to welded braid groups associating to any generator σ i the Wada representation of type k and to any generator τ i the corresponding permutation automorphism. We will say that a Wada representation extends to wB n if the map defined as above on generators of wB n is actually a homomorphism. Type 1 does not extend, while Type 2 and Type 3 extend to wB n but these extensions are not interesting since the image of the group generated by σ i 's in Aut (F n ) is trivial or isomorphic to S n . Between the other four representations it is easy to check (see [START_REF] Valeriy | Groups of virtual and welded links[END_REF]) that only Type 4 and Type 5 representations extend this way to a representation; let denote these two representations respectively by χ 1 and χ 2 . These two representations are not equivalent meaning that there are no automorphisms φ ∈ Aut (F n ) and µ :

wB n → wB n such that φ -1 χ 1 (β) φ = χ 2 (µ(β)),
for any β ∈ wB n (see [START_REF] Valeriy | Groups of virtual and welded links[END_REF]).

Known linear representations for welded braid groups

We present in this section the linear representations of welded braid groups which can be straightforwardly derived from those of braid groups. In particular, we extend heuristic procedure on matrices. For the remainder of §1.3, fix a natural number n 2.

Beforehand we indicate that we study here in the "specific" representations of welded braid groups, not those which factor through symmetric groups. Namely, recall that sending both σ i and τ i in the transposition (i, i + 1) for i ∈ {1, . . . , n -1} we obtain the short exact sequence:

1 / / wP n / / wB n / / S n / / 1.
Therefore, all representations of the symmetric group S n lift to wB n , but we will not consider them since we loose too many informations on welded braid groups.

The Burau representations

The family of Burau representations for braid groups were first introduced in [START_REF] Burau | Über zopfgruppen und gleichsinnig verdrillte verkettungen[END_REF] and has been intensively studied. We refer the reader to [START_REF] Kassel | Braid groups, volume 247 of Graduate Texts in Mathematics[END_REF] for a complete presentation.

These representations can be extended to welded braid groups in the following way.

Proposition 1. [START_REF] Valeriy | The structure of a group of conjugating automorphisms[END_REF] The following assignment defines a representation Bur :

wB n → GL n Z[t ±1 ]
called the Burau representation of the welded braid group wB n :

σ i → Id i-1 ⊕ 0 t 1 1 -t ⊕ Id n-i-1 and τ i → Id i-1 ⊕ 0 1 1 0 ⊕ Id n-i-1
for all natural numbers i ∈ {1, . . . , n -1}.

Proof. The matrices for σ i and τ i define representations of braid groups and symmetric groups respectively. It follows from the relations of wB n of (1.1) that we just have to check the mixed relations between the braid and symmetric generators: this is done by straightforward computations.

Reduced version.

As for the case of braid groups, the Burau representation is reducible and we can define an irreducible version.

Proposition 1.2

The following assignment defines a representation Bur :

wB n → GL n-1 Z[t ±1
] called the reduced Burau representation of the welded braid group wB n :

σ 1 → -t t 0 1 ⊕ Id n-3 and τ 1 → -1 1 0 1 ⊕ Id n-3 ; σ n-1 → Id n-3 ⊕ 1 0 1 -t and τ n-1 → Id n-3 ⊕ 1 0 1 -1 ; σ i → Id i-2 ⊕   1 0 0 t -t 1 0 0 1   ⊕ Id n-i-2 and τ i → Id i-2 ⊕   1 0 0 1 -1 1 0 0 1   ⊕ Id n-i-2
for all natural numbers i ∈ {2, . . . , n -2}. Moreover we have a short exact sequence of wB nrepresentations

0 / / Bur / / Bur / / Z t ±1 / / 0 (1.2)
where wB n acts trivially on the kernel Z t ±1 .

Proof. The matrices for σ i define the reduced Burau representations of braid groups (see [15, Section 3.3]) and those for τ i define the standard representation of symmetric groups. Again the compatibility with respect to the mixed relations of wB n are checked by straightforward computations.

Let r n be the n × n-matrix with coefficients r i,j = 1 if j i and r i,j = 0 otherwise. Then

r n • Bur (σ i ) • r -1 n = Bur (σ i ) C i 0 1
where

C i = 0 • • • 0 δ i,n-1 1
T (where the label T means the transpose matrix) and δ i,n-1 denotes the Kronecker delta: this defines the short exact sequence (1.2).

Remark 1.3 Since Burau representation for B n is not faithful when n ≥ 5, it follows that also its extension is not faithful; indeed, the Burau representation for wB n has non trivial kernel even for n = 2 (see Lemma 6 of [START_REF] Valeriy | Extending representations of braid groups to the automorphism groups of free groups[END_REF]).

Dual versions. Actually, there are two non-equivalent versions of the Burau representation for braid groups: the one which matrices are given in Proposition 1.1, and its dual which matrices are the transpose of the inverse of these matrices. This dual version also lifts to the welded braid group:

Proposition 1.4 Assigning Bur * (σ i ) = Bur T σ -1 i and Bur * (τ i ) = Bur T τ -1 i for all natural numbers i ∈ {1, . . . , n -1} defines a representation Bur * : wB n → GL n Z[t ±1 ] called the dual Burau representation.

It induces a dual reduced Burau representation Bur

* : wB n → GL n-1 Z[t ±1
] which is defined by

Bur * (σ i ) = Bur T σ -1 i and Bur * (τ i ) = Bur T τ -1 i
for all natural numbers i ∈ {1, . . . , n -1}, and defines the short exact sequence of wB n -representations

0 / / Z t ±1 / / Bur * / / Bur * / / 0 (1.3)
where wB n acts trivially on the kernel Z t ±1 .

Proof. Taking the transpose of the inverse of a multiplication of matrices keeps the order of the multiplication. Therefore all the relations of (1.1) are therefore straightforwardly satisfied and proves that Bur * is a representation. The reduced version Bur * is induced by taking the transpose of the inverse of the short exact sequence (1.2): taking the inverse keeps the direction of the arrows, whereas the transpose reverses this direction (since it exchanges lines and columns of matrices) and we obtain (1.3).

The dual Burau representation was already introduced in [26, Section 4].

1.3.2

The Tong-Yang-Ma procedure and associated representations.

In 1996, Tong, Yang and Ma [START_REF] Tong | A new class of representations of braid groups[END_REF] investigated the representations of B n where the i-th generator is sent to a non-singular matrix of the form

Id i-1 ⊕ a b c d ⊕ Id n-i-1 .
In particular, they proved that there exist (up to equivalence and dual) only two non trivial representations of this type: the unreduced Burau representations and a new irreducible representation, called the Tong-Yang-Ma representation. This last family lifts to define two families of linear representations of the welded braid group wB n .

Proposition 1. [START_REF] Valeriy | On representations of braids as automorphisms of free groups and corresponding linear representations[END_REF] The following assignment defines a representation T Y M : wB → GL n Z[t ±1 ] called the Tong-Yang-Ma representation of the welded braid group wB n :

σ i → Id i-1 ⊕ 0 1 t 0 ⊕ Id n-i-1 and τ i → Id i-1 ⊕ 0 1 1 0 ⊕ Id n-i-1
for all natural numbers i ∈ {1, . . . , n -1}. Taking the transpose of the inverse of the above matrices defines the dual Tong-Yang-Ma representation T Y M * :

wB n → GL n Z[t ±1 ] (which is obviously not equivalent to T Y M ).
Proof. As above for the Burau representations, we just have to check the compatibility with respect to the mixed relations (1.1) between the braid and symmetric generators of wB n : this is done by straightforward computations.

In addition, we can carry out the analogous heuristic approach to [START_REF] Tong | A new class of representations of braid groups[END_REF] for the welded braid groups. For all i ∈ {1, . . . , n -1}, we denote by incl n i : wB 2 ∼ = Z × Z/2Z ֒→ wB n the inclusion morphism induced by incl n i (σ 1 ) = σ i and incl n i (τ 1 ) = τ i . Proposition 1.6 Let η n : wB n -→ GL n be a representation. Assume that for all i ∈ {1, . . . , n -1} the following diagram is commutative:

wB n ηn / / GL n Z t ±1 wB 2 η2 / / incl n i O O GL 2 Z t ±1 . idi-1⊕-⊕idn-i-1 O O

Then η n is equivalent or dual to the trivial representation, or to the (unreduced) Burau representation Bur, or to the Tong-Yang-Ma representation T Y M , or else to the specialisation at t = 1 of the Burau representation.

Proof. Restricting along the natural inclusion B n ֒→ wB n , it follows from [25, Part II] that three only possible matrices (up to equivalence and dual) on which the braid generators {σ i } i∈{1,...,n-1} can be sent to are the trivial, Burau and Tong-Yang-Ma matrices. Note that the only non-trivial representation Z/2Z → GL 2 Z t ±1 is the permutation sending the non-trivial element of Z/2Z to the permutation matrix.

There are thus two choices for the symmetric generators {τ i } i∈{1,...,n-1} . First the relation τ i τ i+1 τ i = τ i+1 τ i τ i+1 of (1.1) implies that all these generators are sent to either a non-trivial matrix or the trivial matrix. Then it follows from the relation τ i σ i+1 σ i = σ i+1 σ i τ i+1 of (1.1) that the symmetric generators are sent to the identity matrix if the braid generators are, and to the permutation matrices otherwise.

The Long-Moody construction for welded braid groups

In 1994, Long and Moody [START_REF] Long | Constructing representations of braid groups[END_REF] gave a method to construct a new linear representation of B n from a representation of B n+1 , complexifying in a sense the initial representation. For instance, it reconstructs the unreduced Burau representation from a one dimensional representation. It was studied from a functorial point of view and extended in [START_REF] Soulié | The Long-Moody construction and polynomial functors[END_REF] and then generalised to other families of groups [START_REF] Soulié | Generalized Long-Moody functors[END_REF]. In particular, the underlying framework of this method, called the Long-Moody construction, naturally arises considering representations of welded braid groups: the aim of this section is the study of this construction in this case. We fix a natural number n 3 all along §2.

The theoretical setting of the Long-Moody construction

We detail here the required tool and present the abstract definition of the Long-Moody construction.

Tool. Recall that F n = x 1 , . . . , x n is the free group on n generators. The key ingredient to define the Long-Moody construction for welded braid groups is to find a group morphisms α n : wB n → Aut (F n ) and ξ n : F n → wB n+1 such that:

• the morphism F n * wB n → wB n+1 given by the coproduct of ξ n and id 1 * -factors across the canonical surjection to the semidirect product

F n ⋊ αn B n ;
• the following diagram is commutative

F n / / ξn $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ F n ⋊ αn wB n wB n ? _ o o id1 * - y y t t t t t t t t t t t wB n+1 , (2.1)
where the vertical morphism F n ⋊ αn wB n → wB n+1 is induced by the coproduct of ξ n and

id 1 * -morphism F n * B n → B n+1 .
In other words, we require that for all elements λ ∈ wB n and x ∈ F n the morphism ξ n satisfies the following equality in wB n+1 :

(id 1 * λ) • ξ n (x) = ξ n (α n (λ) (x)) • (id 1 * λ) .
(2.2)

Definition. The Long-Moody construction is defined as follows. We fix an abelian group V . We denote by I Fn the augmentation ideal of the group ring Z [F n ]. Note that the action α n canonically induces an action of wB n on I Fn (that we denote in the same way for convenience).

Let ρ : wB n+1 → Aut Z (V ) be a linear representation. Precomposing by the morphism ξ n , ρ gives the module V a F n -module structure. Then the Long-Moody construction

LM (ρ) : wB n → Aut Z I Fn Fn V
is the map defined by:

LM (ρ) (λ) i Fn v = α n (λ) (i) Fn ρ (id 1 * λ) (v)
for all λ ∈ wB n , i ∈ I Fn and v ∈ V . For sake of completeness, we detail that:

Lemma 2.1 [START_REF] Soulié | Generalized Long-Moody functors[END_REF]Section 2.2.4] The representation LM (ρ) is well-defined.

Proof. We consider elements λ ∈ wB n , x ∈ F n , v ∈ V and i ∈ I Fn . First, since ρ is a morphism, we deduce from (2.2) that

LM (ρ) (λ) i Fn ρ (ξ n (x)) (v) = LM (ρ) (λ) i • x Fn v ,
which gives the compatibility of the assignment LM (ρ) with respect to the tensor product over Z [F n ]. Then this assignment LM (ρ) defines a morphism on wB n follows from the fact that α n and ρ are themselves morphisms.

The natural candidate for the morphism α n is the Artin homomorphism a n recalled in §1.2 and we fix the assignment α n = a n from now on. We could use another Wada representation for a n (recalled in §1.2), but we prove in §2.7 that this is actually not relevant for welded braid groups.

There is always the choice of the trivial morphism F n → 0 → wB n+1 as ξ n so that relation (2.2) is satisfied. However the construction is much more interesting using a non-trivial morphism for this parameter. Indeed applying the Long-Moody construction with the trivial ξ n to a onedimensional representation provides the permutation representation of wB n (sending both the symmetric and braid generators on the permutation matrix). Moreover, the iteration of this Long-Moody construction gives the tensor powers of that permutation representation. We refer the reader to [23, Section 2.2.5] for further details on that point. For that reason, the main point consists in finding non-trivial ξ n such that the diagram (2.1) is commutative, which is the aim the following section.

The natural example for welded braids

We give in this section a first example of a (non-trivial) Long-Moody construction for welded braid groups. It relies on the following natural candidate for the choice of the required morphism ξ n .

Let ξ n,1 : F n ֒→ wB n+1 be the injective morphism defined by

x i -→ (τ i-1 • • • τ 2 τ 1 ) -1 (σ i τ i ) (τ i-1 • • • τ 2 τ 1 ) .
It is natural in the sense that it identifies the free group F n with the free subgroup of order n of D n (see §1.2) generated by the elements {ǫ j,1 } 2 j n+1 . They are also similar to the analogous parameter for the original construction for braid groups (see [START_REF] Soulié | The Long-Moody construction and polynomial functors[END_REF]Example 2.7]). Moreover they satisfy the key condition to define a Long-Moody construction:

Lemma 2.2 The morphism ξ n,1 satisfies the equality (2.2).

Proof. Since

σ i+1 σ i τ i σ -1 i+1 = τ i σ i+1 σ i τ i+1 τ i σ -1 i+1 = τ i σ i+1 τ i+1 τ i then ξ n,1 (a n (σ i ) (x i )) = σ i+1 ξ n,1 (x i ) σ -1 i+1 . Also, note that τ i σ i+1 τ i+1 τ i σ i+1 τ i σ i+1 τ i+1 τ i = σ i+1 σ i σ i+1 τ i+1 τ i = σ i σ i+1 τ i+1 τ i σ i+1 then τ i τ i+1 σ -1 i+1 τ i σ i σ i+1 τ i+1 τ i = σ i+1 τ i σ i+1 τ i+1 τ i σ -1 i+1 hence ξ n,1 (a n (σ i ) (x i+1 )) = σ i+1 ξ n,1 (x i+1 ) σ -1 i+1 .
We thus define a Long-Moody construction associated with this parameter ξ n,1 and denote it by LM 1 . The condition (2.2) is however quite restrictive and the current example follows the spirit of the original Long-Moody construction [START_REF] Long | Constructing representations of braid groups[END_REF]: we therefore focus on the study of LM 1 in this paper. By the way we do not claim to be exhaustive here concerning the study of other possible Long-Moody constructions defined by some other compatible parameters α n and ξ n .

Applications

In addition to recovering the unreduced Burau representation, the second iteration of the original Long-Moody construction recovers the Lawrence-Krammer representation [START_REF] Stephen | Braid groups are linear[END_REF][START_REF] Krammer | Braid groups are linear[END_REF][START_REF] Lawrence | Homological representations of the Hecke algebra[END_REF] as a subrepresentation (see [START_REF] Long | Constructing representations of braid groups[END_REF]Corollary 2.10] or [24, Section 2.3.1]). The linearity of the welded braid groups being an open problem, there was a hope to construct an analogue of the Lawrence-Krammer representation for these groups. Unfortunately, if we can reconstruct the unreduced Burau representation (see §2.3.1), the construction LM 1 does not produce something new at the second iteration (see §2.3.2).

On another note, the variants of the Long-Moody construction introduced in [START_REF] Soulié | The Long-Moody construction and polynomial functors[END_REF] allow to recover the Tong-Yang-Ma representation for braid groups (see [START_REF] Soulié | The Long-Moody construction and polynomial functors[END_REF]Section 2.3.2]). However, we prove in §2.3.2 that it is impossible to reconstruct the Tong-Yang-Ma representation for welded braid groups using any Long-Moody construction.

Notation 2.3

Let R be a ring and r an invertible element of R. We denote by r : wB n → R × the one-dimensional representation defined by r (σ i ) is the multiplication by r and r (τ i ) is the identity for all i ∈ {1, . . . , n -1}. Also for ρ : wB n → GL R (V ) a representation, we denote by rρ : wB n → GL R (V ) the tensor representation r ⊗ R ρ.

Recovering the Burau representation

We start from the one-dimensional representation t : wB n → Z t ±1 × . We obtain that:

Theorem 2.4 The representation t -1 LM 1 (t) is equivalent to Bur.
Proof. The key point to determine the form of the matrices of t -1 LM 1 (t) is to understand the Artin homomorphism on the augmentation ideal. We compute that for all i ∈ {1, . . . , n -1} and j ∈ {1, . . . , n}, a n (σ i ) sends x j -1 to

     x i+1 -1 if j = i (x i -1) x i+1 + (x i+1 -1) 1 -x -1 i+1 x i x i+1 if j = i + 1 x j -1 if j / ∈ {i, i + 1} and a n (τ i ) sends x j -1 to      x i+1 -1 if j = i x i -1 if j = i + 1 x j -1 if j / ∈ {i, i + 1} .
Moreover, for all k ∈ {1, . . . , n}, we denote

Z t ±1 k = Z [(x k -1)] Z[Fn]
Z t ±1 associated to the generator x k of F n . Since the augmentation ideal I Fn is a free F n -module on the set {x k -1, k ∈ {1, . . . , n}}, we thus define an isomorphism

Λ : I Fn Fn Z t ±1 -→ n k=1 Z t ±1 k (x k -1) Fn v -→   0, . . . , 0, k-th v , 0, . . . , 0   .
Note that t (ξ n,1 (x i )) = t for all i ∈ {1, . . . , n}. Then result then directly follows from writing the obtained matrix through Λ.

Iteration on the Burau representations

We follow here the iteration procedure of [START_REF] Long | Constructing representations of braid groups[END_REF]. In particular, an attempt to define an analogue of the Lawrence-Krammer requires to consider one more variable compared to the Burau representation, i.e. to work on the ring of Laurent polynomials in two variables Z t ±1 , q ±1 . For that purpose, we iterate LM 1 on the tensor product of the Burau representation with a one-dimensional representation in the new variable.

Also, from now on, we specify by an index in the notation which parameter we consider (i.e. t or q) and the dimension of the welded braid group we consider for the Burau representation.

Finally, for convenience of computations reasons, we prefer to use the dual version of Burau representation Bur * n+1,t as input representation and consider q -1 LM 1 qBur * n+1,t . We explain below why this choice does not impact the results presented here. We denote by {e i } i∈{1,...,n+1} the basis for the matrices of the representation Bur * n+1,t . First we prove that this iterate of the Long-Moody construction automatically admits the Burau representation as subrepresentation.

Proposition 2. [START_REF] Valeriy | On representations of braids as automorphisms of free groups and corresponding linear representations[END_REF] The submodule of I Fn Fn Z t ±1 , q ±1 ⊕n generated by the elements {(x k -1) ⊗ e 1 } k∈{1,...,n} is closed under the action of wB n . The induced subrepresentation is isomorphic to Bur n,qt .

Proof. First of all, note that qBur * n+1,t (σ i+1 ) (e 1 ) = qBur * n+1,t (τ i+1 ) (e 1 ) = e 1 for all i ∈ {1, . . . , n}. Therefore the action of Artin homomorphism gives that q

-1 LM 1 qBur * n+1,t (σ i ) ((x k -1) ⊗ e 1 ) is equal to          (x i+1 -1) ⊗ e 1 if j = i (x i -1) ⊗ qBur * n+1,t (ξ n (x i+1 )) e 1 + (x i+1 -1) ⊗ qBur * n+1,t ξ n 1 -x -1 i+1 x i x i+1 e 1 if j = i + 1 (x j -1) ⊗ e 1 otherwise
and

q -1 LM 1 qBur * n+1,t (τ i ) ((x k -1) ⊗ e 1 ) =      (x i+1 -1) ⊗ e 1 if j = i (x i -1) ⊗ e 1 if j = i + 1 (x j -1) ⊗ e 1 if j / ∈ {i, i + 1} .
The result thus follows from the fact that qBur * n+1,t (ξ n (x j )) = qt for all j ∈ {1, . . . , n} and the use of the canonical isomorphism Λ (see the proof of Theorem 2.4).

It remains to identify the quotient of q -1 LM 1 qBur * n+1,t by the subrepresentation Bur n,qt . For that purpose, let us first study the case of n = 3. The matrix of the morphism q -1 LM 1 qBur * 4,t (σ 1 ) is

                    0 0 0 0 qt q (1 -t) 0 0 0 0 0 0 0 0 0 0 0 q (1 -t) qt 0 0 0 0 0 0 0 0 0 0 q 0 0 0 0 0 0 0 0 0 0 0 0 0 q 0 0 0 0 1 0 0 0 1 -qt qt -1 1 -t -t 2 + t 3 qt (1 -t) 0 0 0 0 0 0 1 -t t 0 0 (1 -q) (1 -t) t (1 -q) 0 0 0 0 0 0 1 0 0 0 1 -q 0 0 0 0 0 0 0 0 0 1 0 0 0 1 -q 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -t t 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1                    
and the one of q

-1 LM 1 qBur * 4,t (σ 2 ) is                     1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -t t 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 qt 0 q (1 -t) 0 0 0 0 0 0 0 0 0 0 q 0 0 0 0 0 0 0 0 0 0 0 0 q (1 -t) qt 0 0 0 0 0 0 0 0 0 0 q 0 0 0 0 0 1 0 0 0 1 -qt 0 qt -1 1 -t -t 2 + t 3 qt (1 -t) 0 0 0 0 0 1 0 0 0 1 -q 0 0 0 0 0 0 0 0 1 -t t 0 0 (1 -q) (1 -t) t (1 -q) 0 0 0 0 0 0 1 0 0 0 1 -q 0                     .
Those for τ 1 and τ 2 are the analogues with t = q = 1. Then the quotient by the subspace {e 1 , e 5 , e 9 } gives the matrices

              0 0 0 q (1 -t) qt 0 0 0 0 0 0 0 q 0 0 0 0 0 0 0 0 0 0 q 0 0 0 1 -t t 0 (1 -q) (1 -t) t (1 -q) 0 0 0 0 1 0 0 1 -q 0 0 0 0 0 0 0 1 0 0 1 -q 0 0 0 0 0 0 0 0 0 1 -t t 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1                             1 0 0 0 0 0 0 0 0 0 1 -t t 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 q 0 0 0 0 0 0 0 0 0 q (1 -t) qt 0 0 0 0 0 0 0 q 0 0 0 0 1 0 0 1 -q 0 0 0 0 0 0 1 -t t 0 (1 -q) (1 -t) t (1 -q) 0 0 0 0 1 0 0 1 -q 0               .
Writing down the matrices of the tensor product of representations

Bur * 3,t ⊗ Z[t ±1 ,q ±1 ] Bur 3,q
for the generators of the welded braid group, we then recognise the above matrices: we thus identify the quotient of the Long-Moody construction to a tensor product of two Burau representations.

The situation for any n 3 works in the same way: indeed consecutive generators (σ i , τ i ) and (σ i+1 , τ i+1 ) in wB n can be identified to wB 3 . Then the matrices for the quotient of q -1 LM 1 qBur * n+1,t are similar to the above ones, except that there are more diagonal matrix blocks given by Bur 3,t . Hence we proved that the iteration of the Long-Moody construction does not construct any new representation of wB n ; more precisely we have that: = σ 2 ξ n (x 1 ) σ -1 2 . We deduce that ξ n x -1 2 = σ -2 2 ξ n (x 1 ) σ 2 2 and that ξ n (x 1 ) = (τ 2 σ 2 ) -1 ξ n (x 1 ) τ 2 σ 2 . Hence we obtain that:

ξ n (x 1 ) = σ -1 2 τ 2 2 σ -1 2 ξ n (x 1 ) σ 2 τ 2 2 σ 2 = ξ n x -1 1 .
Then (t • ξ n ) 2 = 1 since t and ξ n are morphisms. The only order 2 elements of Z t ±1 are 1 and -1 and it follows from the definition of t that -1 is not in its image. A fortiori the composite t • ξ n is the trivial morphism F n → wB n+1 . Then it follows from a straightforward computation of the matrices that t -1 LM (t) is equivalent to the permutation representation extended to wB n .

Note that a Long-Moody construction multiplies by n the dimension of an input representation. Therefore the only way to construct the Tong-Yang-Ma representation would be to start from a one-dimensional representation. Also, the form of the matrix

Id i-1 ⊕ 0 1 t 0 ⊕ Id n-i-1
for each braid generator σ i implies that the parameter α n of the Long-Moody construction which would recover the Tong-Yang-Ma representation has to be a Wada representation. Therefore the above study of t -1 LM (t) proves that no Long-Moody construction can recover the Tong-Yang-Ma representation.

Therefore, Artin and, more generally, Wada representations do not seem to be a useful tool to obtain interesting linear representations for welded braid groups, at least in the framework of Long-Moody procedure; a possible lead should be to consider other free groups embedded in wB n or other actions. For instance, recently in [START_REF] Damiani | On a canonical lift of artin?s representation to loop braid groups[END_REF] was constructed a lift of Artin representation to an action at the π 2 -level (remind that B 3 less n trivial circles is not aspherical) and it seems interesting to adapt Long-Moody procedure in this framework. It seems also clear that a deeper understanding of wB n and of its subgroup D n-1 in terms of motion groups of circles [START_REF] Damiani | A journey through loop braid groups[END_REF] could provide new perspectives. On another hand, an homological approaches to construct linear representations of welded braid groups is set in [START_REF] Palmer | A unified functorial construction of homological representations of families of groups[END_REF]: the connection with the representations of the present paper would deserve to be explored.

  Theorem 2.[START_REF] Stephen | Braid groups are linear[END_REF] Applying the Long-Moody construction LM 1 to the representation Bur * n+1,t gives the short exact sequence of wB n -representations: Section 2.3.2] uses the Type 2 Wada representation to obtain the representation T Y M for braid groups. Nevertheless, with the extension to wB n , it follows from the compatibility condition (2.2) that τ 2 ξ n (x 1 ) τ 2 = ξ n (x 2 ) = σ -1 2 ξ n (x 1 ) σ 2 and that ξ n x -1

	[24, 2
	Bur * 3,q	/ / 0 .	(2.3)

0 / / Bur n,qt / / q -1 LM 1 qBur * n+1,t / / Bur 3,t ⊗ Z[t ±1 ,q ±1 ]

In[START_REF] Valeriy | On representations of braids as automorphisms of free groups and corresponding linear representations[END_REF] and[START_REF] Wada | Group invariants of links[END_REF], the action described here is the one for σ -1 i . However, we choose to write the inverse symmetry equivalent here to be consistent with respect to the above Artin representation an and with the fact that we compose elements from the right to the left.

Acknowledgements

The authors were partially supported by the ANR Project AlMaRe (ANR-19-CE40-0001-01). The first author was also partially supported by the project ARTIQ (ERDF/RIN). The second author

Cases of the dual and reduced versions. The main conclusion on the iteration of the Long-Moody construction of Theorem 2.6 remains true if we apply LM 1 to the other versions Bur n+1,t and Bur n+1,t of the Burau representation.

For the dual Burau representation Bur n+1,t , it is defined at the level of the matrices by the transpose of the inverse of Bur * n+1,t . Therefore the (n + 1)×(n + 1)-blocks of q -1 LM 1 (qBur n+1,t ) are the transpose of those of q -1 LM 1 qBur * n+1,t . Recall that {e i } i∈{1,...,n+1} denotes the basis for the matrices of the representation Bur * n+1,t . Then the analogue of Proposition 2.5 shows that the submodule of I Fn Fn Z t ±1 , q ±1 ⊕n generated by the elements {(x k -1) ⊗ e j } (k,j)∈{1,...,n}×{2,...,n} is closed under the action of wB n by q -1 LM 1 (qBur n+1,t ). Moreover repeating mutatis mutandis the proof of Theorem 2.6, we prove that there is a short exact sequence of wB n -representations:

For the reduced Burau representation Bur n+1,t , note that we have for all i ∈ {1, . . . , n -1}

where

1 and r n+1 is the n × n-matrix with coefficients r i,j = 1 if j i and r i,j = 0. Therefore, along the injection id 1 * -: wB n ֒→ wB n+1 , the short exact sequence (1.3) splits as a short exact sequence of wB n -representations. Then it follows from the freeness (and a fortiori flatness) of the augmentation ideal I Fn as a Z [F n ]-module that we have an isomorphism of wB n -representations:

The representation q -1 LM 1 qBur * n+1,t is thus determined by the short exact sequence (2.3).

On the impossibility to recover the Tong-Yang-Ma representation

We detail in §2.1 that a Long-Moody construction is equivalent to the setting of two parameters a n : wB n → Aut (F n ) and ξ n : F n → wB n+1 satisfying the compatibility condition (2.1). So far, we have used the Artin homomorphism for the action a n and this parameter determines the shape of the obtained matrices (see §2.3.1 and §2.3.2).

In [24, Section 2.3.2], the Tong-Yang-Ma representation for braid groups is recovered by playing on the choice of this morphism. However, we prove that it is not the case for welded braid groups in the following result. In particular, we call the representation which space is Z t ±1 ⊕n and on which wB n acts trivially the trivial n-dimensional representation of wB n .

Theorem 2.7

Let LM be the Long-Moody construction associated with a Wada representation for a welded braid groups and an abstract compatible ξ n . Then t -1 LM (t) is equivalent either to the Burau representation Bur (or its dual Bur * ), or to the permutation representation extended to wB n , or to the trivial n-dimensional representation of wB n .

In particular, the Tong-Yang-Ma representation (or its dual) cannot be recovered by any Long-Moody construction for welded braid groups.

Proof. We recall from §1.2 that we can only consider four Wada representations (Types 2, 3, 4 and 5). First of all, restricting to B n , [24, Section 2.3.2] automatically implies that:

• with the Type 3 Wada representation, t -1 LM (t) is equivalent to the permutation representation extended to wB n ;

• with the Type 4 or 5 Wada representation, t -1 LM (t) is equivalent the Burau representation.


Kalliopi Kontiza
email: kalliopi.kontiza@ng-london.org.uk

Olga Loboda
email: olga.loboda.13@ucl.ac.uk

Louis Deladiennee
email: louis.deladiennee@list.lu

Sylvain Castagnos
email: sylvain.castagnos@loria.fr

Yannick Naudet
email: yannick.naudet@list.lu

A Museum App to Trigger Users' Reflection

Keywords: Mobile Guide, Reflective Topics, Recommender Systems

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The CrossCult (www.crosscult.eu) aims to spur a change in the way European citizens appraise History, fostering the re-interpretation and reflection upon what they may have learned in the light of cross-border interconnections among pieces of cultural heritage, other citizens viewpoints and physical venues. Within the 4 pilots implemented in the project, there are a number of ways in which reflection is expected, either by considering our personal experiences by our examination and interpretation of history to understand current social phenomena and different viewpoints and/or to examine and reason by exploring reflective topics and stories which facilitate consideration of historical content and narratives. We present here results from one pilot, which relies on the latter approach and where personalised guidance is explored for reflection triggering.

After a short related work in Section 2, Section 3 presents the smart Guide Mobile application that has been designed and developed, highlighting the features involved in reflection triggering. In Section 4, the first user study we have conducted is presented, followed by Section 5 that discusses preliminary results. Future work is presented in Section 6.

Related Work

Even though 2018 has been declared the European Year of Cultural Heritage, many empirical studies show that the technological revolutions of the last 2 decades have upset visitor expectations in museums. In [START_REF] Cohen | Culture track 2017 report[END_REF], only 63% of respondents saw art and design museums as culture. The main mentioned motivations for visiting a museum were to have fun, to experience and learn new things, to feel inspired and to interact with others. If the will of visitors to deepen their knowledge of art is not new [START_REF] Falk | Identity and the Museum Visitor Experience[END_REF], their wish to no longer be mere spectators seems much more recent [START_REF] Museums | Digital transformation in the museum industry[END_REF]. Among the tracks and strategies mentioned in the report, mobile guides and cross-platform services are cited, especially for the educational opportunities that the experiences they generate provide among diverse audiences. Helal et al. showed that visitors expect from multimedia guides to get access to additional information and learn more about artists and exhibits [START_REF] Helal | Lessons learned: Evaluating the whitney's multimedia guide[END_REF]. Mann and Tung even pointed out that most of visitors felt that they explored the museum more when using a mobile guide [START_REF] Mann | A new look at an old friend: Reevaluating the mets audio-guide service[END_REF]. In the same study, the authors highlighted the expectations as regards the guide to render the visit more entertaining and to provide personalised routing instructions in the museum.

To address these issues, recommender systems are becoming more and more prevalent in museums to personalise the visitor experience by providing interesting suggestions at the right moment, in the right order and through the appropriate interaction. As an example, Aroyo et al. introduce semantically driven recommendations for museums and semi-automatic generation of personalised museum visits based on visitors' model [START_REF] Aroyo | Finding your way through the rijksmuseum with an adaptive mobile museum guide[END_REF]. They propose to detect user preferences and compute from these preferences a personalised visit optimised by walking distance and art objects each visitor perceives as interesting. Osche et al. extend the number of dimensions to be consider in the visitor model, such as the crowd tolerance, the distance tolerance, or the technological intrusiveness [START_REF] Osche | Walk the line: Toward an efficient user model for recommendations in museums[END_REF]. Zheng et al. investigate ways to provide location-aware recommendation by exploiting the WiFi logs [START_REF] Zheng | Location-aware poi recommendation for indoor space by exploiting wifi logs[END_REF].

Although recommender systems have been exploited to provide personalised experiences to museum visitors, they mainly focus on precision and user satisfaction [START_REF] Keller | Recommender systems for museums: Evaluation on a real dataset[END_REF]. None of them has so far addressed the question of the reflection this experience can trigger on users.

The CrossCult App for the National Gallery

Pilot 1 takes place in the National Gallery (NG) in London, a large multithematic venue. The developed application is expected to engage users and encourage them to reflect on the information presented and the diverse works of art in the collection. This serves two purposes: (i) development of personalised gallery experiences based on user preferences, and (ii) self-discovery of the collection through three main reflection topics: 'Materials', 'Historical Events' and 'Social Connections' will reveal the interconnections at play. We provide here a quick overview of the Pilot 1 smart museum guide mobile app screens. After the user installs the app for the first time, she is presented with a short sideshow detailing the app main features. After logging, the user lands on the home page which provides access to different features. An initial profiling step has been included (Fig. 1), so as to offer users the ability either to prepare or to tailor a visit; the user is initially invited to fill-in a form with questions about: (i) their general interests, (ii) their tolerances to different factors that might impact his/her visit experience (e.g. to the crowd). The second step of the profiling is delivered as a carousel of images, inspired by the UX of the famous Tinder app, where the user can swipe her way through a stack of pictures depicting paintings from the collection indicating their preferences; if the user likes a painting (swipe right) or not (swipe left).

The set of paintings presented to the user is curated to be representative of the NG collection diversity and is used to build a first a priori profile. The carousel of paintings constitutes a main element of the profiling screens (Fig. 1, left). When users choose to like, dislike or favour a painting shown in the carousel, their user profile is updated in the background with captured interests and preferences extracted from the painting meta-data. Once the user has filled in enough of the profile data, personalised recommendations are computed, delivering suggestions for groups of paintings to visit. When selected by the user for a visit, paintings from the chosen group are available on the map; however the route was not displayed at the time of experiments.

To help the visitor get a grasp of the Gallery size, the app includes a digital map of the different floors (Fig. 2, top right and bottom left). From the map, the user indicates her current position which is then used to generate routes to the different paintings of the collection. The map can be used to locate paintings within the gallery and ask for routing information. The app displays a picture of the painting along with core information such as the painting title, its production date and its author (bottom row). By clicking an information icon, the user can access more detailed information about the work of art and a textual description is available to provide some narrative storytelling about the painting itself. In visit mode, this description is complemented by extra narratives that are related to the selected reflective topic.

By clicking on a room on the map, the user receives a list of all the paintings currently on display, presented as a grid of pictures (Fig. 2, bottom right). Clicking on a picture leads to the paintings details. This layout is reused throughout the entire app when it comes to displaying "groups" of paintings. Currently managed groups are paintings from the same room, paintings from Fig. 2. The map, collection browsing and group presentation the same painter, recommended paintings (from the user preferences) and narratives/stories. These groups are also accessible from the Home page, by clicking the 'Browse the collection' button, which allows the user to browse the whole collection (Fig. 2, top left).

Finally, recommendations are accessible from the Browse collection screen or after configuring the visit. In both cases, a button allows to preview on the map where the recommended paintings are located in the museum and to start the visit. Once the visit started, the user is invited to indicate his current location to get access to the content present around him. An automated in-door location system is foreseen, but not yet operational.

User study

Data Collection and Evaluation Methodology

We have designed a series of controlled user experiments involving participants observation and post-experiment questionnaires, set around the different aspects of the app that needed to be tested and evaluated. Here, we were interested in [START_REF] Antoniou | Micro-augmentations: Situated calibration of a novel non-tactile, peripheral museum technology[END_REF] determining the extent to which the users are satisfied with the recommended groups of paintings, with recommendations given only by explicit profiling and (2) to begin to understand more about how and if the app stimulates reflections resulting from discovery of art via the group recommendations.

We grouped under this user study two small-sized experiments conducted out of the museum, using only the mobile app, to gather first feedback about usability and user experience, following the same experimental protocol. In total, 35 participants (valid questionnaires, all students, 20% postgraduate, 80% undergraduate, with one-third male and two-third female) were collectively given a short 15-20 minute introduction to the project. They were then asked to imagine they would start a visit at the National Gallery and think about the objective of their visit and what they would expect from it. During the second experiment, 4 participants were then engaged in a round table discussion for 30 minutes. Additionally, they were also shown around to the National Gallery and performed a physical visit following the recommended group of paintings. The purpose of visiting the museum was to evaluate the user experience while following the recommendation during the physical visit and see whether it influences reflection more than with only access to digital content brought by the App. All participants were then asked to complete the post-experiment questionnaire.

The post-experiment questionnaire we used took into account the notions of ease of use, user satisfaction and learnability of the app. We used a Likert scale to determine if users agreed or disagreed with statements associated to main constructs of the analysis. We also asked a number of open-ended questions to gather more details related to the perceptions of the user towards the app and understand where the value of the app lies. The items in the questionnaire were adapted from the user acceptance of technology model (UTAUT) [START_REF] Venkatesh | User acceptance of information technology: Toward a unified view[END_REF] and supplemented with questionnaire items common to the approach of evaluating technology such as the USE questionnaire [START_REF] Lund | Measuring usability with the use questionnaire[END_REF]. The questionnaire was divided into four sections and asked the participants about their: (1) basic demographic data (e.g. age group, gender, educational attainment, residential status and nationality), data useful for the profiler and recommender systems (e.g. command of the English language, involvement in and knowledge of Art, use of recommender systems in everyday life), visiting preferences (e.g. frequency of museums visits, use of audio guides, objectives and motivations) and mobile habits (use of mobile apps, use of museum mobile apps, self-reported map reading abilities); ( 2) general thoughts about the app (whether they recommend the app and why);

(3) overall user experience, the ease of learning, features assessment and learning outcomes and (4) final thoughts.

In order to begin to understand more about how and if the primary app stimulates reflections and to evaluate the reflection triggers, a set of questions from section (3) of the questionnaire have been designed and categorised using Blooms taxonomy and its revised version [START_REF] Krathwohl | A revision of Bloom's taxonomy: An overview[END_REF]. In particular, the taxonomy describes 6 levels for high-level cognitive processes that are hierarchically structured from lower (memory) to higher (creativity). According to Daudelin [START_REF] Wood Daudelin | Learning from experience through reflection[END_REF], reflection could also be treated as an upper level learning process. Finally, trackers were also implemented to capture the users interactions with the app.

Results

Generally, the participants recognised the value of the App and had a positive perception about it (90%). The testers mainly liked the user experience with the recommendations, with 74% of respondents slightly /strongly agreeing and agreeing that they were satisfied with the group of paintings recommended to them and 69% of users slightly /strongly agreeing and agreeing with the statement [Each painting in the recommended group was relevant regarding my interests]. With respect to the personalisation of the content during the museum visit, one of the key motivations for using the app was associated with the discovery and exploration of the collection in a free, user personalised way, i.e. not choreographed by the curator; a few users found the personalised content delivery mechanisms to be a more time efficient approach: "you dont have to spend The perceived value of the recommendations received a mixed responses, especially during the early experiments. One likely explanation for this is that this experiment occurred early on and purposely used a very small sample of paintings to determine efficiency of the recommender. At the same time, this highlights the need to evaluate with a more qualitative approach the perception of the recommendation system. Answers representative of those who found the recommendation process likeable were "I liked that the paintings matched my preferences after a while" and "It recommends to you few masterpieces", showing the user's appreciation that it integrated lesser known works as well as the well-known masterpieces. Some users perceived a value in the recommendation system because it streamlined the process of discovering and viewing an interesting painting.

Concerning reflection, we consider four constructs that stimulate and contribute to reflection processes in our pilots, including influences from our prior learning processes [START_REF] Tobias | Interest, prior knowledge, and learning[END_REF], our personal experiences [START_REF] David | Using Experience For Learning[END_REF], our emotional responses [START_REF] Antoniou | Micro-augmentations: Situated calibration of a novel non-tactile, peripheral museum technology[END_REF] to both the app and its content as well as our personal interpretation of the content based on a users world view [START_REF] Kafai | Constructionism in practice : designing, thinking, and learning in a digital world[END_REF]. Reflection involves linking a current personal experience to acquire information and knowledge for new or existing learning and is driven by the process of combing cognitive and emotional information from different sources. It is a process that occurs when we act upon different information to synthesise and evaluate it. In order though to begin to understand more about how and if the primary app stimulates reflections and to evaluate the reflection triggers, the participants were asked the questions presented in Figure 3 and expressed their agreement or disagreement with the statements using a 6 point Likert scale, to trigger their reflection in regards to the outcome of the Recommender System and the feature of the recommended group of paintings in the App.

Both the diversity and coherence of the recommended group of paintings was understood by the participants with 20 out of 35 and and 22 out of 35 slightly/strongly and agreed with [The paintings in the recommended group are all part of a coherent whole] and [The paintings in the recommended group were diverse] respectively. In terms of the question related to the users memory processes 26 out 35 of participants slightly/strongly and agreed with the statement [The paintings in the recommended group were new] having quite successfully monitored that the Recommender System was updating the recommended group of paintings in the app according to their interaction. The participants were able to provide the following evaluations related with the recommended group of paintings: 28 out 35 of participants slightly /strongly and agreed with the statements that [Each painting in the recommended group was according to my visit objectives] and 24 out of 35 participants slightly /strongly with the self-observation statement that [I reached my objectives following the app rec.].

For the two higher level processes of 'create' and 'apply' information in new situations almost half of the participants provided a title for the recommended group of their choice, with different levels of creativity and details. Almost half of the participants matched the recommended group of paintings with the one of the two categories that were given to them; 'Materials and Techniques' and 'Social Connections'. When asked to provide an explanation for their choice some of the comments that describe the reflection process of the testers were "Techniques probably, because I was able to find the kind of art I was interested in without having to roam around", "I really don't understand these categories (...)", although almost half of the participants just answered that they did not know. Finally one participant reflected on his previous knowledge ("A selection by William Hogarth that I didn't know about before"), when asked [Which painting did you discover today that was recommended to you].

Conclusion and Perspectives

Due to the maturity of the app, we have been mainly concerned with evaluating the interactions that we believe will trigger the process of reflection more so than the reflections themselves. So far, we attempted to trigger reflection through the following user interactions with the app: (1) a reaction to art through likes of painting in the carousel, (2) personalised recommendations that stimulate interest in the artworks and their topics and (3) engaging with the paintings, group of paintings and stories descriptions. A promising number of participants were able to remember, consider, examine, compare and reason as a response to discover content and narratives, which is the underlying reflection purpose of Pilot 1. These preliminary results are encouraging to further investigate whether visitors are more prone to reflection and more interested by the collection in a museum if they can discover it through other facets than those highlighted solely in the museum and if this discovery is personalised for each of them.

In the next stage of the evaluation it is evident that there is a need to focus on the evaluation of the app to understand in more depth how it stimulates the process of reflection from the perspective of the recommender system. The results will now feed into how we refine the evaluation protocol for the next phase, which will appraise the interactions that contribute to reflection and to qualitatively evaluate the form of reflection that is stimulated by the app in order to further investigate and analyse the users perceptions of the app and its ability to encourage them to think.

Fig. 1 .

 1 Fig. 1. The application profiling screens

Fig. 3 .

 3 Fig. 3. Answers to statements related to reflection triggering by recommendations

Acknowledgement

This project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 693150.


D Siret

INVERSE SIMULATION OF SUNLIGHTING : A GEOMETRICAL FRAMEWORK FOR ARCHITECTURAL AND URBAN DESIGN

We propose a framework based on integral geometry that enables to express and to solve all sunlighting problems : the direct problems for assessing sunlighting in a scene, and the inverse ones for achieving sunlighting constraints in a design context. The heart of our method is the sunlighting volume that we denote ∏(P, T). We describe this notion and we show how to implement it in an architectural and urban design process.

RÉSUMÉ

Nous proposons un formalisme géométrique qui permet d'exprimer et de résoudre tous les problèmes d'ensoleillement : les problèmes directs pour évaluer l'ensoleillement d'une scène, et les problèmes inverses pour satisfaire des contraintes d'ensoleillement en situation de conception. Le coeur de notre formalisme est la pyramide d'ensoleillement. Nous décrivons cette notion et nous précisons sa mise en oeuvre pour la conception architecturale ou urbaine.

INTRODUCTION

We distinguish the direct sunlighting problems from the inverse ones. The former consist in determining the sunlighting states of a given scene using both shadows and shadings ; the latter consist in resolving the geometrical conditions that enable a scene to achieve a given sunlighting constraint in a design context (sunlighting as formgiver [START_REF] Lam | Sunlighting as Formgiver for Architecture[END_REF]).

In this paper, we propose a geometrical framework based on integral geometry [START_REF] Serra | Image analysis and mathematical morphology[END_REF] for managing all sunlighting problems. This framework offers three main benefits : it gives homogeneous expressions and solutions of the direct sunlighting problems of shadings and shadows ; it enables the generalisation of these problems either in space or time ; it offers solutions to the inverse problems of sunlighting.

The heart of our framework is the sunlighting volume that we denote ∏(P, T). We define this notion in the first section, within a set of geometrical notions that make up our method. In the second and third sections, we give solutions to the direct and inverse problems. Finally, we show how these methods can be implemented in an existing CAD system, and why they offer a powerful framework to cope with sunlighting problems in architectural and urban design.

THE INTEGRAL GEOMETRY OF SUNLIGHTING

Let ∑ be a geometrical scene located at the latitude l. We denote p any point of ∑ and P any continuous set of points in ∑. We consider D l the set of all the apparent solar directions at the latitude l, and we denote t any direction included into D l (i.e. any instant of the solar year). We define a time period T as any continuous set of instants t, that is consequently a composition of intervals of instants among days and months. T can likewise be seen as a geometrical patch of D l . Finally, we define the sunbeam R(p, t) as the half line starting from p in the direction of t. We define the inverse sunbeam R(p,-t) starting from p in the opposite direction of t (which is the perceived natural sunbeam).

Sunlighting volumes

Let us consider now sunlighting volumes settled as continuous set of sunbeams. First, we define the sunlighting pyramid (p, T), that is the set of sunbeams starting at the point p for all the instants t of the time period T : (p, T) = {R(p,t), t OE T}. In the same way, we define the sunlighting prism v(P, t), that is the set of sunbeams defined for all the points p of P and for the single instant t : v(P, t) = {R(p,t), p OE P}. Finally, we define the complex sunlighting volume ∏(P, T) as the set of sunbeams starting from all the points p of the set P for all the directions t of the time period T, that is : ∏(P, T) = {R(p,t), p OE P, t OE T}. Notice that ∏(P, T) can be equally considered as the union of simple sunlighting pyramids (p, T) when p draws P, or as the union of prisms v(P, t) when t draws T, that is (fig. 1) : We define ∂∏(P, T) the core of ∏(P, T) as the intersection of the sunlighting prisms v(P, t) when t draws T, that is :

∏(P, T) = pOEP U (p, T) = t OET U v(P, t) (1)
∂∏(P, T) = t OET I v(P, t) (2)
The core ∂∏ embodies the set of all the points of ∑ that are affected by all the sunbeams defined by all points of P during all instants of T. In other terms, any point of ∑ within ∂∏ will have its shadow within P whatever is the instant of T. This means that the shadows of any volume built in ∂∏ will never spill over the edge of P during T (fig. 2). This is the definition of Ralph Knowles' solar envelopes, as a container to regulate development within limits derived from the sun's relative motion [START_REF] Knowles | Solar Access and Urban Form[END_REF]. We offer here a geometrical framework for defining and using this notion usually handled with empirical geometrical methods.

DIRECT SUNLIGHTING PROBLEMS : EXPRESSION AND SOLUTIONS

Direct problems of shadows and shadings can be easily expressed, generalised either in time and space, and geometrically solved using integral geometry.

Generalised shadows

Let Ω(P, t) be the shadow of P in the scene ∑ for the instant t. By definition, Ω(P, t) is the intersection between the sunlighting prism v(P,-t) and ∑, that is : Ω(P, t) = v(P,-t) « ∑. If we consider the time period T instead of the instant t, we can express the generalised shadow of P during T, that is the union of all the shadows of P for all the instants t of T :

Ω(P, T) = t OET U Ω(P, t) = t OET U (v(P,-t) « ∑) = ∏(P,-T) « ∑ (3)
Therefore, the generalised shadow of P during T is given by the intersection between the sunlighting volume ∏(P,-T) and the objects of ∑. This is the exact set of points of ∑ that are exposed to the shadow of P during at least one instant of T.

In the same way, we prove that the exact set of points of ∑ that are exposed to the shadow of P during all the instant of T are those that are in the core of the sunlighting volume ∂∏(P,-T).

Then we denote the core of shadow : ∂Ω(P, T) = ∂∏(P,-T) « ∑ (fig. 3

left).

Obviously, if P is an opening instead of a shading, then the generalised shadow is the union of all sunspots generated by P during T (generalised sunspot), and the core of shadow is the set of points that are always sunlit through P during T (core of sunspot, see fig. 5).

Generalised shading

Let M(p, T) be the shadings of the scene ∑ on the point p for the period T. M(p, T) is the intersection between the simple sunlighting pyramid (p, T) centred in p and the volumes of the scene : M(p, T) = (p, T) « ∑. If we want to determine the shadings of the scene ∑ on the set of points P for the time period T, that is all the points of ∑ that may shade at least one point of P during T, then we obtain :

M(P, T) = pOEP U M(p, T) = pOEP U ( (p, T) « ∑) = ∏(P, T) « ∑ (4)
The generalised shadings on P during T are given by the intersection between the sunlighting volume ∏(P, T) and the objects of the scene (fig. 3 right). We also prove that any point within the core of ∏(P, T) shade at least one point of P for all the instants of T. In other words, any object within ∂∏ is a permanent shading on P during T.

Discussion

Geometrically speaking, sunlighting problems have to be considered in a 4D space (3D plus time). However, graphical simulation means have led to solve these problems by reducing them to 3D volumes, using a discretization of space or of time, and then, by projecting this 3D volumes onto 2D charts (axonometric projections for shadows at a discrete instant, and perspective projections for shadings on a discrete point). When computer is used, the same methods are usually transposed in a digital manner.

We have shown in this section that both problems of shadows and shadings i) can easily be generalised either in time and space, and ii) can be expressed and solved in an homogeneous way, as an intersection between the scene and a given volume ∏(P, T). Whatever are P and T (single point, single instant or continuous sets of points and instants), we define the sets of points of ∑ that involve simultaneously the points of P and the instants of T. Only the sign of T changes (the direction of the sunbeams).

INVERSE PROBLEMS OF SUNLIGHTING

Direct simulation tools developed for integrating solar constraints within architectural and urban design appraise the solar properties of geometrical forms that are fully ascertained. Supposing the forms do not achieve previous fixed solar goals, the process of simulation must be fully repeated until a good solution is found. One should consider two major drawbacks of such an analytic approach : it imposes a trial and error process inappropriate for design purpose ; it can neither provide the best solution that may achieve the goals, neither indicate the whole set of solutions if any, neither specify if there are no solution.

The paradigm of inverse simulation has been proposed in order to overtake these drawbacks [START_REF] Schoeneman | Painting with Light[END_REF]. The inverse problems of sunlighting can be expressed in a general way : given a set of points P and a time period T, these consist in achieving the following constraint : « P must check on the sunlighting state S during T ». In a more formal way, let us consider a sunlighting constraint as a logical proposal (P, S, T). Solving the inverse problem consists in making the sunlighting proposal true. Therefore, the solutions of a proposal (P, S, T) are the transformations of the scene ∑ that make P be S during T. These are openings and closings of the scene, that is geometrical transformations handled by architects during the design process.

Let's study the solutions for the elementary proposals (P, sunlit, T) and (P, sunless, T).

Elementary proposals (P, sunlit, T)

The solutions for these proposals are the openings of the scene ∑ such as all the points of P become sunlit for all the instants of T. Therefore, the set of solutions of the proposal (P, sunlit, T) is the set of all the openings of the scene that contain the volume ∏(P, T). It exists a single minimal solution O = ∏(P, T) « ∑, that is the opening by the exact intersection between the scene and the sunlighting volume. Obviously, if this intersection is already empty, the proposal is already true and the opening has no effect.

Elementary proposals (P, sunless, T)

The solutions for these proposals are the closings of the scene ∑ such as all the points of P become sunless for all the instants of T. There exists an infinite set of solutions that are all the objects that intersect all the sunbeams of ∏(P, T) (fig. 4). Therefore, as well as a generator of sunlighting volumes, an implementation of the method within a 3D CAD engine should consider a method for exploring this set of solutions. In [START_REF] Siret | Propositions pour une approche déclarative des ambiances dans le projet architectural[END_REF], we propose such a method by handling a plane that interactively intersects the volume ∏(P, T) and displays the shadings.

A point should be discussed here. If P is exactly the same for all the closings M defined with ∏(P, T), we notice that each generalised shadow Ω( M, T) is distinct. Obviously, the shadings M close to the plane of P create generalised shadows close to P, while the shadings far from the plane of P create very large generalised shadows. That means that each solution M of the proposal (P, sunless, T) can be distinguished by its own generalised shadow (intersection between ∏( M,-T) and the scene ∑). This gives a useful means for checking up the effects of each closing achieving a (P, sunless, T) proposal while exploring the set of its solutions.

IMPLEMENTATION

Computing 3D boundaries of sunlighting volumes

This geometrical problem is not self-evident. In [START_REF] Siret | Propositions pour une approche déclarative des ambiances dans le projet architectural[END_REF] we propose a solution for computing the boundary of ∏(P, T) when P is a convex plane polygon. In that case, we show that any section between ∏(P, T) and a plane Q parallel to the plane of P can be figured as the Minkowski sum between P and the projection of T onto Q (that is, a computable geometrical operation). Moreover, the intersection between ∏(P, T) and Q owns a geometrical structure that enables to determine all the external beams of ∏. Using this approach, we have developed a generator of 3D sunlighting volumes usable in a CAD system [START_REF] Siret | Propositions pour une approche déclarative des ambiances dans le projet architectural[END_REF].

Using sunlighting volumes within a CAD system

A suitable CAD system for helping designers to cope with sunlighting problems only needs to compute the intersections between some defined sunlighting volumes and the objects of the scene. No more charts or geometrical constructions are needed. Only intuitive orders or questions are used such as : « Is this face sunlit during the afternoon in winter ? », or « What is the shadings of these buildings in the morning in summer ? », and « I want this room to be sunless during T ». Such a system can make a diagnosis and it can provide an intelligent way for managing sunlighting constraints throughout the evolution of the design. The constraints are visualised as any other 3D volumes and their solutions can be explored interactively.

For instance, figure 5 shows how to design a window by controlling is total sunspot within a room for a given time period. Moreover, we can simultaneously adjust the whole limit of the spot given by the window on T (generalised sunspot) and the limit of the set of points that are always sunlit by the window during T (core of sunspot). In addition, the framework enables to solve the composition of sunlighting proposals in a geometrical way. The point is to consider that the volumes ∏(P i , T k ) associated to the proposals (P i , S j , T k ) embody a geometrical representation of the corresponding constraints.

The set of solutions of two constraints is found at the intersection or in the geometrical difference between the associated volumes. In figure 6 (left), we show an opening O defined such as P is sunlit for all instants of T, and a shading M such as the same P is sunless during T' (disconnected from T). We verify the achievement of these constraints for all instants of T and T' with direct simulation (figure 6, middle and right).

Rendering and meshing

Other possibilities of the framework should be mentioned. For instance, it can help the preprocessing for rendering very large scenes, like urban scenes. Supposing that T is the entire solar year, and P any face of the scene, then only the faces that intersects ∏(P, T) have to be taken into account for computing shadows on P (whatever is the considered instant). In the same way, the sunlighting volumes may be helpful for computing the discontinuity meshing of a scene. Indeed, the volume ∏(P, T) gives the limit of the points that are always exposed to P during T (the core) and the limit of the points that are never exposed to P during T. For solar assessment, the only required meshing is between these two limits.

Figure 1 : 1 . 2

 112 Figure 1: The three kinds of sunlighting volumes

Figure 2 :

 2 Figure 2: M defined in the core of a sunlighting volume (left) and its shadows during T (right)

Figure 3 :

 3 Figure 3: Generalised shadows (left) and generalised shadings (right)

Figure 4 :

 4 Figure 4: M is a section of ∏(P,T) (left and middle). The intersection of the shadows of M for all instants of T is equal to P (right).

Figure 5 :

 5 Figure 5: Interactive design of a window considering the generalised sunspot and its core

Figure 6 :

 6 Figure 6: Solutions of two sunlighting proposals (left) and verifications (middle and right)