The monoid algebra of all relations on a finite set

Serge Bouc and Jacques Thévenaz

Abstract : We classify all the simple modules for the algebra of relations on a finite set, give their dimension, and find the dimension of the Jacobson radical of the algebra.

AMS Subject Classification : 06A06, 13C05, 16B50, 16D60, 16N20, 18B05, 18B10, 18B35, 18E05.

Key words : finite set, relation, simple module, Jacobson radical.

1. Introduction

J.A. Green was a pioneer in the representation theory of algebras, in particular group algebras, but he also produced important work in the theory of monoids. We wish to contribute to this memorial issue by presenting some results on the k-algebra R_{X} of a specific monoid, namely the monoid of all relations on a finite set X, where k is a field. This is an algebra of dimension 2^{n^{2}}, where n = |X|, hence growing very fast in terms of n.

In a previous work [BT1], we described all simple modules for the algebra E_{X} of essential relations on X, which is a quotient of R_{X}, but it was then not clear how to extend this result. We do so here and find all the simple modules for R_{X}. They are classified by isomorphism classes of triples (E, R, V), where E is finite set with |E| ≤ |X|, R is a partial order relation on E, and V is a simple module for the group algebra k Aut(E, R). When E = X, we recover the simple modules for the essential algebra E_{X}.

We also give a formula for the dimension of each simple module and describe the action of every relation. From this, we deduce a formula for the dimension of the Jacobson radical of R_{X} (in characteristic zero). The formulas behave exponentially with respect to n.

Apart from [BT1], the other main ingredient for this work is our paper [BT2] about correspondence functors, in which all simple correspondence functors are classified and a formula is obtained for the dimension of each of their evaluations. It turns out that the evaluation at X of a simple functor is either zero or a simple module for the algebra R_{X}. Conversely every simple module for the algebra R_{X} occurs as the evaluation at X of a simple functor. This provides a way to handle simple modules for the algebra R_{X} by using simple correspondence functors. This method sheds some new light on the structure of the algebra R_{X} and allows us to prove our results.

Throughout this paper, X and E denote finite sets and k denotes a field.

2. Simple modules for the essential algebra

A correspondence from a finite set X to a finite set Y is a subset of Y × X. The set of all correspondences from X to Y is written C(Y, X). We reverse the order of X and Y in order to have later a left action of correspondences. Correspondences can be composed as follows: if R ⊆ Z × Y and S ⊆ Y × X, then

RS = {(z, x) ∈ Z × X | ∃y ∈ Y, (z, y) ∈ R, (y, x) ∈ S} ⊆ Z × X .

A correspondence from a finite set X to itself is called a relation on X. The set C(X, X) of all relations on X is a monoid for the composition above and forms a basis of the finite-dimensional k-algebra R_{X} := kC(X, X). This algebra is called the algebra of all relations on X.

A relation R on X is called essential if it does not factor through a set of cardinality strictly smaller than |X|. The k-submodule generated by the set of inessential relations is a two-sided ideal

I_{X} = ∑_{|Y |<|X|} kC(X, Y)kC(Y, X)

and the quotient

E_{X} := kC(X, X)/I_{X}

is called the essential algebra.

By an order relation, we always mean a partial order. Let R be an order relation on a finite set E and let σ ∈ Σ_{E}, the group of all permutations of the set E. Define the conjugate relation ^{σ}R = Δ_{σ}RΔ_{σ^{−1}}, where

Δ_{σ} = {(σ(x), x) ∈ E × E | x ∈ E} .

Then ^{σ}R is an order relation on E and this defines an action of the group Σ_{E} on the set of all order relations on E. The stabilizer of R under this action is written Aut(E, R). The group Σ_{E} acts by conjugation on the set of all pairs (R, V), where R is an order relation on E and V is a simple k Aut(E, R)-module. The following result is Theorem 8.1 in [BT1].

2.1. Theorem. Let E be a finite set. The set of isomorphism classes of simple E_{E}-modules is parametrized by the set of conjugacy classes of pairs (R, V), where R is an order relation on E and V is a simple k Aut(E, R)-module.

We let T_{R,V} be the simple E_{E}-module parametrized by (the conjugacy class of) the pair (R, V).

3. Simple correspondence functors

We define the category C as follows :

• The objects of C are the finite sets.

• For any two objects X and Y, the set of morphisms from X to Y is the set C(Y, X) of all correspondences from X to Y, namely all subsets of Y × X.

• The composition of morphisms is defined as above.

The k-linearization of the category C is defined as follows :

• The objects of kC are the objects of C.

• For any two objects X and Y, the set of morphisms from X to Y is the k-vector space kC(Y, X) with basis C(Y, X).

• The composition of morphisms in kC is the k-bilinear extension

kC(Z, Y) × kC(Y, X) ⟶ kC(Z, X)

of the composition in C.

A correspondence functor is a k-linear functor from kC to the category k-Vect of k-vector spaces. We could define a correspondence functor to be a functor from C to k-Vect, but it is convenient to linearize first the category C (just as for group representations, where one can first introduce the group algebra). The category of correspondence functors is an abelian category and we are particularly interested in its simple objects.

We now describe the parametrization of simple correspondence functors. If S is a simple correspondence functor, we let E be a minimal set such that S(E) ≠ {0}.

Then S(E) is a module for the algebra R_{E} = kC(E, E), and actually a module for the essential algebra E_{E}, because the ideal I_{E} of inessential relations acts by zero, by minimality of E. Moreover, S(E) is easily seen to be simple (see Proposition 2.6 in [BT2]). Therefore, by Theorem 2.1, S(E) ≅ T_{R,V} for some order relation R on E and some simple k Aut(E, R)-module V. Associated to the simple functor S, this defines a triple (E, R, V) which is uniquely defined up to isomorphism. The following result is Theorem 3.12 in [BT2].

3.1. Theorem. The set of isomorphism classes of simple correspondence functors is parametrized by the set of isomorphism classes of triples (E, R, V), where E is a finite set, R is an order relation on E, and V is a simple k Aut(E, R)-module.

We let S_{E,R,V} be the simple correspondence functor parametrized by (the isomorphism class of) the triple (E, R, V). Thus we have

S_{E,R,V} (E) ≅ T_{R,V} and S_{E,R,V} (F) = {0} if |F | < |E| .

One of the main results of [BT2] provides a closed formula for the dimension of any evaluation of a simple functor. The following result is Theorem 17.19 in [BT2].

3.2. Theorem. Let S_{E,R,V} be the simple correspondence functor parametrized by the triple (E, R, V) and let X be a finite set. Then

dim S_{E,R,V} (X) = \frac{dim V}{| Aut(E, R)|} ∑_{i=0}^{|E|}(−1)^{i}\binom{|E|}{i}(g_{E,R} − i)^{|X|} ,

where g_{E,R} is a positive integer canonically associated to (E, R).

We only sketch here where the integer g_{E,R} comes from and give references. We can associate to the poset (E, R) a finite lattice T having (E, R^{op}) as the full subset of its join-irreducible elements, where R^{op} denotes the opposite relation. For instance, we can take T to be the lattice of all lower ideals in the poset (E, R^{op}) and identify E with the set of principal ideals. Then there are two operations s^{∞} and r^{∞} in T, defined in Section 16 of [BT2], and we let

G_{E,R} = E ⊔ {a ∈ T | r^{∞}(s^{∞}(a)) = a} .

Clearly E ⊆ G_{E,R} ⊆ T and we let g_{E,R} = |G_{E,R}|. Since dim S_{E,R,V} (X) does not depend on the choice of T, the integer g_{E,R} turns out to be independent of the choice of T. We refer to Section 16 of [BT2] for details.

4. Simple modules for the algebra of relations

We now give the parametrization of all simple modules for the algebra R_{X} of relations on a finite set X. They are described in terms of simple correspondence functors.

4.1. Theorem. Let X be a finite set.

(a) The set of isomorphism classes of simple R_{X}-modules is parametrized by the set of isomorphism classes of triples (E, R, V), where E is a finite set with |E| ≤ |X|, R is an order relation on E, and V is a simple k Aut(E, R)- module.

(b) The simple module parametrized by the triple (E, R, V) is S_{E,R,V} (X), where S_{E,R,V} is the simple correspondence functor corresponding to the triple (E, R, V).

Proof : We first claim that any simple R_{X}-module S occurs as the evaluation of some simple functor S_{E,R,V}, that is, S ≅ S_{E,R,V} (X). This forces |E| ≤ |X| because S_{E,R,V} vanishes on sets Y with |E| > |Y |. A proof of this first claim appears in Proposition 3.2 of [We] and is attributed to Green (6.2 in [Gr]). This requires to view R_{X} = kC(X, X) as a category with a single object X, hence a full subcategory of kC. Another proof of the claim appears in Corollary 4.2.4 of [Bo], but only for subcategories of the biset category. However, the result and its proof hold for any small category.

Our second claim is that, conversely, the evaluation of a simple functor S_{E,R,V} at a finite set X is either zero or a simple R_{X}-module. A proof of this second claim again appears in Proposition 3.2 of [We], or in Corollary 4.2.4 of [Bo] (for subcategories of the biset category, but the result is again quite general). Moreover, S_{E,R,V} (X) is zero if |E| > |X| and is nonzero if |E| ≤ |X| because S_{E,R,V} (E) = T_{R,V} is nonzero, hence S_{E,R,V} (X) is also nonzero, by Corollary 4.4 in [BT2].

The given references also indicate that the simple functor S_{E,R,V} associated to a simple R_{X}-module S, i.e. such that S ≅ S_{E,R,V} (X), is unique up to isomorphism. This provides the parametrization of the statement and completes the proof.

Note that we used in the proof above the non-vanishing of S_{E,R,V} (X) when |E| ≤ |X|. This is a special property of correspondence functors (Corollary 4.4 in [BT2]) and it may not hold for representations of other small categories. For instance, it does not hold for the biset category (see [BST]).

In view of Theorem 4.1, a formula for the dimension of any simple R_{X}-module is now given by Theorem 3.2.

An explicit description can be given for every simple R_{X}-module and for the action of relations. This is rather technical and is an application of Theorem 18.4 in [BT2]. It can be summarized as follows. We fix a simple R_{X}-module S_{E,R,V} (X). As already noticed, we can associate to the poset (E, R) a finite lattice T having (E, R^{op}) as the full subset of its join-irreducible elements, and we can also choose T such that Aut(T) = Aut(E, R^{op}) by taking for instance T to be the lattice of all lower ideals in the poset (E, R^{op}). Then there is a correspondence functor F_{T} defined by F_{T} (X) = k(T^{X}), the k-vector space with basis the set T^{X} of all maps from X to T. The action of a correspondence U ∈ C(Y, X) on a map φ ∈ T^{X} is a map Uφ ∈ T^{Y} defined by

(Uφ)(y) = ⋁_{(y,x)∈U} φ(x) ,

and this makes F_{T} into a correspondence functor, by Proposition 11.2 in [BT2]. Define a subset

B_{E,R,X} = {φ ∈ T^{X} | E ⊆ φ(X) ⊆ G_{E,R}} ,

where G_{E,R} is defined as in Section 3. Note that the cardinality of B_{E,R,X} is given by the formula

|B_{E,R,X}| = ∑_{i=0}^{|E|}(−1)^{i}\binom{|E|}{i}(g_{E,R} − i)^{|X|} .

This gives an interpretation of the sum appearing in Theorem 3.2. The formula is proved in Lemma 8.1 of [BT2]. Let kB_{E,R,X} be the k-subspace of F_{T} (X) with basis B_{E,R,X}. The family of subspaces kB_{E,R,X} do not form a subfunctor of F_{T}, but they can be used to describe the evaluations of the simple functor S_{E,R,V}. Indeed, by Remark 4.8 and Corollary 17.17 in [BT2], we have

S_{E,R,V} (X) ≅ kB_{E,R,X} ⊗_{k Aut(E,R)} V ,

using the right action of Aut(E, R) on B_{E,R,X} defined as follows: given φ ∈ B_{E,R,X} and a permutation σ ∈ Aut(E, R), then σ yields an automorphism of T since Aut(E, R^{op}) = Aut(E, R) and we let φ · σ := σ^{−1} ∘ φ. This provides an explicit description of the simple R_{X}-module S_{E,R,V} (X) as a k-vector space. It remains to describe the action of every relation U ∈ C(X, X).

Let π_{T,X} be the k-linear idempotent endomorphism of k(T^{X}) defined by

∀φ ∈ T^{X}, π_{T,X}(φ) = \{ \begin{matrix}φ & if E ⊆ φ(X) , \\ 0 & otherwise . \end{matrix}

By Theorem 17.7 in [BT2], there is an element u_{T} ∈ k(T^{T}) which has the property that, for any φ ∈ T^{X}, the composition u_{T} ∘ φ is a k-linear combination of maps ψ ∈ T^{X} such that ψ(X) ⊆ G_{E,R}. (Actually, u_{T} is idempotent, by Theorem 17.9 in [BT2].) Then, for any map φ ∈ T^{X}, we obtain

π_{T,X}(u_{T} ∘ φ) ∈ kB_{E,R,X} ,

that is, a k-linear combination of maps ψ ∈ T^{X} such that E ⊆ ψ(X) ⊆ G_{E,R}. Now we can describe the action of relations on the simple R_{X}-module S_{E,R,V} (X).

4.2. Theorem. Fix the notation above.

(a) S_{E,R,V} (X) ≅ kB_{E,R,X} ⊗_{k Aut(E,R)} V as k-vector spaces.

(b) The action of a relation U ∈ C(X, X) on an element

φ ⊗ v ∈ kB_{E,R,X} ⊗_{k Aut(E,R)} V , (φ ∈ B_{E,R,X}, v ∈ V)

is given by

U · (φ ⊗ v) = π_{T,X}(u_{T} ∘ Uφ) ⊗ v .

The proof appears in Theorem 18.4 of [BT2]. Theorem 4.2 provides a computational method for the description of any simple R_{X}-module.

5. The Jacobson radical of the algebra of relations

In this section, we assume for simplicity that the field k has characteristic zero. Our purpose is to give the dimension of the Jacobson radical J(R_{X}) of the k-algebra R_{X}.

5.1. Theorem. Assume that k is a field of characteristic zero. Let J(R_{X}) be the Jacobson radical of the k-algebra R_{X} and let n = |X|. Then

dim J(R_{X}) = 2^{n^{2}} − ∑_{e=0}^{n} ∑_{R} \frac{1}{| Aut(E, R)|}( ∑_{i=0}^{e}(−1)^{i}\binom{e}{i}(g_{E,R} − i)^{n})^{2} ,

where R runs over a set of representatives of Σ_{e}-conjugacy classes of order relations on the set E = {1, …, e}. The integer g_{E,R} is as in Theorem 3.2.

Proof : Since k has characteristic zero, the semi-simple algebra R_{X}/J(R_{X}) is separable, that is, it remains semi-simple after scalar extension to an algebraic closure \overline{k} of k. In other words, dim J(R_{X}) does not change after this scalar extension. Therefore, we can assume that k = \overline{k}.

By Theorem 4.1, every simple R_{X}-module has the form S_{E,R,V} (X) with |E| ≤ |X|, where S_{E,R,V} is the simple correspondence functor parametrized by the triple (E, R, V). In order to have a parametrization, we take E = {1, …, e} with 0 ≤ e ≤ n, we take R in a set of representatives as in the statement, and finally we take V in a set of representatives of isomorphism classes of simple k Aut(E, R)- modules.

Since the endomorphism algebra of a simple module is isomorphic to k, by Schur’s lemma and the assumption that k is algebraically closed, the dimension of the semisimple algebra R_{X}/J(R_{X}) is equal to the sum of the squares of the dimensions of all simple modules, by Wedderburn’s theorem. By Theorem 3.2, we obtain

dim (R_{X}/J(R_{X})) = ∑_{E,R,V} ( dim S_{E,R,V} (X))^{2}

= ∑_{E,R,V} (\frac{dim V}{| Aut(E, R)|})^{2}( ∑_{i=0}^{|E|}(−1)^{i}\binom{|E|}{i}(g_{E,R} − i)^{|X|})^{2}

= ∑_{e=0}^{n} ∑_{R} ( ∑_{V} \frac{(dim V)^{2}}{| Aut(E, R)|^{2}} )( ∑_{i=0}^{e}(−1)^{i}\binom{e}{i}(g_{E,R} − i)^{n})^{2}
= ∑_{e=0}^{n} ∑_{R} \frac{1}{| Aut(E, R)|}( ∑_{i=0}^{e}(−1)^{i}\binom{e}{i}(g_{E,R} − i)^{n})^{2} ,

because ∑_{V}(dim V)^{2} = dim(k Aut(E, R)) = | Aut(E, R)|, by semi-simplicity of the

group algebra in characteristic zero (Maschke’s theorem). Now

dim J(R_{X}) = dim R_{X} − dim (R_{X}/J(R_{X})) = 2^{n^{2}} − dim (R_{X}/J(R_{X}))

and the result follows.

If k is an algebraically closed field of prime characteristic p, the formula has to be modified in a straightforward manner, in order to take into account the Jacobson radical of k Aut(E, R). Then it seems likely that the same formula holds over any field of characteristic p (that is, R_{X}/J(R_{X}) is likely to be a separable algebra), but we leave this question open.

6. Examples

In this final section, we provide examples of the algebra R_{X} for small values of n = |X|.

6.1. Example. Let X = ∅, hence n = 0. There is a single relation on ∅, namely ∅, and R_{∅} ≅ k. The relation ∅ has a trivial automorphism group, which has only the trivial module k as a simple module, and we obtain that B_{∅,∅,∅} has cardinality 1 and kB_{∅,∅,∅} ≅ k, so that

S_{∅,∅,k}(∅) ≅ kB_{∅,∅,∅} ⊗_{k} k ≅ k .

The unique relation ∅ acts as the identity on k.

6.2. Example. Let X = {1}, hence n = 1. There are 2 relations on {1}, namely ∅ and R = {1} × {1}, so R_{{1}} has dimension 2.

For E = ∅, there is a simple correspondence functor S_{∅,∅,k} and again B_{∅,∅,{1}} has cardinality 1. Thus we get a simple R_{{1}}-module

S_{∅,∅,k}({1}) ≅ kB_{∅,∅,{1}} ⊗_{k} k ≅ k .

Both relations act as the identity on k.

For E = {1}, there is a single essential relation R, with a trivial automorphism group, having only the trivial module k as a simple module. It turns out that g_{E,R} = 2 and |B_{{1},R,{1}}| = 2^{1} − 1^{1} = 1. Then there is a simple correspondence functor S_{{1},R,k} and we have a simple R_{{1}}-module

S_{{1},R,k}({1}) ≅ kB_{{1},R,{1}} ⊗_{k} k ≅ k .

The relation ∅ acts by zero, while R acts as the identity.

Since dim R_{{1}} = 2 and there are two simple modules, we have R_{{1}} ≅ k × k, a semi-simple algebra.

6.3. Example. Let X = {1, 2}, hence n = 2. There are 2^{4} = 16 relations on {1, 2}, so R_{{1,2}} has dimension 16.

For E = ∅, we get a simple R_{{1,2}}-module S_{∅,∅,k}({1, 2}), of dimension 1.

For E = {1}, there is a single essential relation R, with a trivial automorphism group, having only the trivial module k as a simple module. It turns out that g_{{1},R} = 2 and |B_{{1},R,{1,2}}| = 2^{2} − 1^{2} = 3. We get a simple R_{{1,2}}-module S_{{1},R,k}({1, 2}) of dimension 3.

For E = {1, 2}, there are two essential relations up to conjugacy, namely the equality relation eq and the usual total order tot. It turns out that g_{{1,2},eq} = 4 and B_{{1,2},eq,{1,2}} has cardinality

|B_{\{1,2\},eq,\{1,2\}}| = ∑_{i=0}^{2}(−1)^{i}\binom{2}{i}(g_{\{1,2\},eq} − i)^{2} = 4^{2} − 2 · 3^{2} + 2^{2} = 2 .

Moreover, Aut({1, 2}, eq) = C_{2} has order 2, with two simple modules k_{+} and k_{−} (assuming that the characteristic of k is not 2). Therefore, we obtain two simple R_{{1,2}}-modules

S_{{1,2},eq,k_{+}}({1, 2}) ≅ kB_{{1,2},eq,{1,2}} ⊗_{kC_{2}} k_{+} ,
S_{{1,2},eq,k_{−}}({1, 2}) ≅ kB_{{1,2},eq,{1,2}} ⊗_{kC_{2}} k_{−} ,

both of dimension 1.

For the other relation tot, we have g_{{1,2},tot} = 3 and B_{{1,2},tot,{1,2}} has cardinality

|B_{\{1,2\},tot,\{1,2\}}| = ∑_{i=0}^{2}(−1)^{i}\binom{2}{i}(g_{\{1,2\},tot} − i)^{2} = 3^{2} − 2 · 2^{2} + 1^{2} = 2 .

Moreover, Aut({1, 2}, tot) is the trivial group and has only the trivial module k as a simple module. We obtain a simple R_{{1,2}}-module

S_{{1,2},tot,k}({1, 2}) ≅ kB_{{1,2},tot,{1,2}} ⊗_{k} k ≅ kB_{{1,2},tot,{1,2}} ,

of dimension 2.

Altogether, there are 5 simple R_{{1,2}}-modules and the dimension of the semisimple quotient is

dim (R_{{1,2}}/J(R_{{1,2}})) = 1^{2} + 3^{2} + 1^{2} + 1^{2} + 2^{2} = 16 ,

so the Jacobson radical has dimension dim J(R_{{1,2}}) = 2^{4} − 16 = 0. Therefore R_{{1,2}} is semi-simple (provided the characteristic of k is not 2).

6.4. Example. For n = |X| = 3, the algebra R_{X} is not anymore semi-simple. Using the computer software GAP ([GAP4]), we have computed the dimension of the Jacobson radical of R_{X} and it is equal to 42. According to Theorem 5.1, this value can be recovered directly as follows:

Size e Poset(E, R) | Aut(E, R)| g_{E,R} ∑_{i=0}^{e}(−1)^{i}\binom{e}{i}(g_{E,R} − i)^{3} total
0  ∅  1  1  1  1
1  •  1  2  7  49
2  ••  2  4  18  162
\begin{matrix}• \\ • \end{matrix}  1  3  12  144
3  • • •  6  5  6  6
\begin{matrix}• \\ • \end{matrix} •  1  5  6  36

{• • \atop \/} ; •  2  5  6  18

• ; {/\ \atop • •}  2  5  6  18

• ; \begin{matrix}• \\ • \end{matrix}  1  6  36  36

In this case, the algebra R_{X} has dimension 2^{3^{2}} = 512. The sum of the last column of this table is equal to 470, so we recover the dimension of the radical 42=512-470.

6.5. Example. For |X| = 4, the algebra R_{X} has dimension 2^{4^{2}} = 65,536. The direct computation of the radical of such a big algebra seems out of reach of usual computers. However, using the formula of Theorem 5.1 and the structure of the 16 posets of cardinality 4, one can show by hand that the radical of R_{X} has dimension 32,616 when |X| = 4.

For larger values of n = |X|, a computer calculation using Theorem 5.1 yields the following values for the dimension of J(R_{X}):

\begin{matrix}n = 5 & n = 6 & n = 7 & n = 8 \\ 29,446,050 & 67,860,904,320 & 562,649,705,679,642 & 18,446,568,932,288,588,616 \end{matrix}

We leave the reader compare these values to the values dim R_{X} = 2^{n^{2}} and deduce the dimension of the semi-simple algebra R_{X}/J(R_{X}).

References

[Bo] S. Bouc. Biset functors for finite groups, Springer Lecture Notes in Mathematics no. 1990 (2010).

[BST] S. Bouc, R. Stancu, J. Thévenaz. Vanishing evaluations of simple functors, J. Pure Appl. Alg. 218 (2014), 218–227.

[BT1] S. Bouc, J. Thévenaz. The algebra of essential relations on a finite set, J. reine angew. Math., to appear.

[BT2] S. Bouc, J. Thévenaz. The representation theory of finite sets and correspondences, preprint, 2015.

[GAP4] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.7.8, 2015.

 (http://www.gap-system.org).

[Gr] J.A. Green. Polynomial representations of GL_{n}, Springer Lecture Notes in Mathematics no. 830 (1980).

[We] P. Webb. An introduction to the representations and cohomology of categories, in: M. Geck, D. Testerman, J. Thévenaz (Eds.), “Group Representation Theory”, EPFL Press, Lausanne, 2007, pp. 149–173.

Serge Bouc, CNRS-LAMFA, Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France.

serge.bouc@u-picardie.fr

Jacques Thévenaz, Section de mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland.

Jacques.Thevenaz@epfl.ch