. 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 symmetrygroups. 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  (see also chapter by Schweizer).
Almost algebraic actions of algebraic groupsand 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 ].
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 representationsand 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.
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 ). See also the work of the physicist S.C. Creagh in . 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:
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 , ) for high energy asymptotics, and later by El Houakmi and Helffer in the semi-classical setting (see , ). 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  and ).
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.
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.
Let us observe that the existence of a decomposition as a sum of finitely many irreducible representationsand 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-
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.
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.
Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP ∗ , LJK, 38000 Grenoble, France
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
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  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 .
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 groupand 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.
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
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.
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
Proof: This is basically the same as Behrend-Noohi  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 , so π 2 (Y ) = 0. In particular, the long