. Attempt will be made in the following to provide an intuitive insight. More detailed **and** technical descriptions may be consulted in the refs. [7–11]. Section 3.1 discusses at the most elementary level the few concepts necessary to understand or to perform analysis of finite **group** linear **representations** with the help of already tabulated or computer generated irreducible **representations** [6, 8–13]. The method is applied to molecule vibrations. Section 3.2 provides a qualitative description of the irreducible **representations** of the 3-dimensional crystal **symmetry** **groups**. No time transform is considered. Notice that if the physical property is dynamical but is not slaved to an external explicitly time dependent field then it ought to be stationary if it is at thermodynamic equilibrium, in which case it does not depend on an initial time, in particular dynamical correlations then involve only time intervals. When time reversal is possibly relevant, it is dealt with in the context of co-**representations** of **groups** [14] (see also chapter by Schweizer).

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Almost algebraic actions of algebraic **groups** **and** applications to algebraic **representations**
Uri Bader, Bruno Duchesne, **and** Jean Lécureux
Abstract. Let G be an algebraic **group** over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic **and** the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic **representations** of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis–Zimmer super-rigidity phe- nomenon [ 2 ].

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In fact we prove a stronger statement that provides a criterion for the surjectivity of the iterated monodromy **group** attached to an even degree polynomial 𝑓 in terms of the critical orbit of 𝑓 . Jamie Juul [28, Proposition 3.2.] proves a version of Theorem 2.1.5 for finite level arboreal **representations** **and** polynomials of arbitrary degree. However, we do not know if there are many polynomials for which Juul’s result implies surjectivity of the infinite level arboreal representation is large. Our result shows that when 𝐹 is a number field most polynomials have surjective infinite level iterated monodromy **groups**, see Remark 2.3.3.

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one of same period, which creates tubes of periodic orbits with same period **and** doesn’t match with a non degenerate situation.
The author expects that one can calculate the density d(t 0 , z, g) in terms of primitive period,
Maslov index, **and** of the energy restricted Poincar´e map of the periodic orbit ¯ γ. This seems to be a non-trivial calculus we have been able to complete only for finite **groups** for the moment (see [7]). See also the work of the physicist S.C. Creagh in [12]. If one omits the assumption of non-degeneracy, under hypothesis of ‘G-clean flow’ (see Definition 4.4), we still get an asymptotic expansion, which depends on the connected components of the set:

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in the early 80’s for globally elliptic pseudo-differential operators, both in cases of compact finite **and** Lie **groups**, by Helffer **and** Robert (see [16], [17]) for high energy asymptotics, **and** later by El Houakmi **and** Helffer in the semi-classical setting (see [10], [11]). Main results were then given in terms of reduced asymptotics of Weyl type for a counting function of eigenvalues of the operator. Here, in a semi-classical study with a finite **group** of **symmetry**, we want to go one step beyond Weyl formulae, investigating oscillations of the spectral density, **and** establishing a Gutzwiller formula for the reduced quantum Hamiltonian. The case of a compact Lie **group** will be carried out in another paper (see [4] **and** [5]).

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For the **group** SO 0 (2, 2), Alessandrini **and** Li [ AL15 ] used Higgs bundle techniques
to construct anti-de Sitter structures on circle bundles over Σ, recovering a result of Salein **and** Guéritaud-Kassel [ Sal00 , GK17 ].
Length spectrum of maximal **representations** in rank 2. Some Anosov rep- resentations of surface **groups**, such as Hitchin **representations** into real split Lie **groups** or maximal **representations** into Hermitian Lie **groups**, have the additional property of forming connected components of the whole space of **representations**. There have been several attempts to propose a unifying characterization of these **representations** (see [ MZ16 ] **and** [ GW17b ]). Note that quasi-Fuchsian representa- tions into PSL(2, C) do not form components; indeed, they can be continuously deformed into **representations** with non-discrete image.

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It turns out that there exist rigid points (i.e. isolated points in the (decorated) unipotent representation variety) together with non-rigid components.
There is a natural map h from the (decorated) representation variety of M to the representation variety of its boundary. It is known that its image is a Lagrangian subvariety **and** the map is a local isomorphism on a Zariski-open set. Our remark 5.8 proves in a combinatorial way these facts. When M is a knot complement **and** one considers the **group** SL(2, C) instead of SL(3, C), this image is the algebraic variety defined by the famous A-polynomial of the knot. In this paper, we explore more precisely the map h **and** exhibit a complicated fiber.

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Let us observe that the existence of a decomposition as a sum of finitely many irreducible **representations** **and** a unitary one is known for any **group** as soon as there is no totally isotropic invariant space [ Ism66 , Sas90 ]. Moreover, this theorem extends the results of [ Na˘ı63 ] where it is proved for SL 2 ( C ) . The strategy to prove Theorem 1.1 is to use the aforemen- tioned mention [ Duc15b , Theorem 1.2] for lattices **and** extended it to the whole ambiant **group**. This strategy may seem surprising since the ambiant Lie **group** has much more structure than its lattices. In particular, it has a differentiable structure. The topology used on PO K ( p, ∞ ) is the coarsest that makes the action on the symmetric space X K ( p, ∞ ) con-

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Let κ : Γ F → Z × p be the p-adic cyclotomic character **and** κ be its reduction modulo p. We will always assume p 6= 2, **and** our main theorems will make stronger hypotheses on p.
We recall here some deformation-theoretic terminology. Given a topo- logically finitely-generated profinite **group** Γ, a finite extension E/Q p with ring of integers O **and** residue field k, a reductive algebraic **group** G de- fined over O **and** a continuous homomorphism r : Γ → G(k), let R O,¯ r be the universal lifting ring representing the functor sending a complete local noetherian O–algebra R with residue field k to the set of lifts r : Γ → G(R) of r. We will always leave the O implicit, writing only R ¯ r , **and** at various points in the argument we enlarge O; see [3, Lemma 1.2.1] for a justification of (the harmlessness of) this practice. We write R ¯ r ⊗ Q p for R O,¯ r ⊗ O Q p for any particular choice of O, **and** again by [3, Lemma 1.2.1], R r ¯ ⊗ Q p is independent of the choice of E.

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In an oligopoly setting, the impact of more uniform prices is ambiguous **and** depends on demand **symmetry**. Demand is symmetric when companies agree about the ranking of high-price markets **and** low-price markets - the strong markets **and** the weak markets. If demand is symmetric, the situation is similar to price discrimination in the monopoly case, a shift to more uniform prices decreases the average price in the strong market **and** increases the average price in the weak market - Corts (1998), Stole (2007), Armstrong (2006). However, these findings do not take into account the possible changes in the bar- gaining powers. To the best of my knowledge, Grennan (2013) is the only paper studying the impact of hospital **group** purchasing using actual data on transaction between firms **and** hospitals. The author study the market for medical devices - the coronary stent in- dustry - in the United States by simulating hospital mergers. The author points out that hospitals’ demands for stents are asymmetric **and** that consequently, more uniform pricing would soften competition **and** increase prices. Therefore, there must be an important gain in bargaining power to compensate this effect **and** for prices to be lower with **group** pur- chasing.

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Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP ∗ , LJK, 38000 Grenoble, France
Abstract
The success of deep convolutional architectures is often attributed in part to their ability to learn multiscale **and** invariant **representations** of natural signals. However, a precise study of these properties **and** how they affect learning guarantees is still missing. In this paper, we consider deep convolutional **representations** of signals; we study their invariance to translations **and** to more general **groups** of transformations, their stability to the action of diffeomorphisms, **and** their ability to preserve signal information. This analysis is carried by introducing a multilayer kernel based on convolutional kernel networks **and** by studying the geometry induced by the kernel mapping. We then characterize the corresponding reproducing kernel Hilbert space (RKHS), showing that it contains a large class of convolutional neural networks with homogeneous activation functions. This analysis allows us to separate data representation from learning, **and** to provide a canonical measure of model complexity, the RKHS norm, which controls both stability **and** generalization of any learned model. In addition to models in the constructed RKHS, our stability analysis also applies to convolutional networks with generic activations such as rectified linear units, **and** we discuss its relationship with recent generalization bounds based on spectral norms. Keywords: invariant **representations**, deep learning, stability, kernel methods

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Q ′ of Q such that α(A) ⊂ A ⊗ Q ′ . Furthermore if τ is a continuous linear functional on A, we say
that α preserves τ if (τ ⊗ id Q ) ◦ α = τ (·)1 Q .
2. Quantum **groups** actions on Hilbert modules
We recall now the deﬁnition of an action of a compact quantum **group** on a Hilbert module (see [11] for background material on Hilbert modules). Then we introduce the notion of orthogonal ﬁltration on a Hilbert module, give some natural examples of such objects, **and** deﬁne what we mean by

2.4. Tetrahedral complexes
Because the T, O **and** I **groups** all have multiple C 3 axes, a natural way to build complexes with these symmetries is to begin by making C 3 trimers, **and** then to assemble an appropriate number of trimeric copies to build the ﬁnal complex. In particular, a tetra- hedron has one threefold **symmetry** axis about each of four lines joining the four face centres **and** vertices, **and** one twofold rotational **symmetry** axis through each of three lines joining pairs of opposite edges. The dihedral angle between two faces is ¼ cos 1 ð1=3Þ ¼ 70:53 . Seen from the origin, the angle between any two face centres is ¼ 109:47 , which is the classical tetrahedral bond angle in methane, for example. The angle between the base **and** the fourth vertex is half of this angle, i.e. ¼ ð Þ=2 ¼ sin 1 ð2 1 =2 =3 1 =2 Þ ¼ 54 :74 .

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progress may let one hope that it is also possible with a larger class of nitely presented **groups**.
The propositions are true for any lattice in any real Lie **group** **and** do not resort to any specic geometric technique like decomposition into pairs of pants or FenchelNielsen parameters. They deal with uniform **and** non uniform lattices simultaneously on the contrary to the theorems of Bowen. There is no limitation on the representation ρ. Besides there is a signicant dierence with the point of view presented by Bowen, since passing to a nite-index subgroup is not needed anymore.

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1980s, Atiyah **and** Bott gave a new impulsion (see [AB83]) to the subject by identifying these spaces as the moduli spaces of flat connections on principal bundles of **group** U on Σ g,l , thereby revealing the
importance of the representation varieties in gauge theory. These spaces also arise in differential Galois theory **and** in operator algebra theory. Finally, it is possible to use these spaces to construct deformations of discrete subgroups of Lie **groups** (see for instance [MG88]). The diversity of the fields which these representation spaces are attached to justifies the fact that they are such an important object of study **and** that their geometry should be investigated. One may for instance find an introduction to the study of geometric structures of moduli spaces in [Gol88]. As for us, we shall focus our attention on studying the symplectic structure of some of these representation spaces. This symplectic structure can be obtained **and** described in a wide variety of ways (see for instance [GHJW97, AM95, AMM98, MW99]), each of which has its own advantages. The description given by Alekseev, Malkin **and** Meinrenken in [AMM98] is in our sense particularly well-suited for studying **representations** of π g,l . This description rests on the notion of

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real hyperbolic plane) are fully prescribed by p, q **and** r (up to conjugation), whereas
in PU(2, 1) (which is the transformation **group** of the complex hyperbolic plane) an additional parameter controls the representation. This additional parameter can be interpreted as follows. One can always set two vertices of the triangle in a same real plane of H 2 C . The last vertex has to be placed at the intersection of the complex geodesic lines issued from the previous vertices. That intersection is a one- dimensional topological space **and** it represents the possible values of the additional parameter. Only one of those values corresponds to the case where the last vertex lies in the same previous real plane (**and** therefore corresponds to a R-Fuchsian representation). This parameter is called the angular invariant.

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maine Universitaire, 38402- Saint Martin d’H`eres, France.
Abstract. We present a construction of **symmetry** plane-**groups** for quasiperi-
odic point-sets named beta-lattices. The framework is issued from beta- integers counting systems. Beta-lattices are vector superpositions of beta-integers. When β > 1 is a quadratic PisotVijayaraghavan alge- braic unit, the set of beta-integers can be equipped with an abelian **group** structure **and** an internal multiplicative law. When β = (1 + √

In this article we study the problem of finding such finite **groups** that the modular forms associated with all elements of these **groups** by means of a certain faithful representation belong to a special class of modular forms (so-called multiplicative η-products). This problem is open: all such **groups** have not been found. It will be interesting to find the complete classification. G.Mason gave an example of such a **group**: the **group** M 24 .

In the ﬁrst part, we deal with the case when the Lie **group** is locally isomorphic to the **group** of isometries of the hyperbolic plane. In this case, integral maximal homomorphisms induce hyperbolizations of the initial surface **and** we relate them to spin structures on Riemann surfaces, that is to line bundles whose tensor power is isomorphic to the tensor product of the canonical bundle **and** a given divisor. Fixing such an integral maximal representation, we associate to each geodesic an integer modulo a ﬁxed integer, its translation

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Proof: This is basically the same as Behrend-Noohi [10] Proposition 4.7.
If X is already an orbicurve then the generic stabilizer **group** G is trivial, BG = Spec(C) **and** Y = X.
In general, let Z be the coarse moduli space for X. It is a normal (hence smooth) curve. Let Y be the orbicurve obtained by setting, for each point z ∈ Z, the orbifold structure at z to be the ramification degree of the map X → Z over z (this is different from 1 for only a finite number of points). There is a unique factorization to a map p : X → Y (note that there is no question of natural transformations here because p is submersive **and** the generic stabilizers on Y are trivial). The map p is etale by the choice of orbifold indices. An etale covering is a fibration in the etale topology, **and** the fiber over a point (say a general point) is a one-point DM-stack, hence of the form BG for a finite **group** G. We obtain a long exact sequence in homotopy. If X is elliptic or hyperbolic, then by definition the same is true of Y . Thus the universal covering of Y is C or a disk [10], so π 2 (Y ) = 0. In particular, the long

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