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We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of the geometric Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. Using results on the reduction of dynamical Belyi maps modulo certain primes, we obtain results on the corresponding arithmetic Galois groups of iterates. These lead to results on the behavior of the arithmetic Galois groups under specialization, with applications to dynamical sequences.

Exploiting several ℓ-factorization results for strictly singular operators, we study the strict singularity of the multiplication operator LR: T → ATB on ℒ(X) for various Banach spaces X., The second-named author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness through grants MTM2016-75196-P, MTM2016-76808-P, and the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). Support of an LMS Research in Pairs grant is also gratefully acknowledged.

In this note we compute the cohomology of the elliptic tangent bundle, a Lie algebroid used to describe singular symplectic forms arising from generalized complex geometry., 16 pages

A coaction of a Hopf algebra on a unital algebra is called homogeneous if the algebra of coinvariants equals the ground field. A coaction of a Hopf algebra on a (not necessarily unital) algebra is called Galois, or principal, or free, if the canonical map, also known as the Galois map, is bijective. In this paper, we establish a duality between a particular class of homogeneous coactions, up to equivariant Morita equivalence, and Galois coactions, up to isomorphism., Comment: 24 pages

In this paper, we develop a semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consists of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensioned representations of the group, even though they are negligible with respect to the Plancherel measure., Comment: 29 pages

Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of the disk ramified over $n = 2g + 1$ points. The main part of such homology is described by the homology of the braid group with coefficients in a symplectic representation, namely the braid group $\mathrm{Br}_n$ acts on the first homology group of a genus $g$ surface via Dehn twists. Our computations shows that such groups have only $2$-torsion. We also investigate stabilization properties and provide Poincar\'e series, both for unstable and stable homology., Comment: 26 pages, missing references added, Table 1 completed with missing computations

Let G be a group and let x ε G be a left 3-Engel element of odd order. We show that x is in the locally nilpotent radical of G. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel’manov. We also give some applications of our main result. In particular, for any given word w = w(x1, . . ., xn) in n variables, we show that if the variety of groups satisfying the law w 3 = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (x 3 n +1 w 3 ) 3 = 1.

We consider generic 2 x 2 singular Liouville systems on a smooth bounded domain in the plane having some symmetry with respect to the origin. We construct a family of solutions to which blow-up at the origin and whose local mass at the origin is a given quantity depending on the parameters of the system. We can get either finitely many possible blow-up values of the local mass or infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials., 35 pages, accepted on Rev. Mat. Iberoam

Let H be a quasi-Hopf algebra. We apply results obtained in [8] to give necessary and sufficient conditions for the forgetful functor from Doi–Hopf modules, two-sided Hopf modules or Yetter–Drinfeld modules over H to representations of the underlying algebra to be Frobenius (resp. separable). We show that in some situations these conditions reduce to the unimodularity and/or (co)semisimplicity of the quasi-Hopf algebra H.

We establish new approximation results, in the sense of Lusin, of Sobolev functions by Lipschitz ones, in some classes of non-doubling metric measure structures. Our proof technique relies upon estimates for heat semigroups and applies to Gaussian and RCD(K,∞) spaces. As a consequence, we obtain quantitative stability for regular Lagrangian flows in Gaussian settings. We establish new approximation results, in the sense of Lusin, of Sobolev functions by Lipschitz ones, in some classes of non-doubling metric measure structures. Our proof technique relies upon estimates for heat semigroups and applies to Gaussian and RCD(K,∞) spaces. As a consequence, we obtain quantitative stability for regular Lagrangian flows in Gaussian settings.

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over Q whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems., Comment: 35 pages. Added more details and fixed typos. Final version

In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $${H^\infty}$$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$ where $${e_{\ell}, {\ell} = 1, 2, 3}$$ are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables $${x = (x_{1}, x_{2}, x_{3})}$$ and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version $${T^{\alpha}, {\rm for} \alpha \in (0, 1)}$$ , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.

Let r : X 2 → X 2 {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra K ⁢ 〈 x ∈ X ∣ x ⁢ y = u ⁢ v ⁢ if ⁢ r ⁢ ( x , y ) = ( u , v ) 〉 {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions r B {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

Let $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih's construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne., Comment: 16 pages

The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge bundle over $A_g$ has its Chern classes in the Chow ring of $A_g$ with rational coefficients. We show that over the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so in the best possible way: despite the highly singular nature of $A_g^*$ they are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements in its bivariant Chow ring. This is in contrast to the situation in the analytic topology, where these Chern classes have canonical lifts to the complex cohomology of the minimal compactification as Goresky-Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology., Comment: To appear in the Tunisian Journal of Mathematics

Aim of this paper is to prove the second order differentiation formula for $H^{2,2}$ functions along geodesics in $RCD^*(K,N)$ spaces with $N < \infty$. This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity. We establish this result by showing that $W_2$-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolation. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain: - equiboundedness of the densities along the entropic interpolations, - local equi-Lipschitz continuity of the Schr��dinger potentials, - a uniform weighted $L^2$ control of the Hessian of such potentials. Finally, the techniques adopted in this paper can be used to show that in the $RCD$ setting the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality., arXiv admin note: substantial text overlap with arXiv:1701.03932

In this paper we study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that H\"ormander type conditions on the Fourier multipliers imply $L^p$-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper., Comment: 23 pages

We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holonomy group causes a decomposition of the twisted cohomology. For SKT manifolds this decomposition is accompanied by an identity between different Laplacian operators and equates different cohomologies defined in terms of the SKT structure. We illustrate our theory with examples based on Calabi–Eckmann manifolds, instantons, Hopf surfaces and Lie groups.

Entwined modules over cowreaths in a monoidal category are introduced. They can be identified to coalgebras in an appropriate monoidal category. It is investigated when such coalgebras are Frobenius (resp. separable), and when the forgetful functor from entwined modules to representations of the underlying algebra is Frobenius (resp. separable). These properties are equivalent when the unit object of the category is a $\otimes$-generator., 34 pages

Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field $k$ of characteristic $p$. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic three-dimensional principally polarised supersingular abelian variety over $k$ of characteristic $\neq 2$ has automorphism group $\{ \pm 1 \}$., published version, 39 pages

We give explicit definitions of the Weierstrass elliptic functions $\wp$ and $\zeta$ in terms of pfaffian functions, with complexity independent of the lattice involved. We also give such a definition for a modification of the Weierstrass function $\sigma$. As immediate applications, we give an explicit uniform zero estimate for $\wp$ and answer a question of Corvaja, Masser and Zannier on additive extensions of elliptic curves., Comment: 38 pages. Fixed a typo, and slightly expanded the introduction. Comments welcome!

As well known, classical catenoids in R 3 possess logarithmic growth at infinity. In this note we prove that the case of nonlocal minimal surfaces is significantly di¤erent, and indeed all nonlocal catenoids must grow at least linearly. More generally, we prove that stationary sets for the nonlocal perimeter functional which grow sublinearly at infinity are necessarily half-spaces.

Given a compact constant scalar curvature Kahler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kahler Ricci-flat resolution, we find sufficient conditions on the position of the singular points to ensure the existence of a global constant scalar curvature Kahler desingularization. We also give complete proofs of a number of analytic results which have been used in this context by various authors. A series of explicit examples is discussed.

The infinite relaxations in integer programming were introduced by Gomory and Johnson to provide a general framework for the theory of cutting planes: the so-called valid functions, and in particular the minimal and extreme functions, can be seen as automatic rules for the generation of cuts. However, while many extreme functions are piecewise linear and therefore easy to describe, the set of extreme functions turns out to have a very complicated mathematical structure, as several extreme functions are known that exhibit a somewhat pathological behavior. In this paper we show that if some smoothness assumption is imposed on an extreme function $$\pi $$, then $$\pi $$ is necessarily piecewise linear. More precisely, we show that if a continuous extreme function for the Gomory–Johnson one-dimensional infinite group relaxation is a piecewise $${\mathcal {C}}^2$$ function, then it is a piecewise linear function.

The geometric structure of the nonparametric statistical model of all positive densities connected by an open exponential arc and its intimate relation to Orlicz spaces give new insights to well-known financial objects which arise in exponential utility maximization problems.

We consider some fully nonlinear degenerate elliptic operators and we investigate the validity of certain properties related to the maximum principle. In particular, we establish the equivalence between the sign propagation property and the strict positivity of a suitably defined generalized principal eigenvalue. Furthermore, we show that even in the degenerate case considered in the present paper, the well-known condition introduced by Keller–Osserman on the zero-order term is necessary and sufficient for the existence of entire weak subsolutions.

When the base ring is not a field, power reductivity of a group scheme is a basic notion, intimately tied with finite generation of subrings of invariants. Geometric reductivity is weaker and less pertinent in this context. We give a survey of these properties and their connections., 6 pages; Dedicated to the memory of T.A. Springer

In the literature, the existence of Darboux polynomials and additional polynomial first integrals has been considered in the case of Hamiltonian systems. In this article such problem is formulated in the more general framework of Poisson structures, which include Hamiltonian systems as a particular case. This generalization allows a natural extension of the previous results, which can now be applied to a larger class of vector fields and is valid for arbitrary diffeomorphisms (instead of canonical transformations). Examples are discussed.

We discuss the optimality of Kahane’s multidimensional Ingham type theorem and we establish a relationship between the absolute continuity of Bernoulli convolutions and some results on expansions in non-integer bases.

It is known that if a group contains an abelian subgroup of finite index, then it also has an abelian characteristic subgroup of finite index. The aim of this paper is to prove that corresponding results hold when abelian subgroups are replaced either by subgroups having a modular subgroup lattice or by quasihamiltonian subgroups.

We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix $A$ by $sM_n(A)=trace(A^3)$. The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent $\omega$., Comment: 14 pages + appendix of 3 pages with numerical decompositions

In [1] , even if f only belongs to L 1 ( Ω ) , we proved that the assumption | f ( x ) | ≤ Q a ( x ) ∈ L 1 ( Ω ) , Q ∈ R + , implies the existence of a bounded solution of boundary value problems, whose simplest example is the linear problem { − div ( M ( x ) ∇ u ) + a ( x ) u = f ( x ) , in Ω ; u = 0 , on ∂ Ω ; where Ω is a bounded open set of R N and M is a bounded elliptic matrix. In this paper, we continue the study of the regularizing effect on the solutions, in some nonlinear Dirichlet problems, when we have L q interaction between the coefficient a ( x ) and the datum f ( x ) (i.e. Q is now a function in L q ( Ω ) ).

In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup. If U is any semigroup, and A is a subset of U, then we denote by the least subsemigroup of U containing A. If B is any other subset of U, then, roughly speaking, the first algorithm we present describes how to use any information about , that has been found using the Froidure-Pin Algorithm, to compute the semigroup . More precisely, we describe the data structure for a finite semigroup S given by Froidure and Pin, and how to obtain such a data structure for from that for . The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm. Postprint

It is well-known that the Jacobson radical of a unital ring is its largest superfluous right ideal. It is recorded here that the result carries over to non-unital rings provided the notion of "superfluous" is taken relative to all regular ideals., 3 pages

In this paper, we will prove that the six-dimensional Gopel variety in P134 is generated by 120 linear, 35 cubic, and 35 quartic relations. This result was already obtained in [Ren et al. 1...

A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a Lie algebroid which allows us to view such structures as symplectic forms. This allows us to construct new examples of stable structures, and also to define period maps for their deformations in which the background three-form flux is either fixed or not, proving the unobstructedness of both deformation problems. We then use the same tools to establish local normal forms for the degeneracy locus and for Lagrangian branes. Applying our normal forms to the four-dimensional case, we prove that any compact stable generalized complex 4-manifold has a symplectic completion, in the sense that it can be modified near its degeneracy locus to produce a compact symplectic 4-manifold.

We give conditions on the exponent function $p(\cdot)$ that imply the existence of embeddings between grand, small and variable Lebesgue spaces. We construct examples to show that our results are close to optimal. Our work extends recent results by the second author, Rakotoson and Sbordone., Comment: Final version to appear in Math. Notes

Minkowski’s question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in the sense of Ullman–Saff–Stahl–Totik and moreover that it belongs to a Nevai class; we provide numerical evidence of the validity of these conjectures. In addition, we study the zeros of its orthogonal polynomials and the associated Christoffel functions, for which asymptotic formulae are derived. As a by-product, we compute upper and lower bounds to the Hausdorff dimension of Minkowski’s measure. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Mobius maps.

Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperkählerian manifolds as stated by Verbitsky. We extract the correct statement and give a short proof of it. We also revisit a few of its consequences, some of which are given new (shorter) proofs, and ask some questions.

We prove structure results for the radial solutions of the semilinear problem Δ ⁢ u + λ ⁢ ( | x | ) | x | 2 ⁢ u + f ⁢ ( u ⁢ ( x ) , | x | ) = 0 , \Delta u+\frac{\lambda(|x|)}{|x|^{2}}u+f(u(x),|x|)=0, where λ is a function and f is superlinear in the u-variable. As particular cases, we are able to deal with Matukuma potentials and with nonlinearities f having different polynomial behaviors at zero and at infinity. We give the complete picture for the subcritical, critical and supercritical cases. The technique relies on the Fowler transformation, allowing to deal with a dynamical system in ℝ 3 {{\mathbb{R}}^{3}} , for which elementary invariant manifold theory allows to draw the conclusions involving regular/singular and fast/slow-decay solutions.

We extend known results concerning the centre of spaces of regular (resp. weakly compact or compact) operators between two Banach lattices to the setting of L-weakly compact and M-weakly compact operators. We also show that the L-weakly compact, M-weakly compact, and compact operators lying in the centre of a Banach lattice coincide. Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK); Research Foundation of Namik Kemal UniversityNamik Kemal University [NKUBAP.01.GA.17.108] Author E. Bayram was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) in the context of 2219-Post Doctoral Fellowship Program and by the Research Foundation of Namik Kemal University (Project No. NKUBAP.01.GA.17.108).

Using the notion of generalized divisors introduced by Hartshorne, we adapt the theory of adjoint forms to the case of Gorenstein curves. We show an infinitesimal Torelli-type theorem for vector bundles on Gorenstein curves. We also construct explicit counterexamples to the infinitesimal Torelli claim in the case of a reduced reducible Gorenstein curve., 17 pages

We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups $\mathcal {N}(G)$ has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups.

We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. (ii) An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete. (iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups. (iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.

Let $\beta\in(1,2)$ and let $H_\beta$ denote Garsia's entropy for the Bernoulli convolution $\mu_\beta$ associated with $\beta$. In the present paper we show that $H_\beta>0.82$ for all $\beta \in (1, 2)$ and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varj\'u, this yields $\dim (\mu_\beta)\ge0.82$ for all $\beta\in(1,2)$. In addition, we show that if an algebraic $\beta$ is such that $[\mathbb{Q}(\beta): \mathbb{Q}(\beta^k)] = k$ for some $k \geq 2$, then $\dim(\mu_\beta)=1$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself., Comment: 8 pages, no figures

A group G is said to be a V-group if every subnormal subgroup of G has only finitely many conjugates. It is proved here that if G is a group admitting an ascending normal series whose factors have finite rank, and all proper subgroups of G have the V-property, then G itself is a V-group, provided that G belongs to a suitable class of generalized soluble groups, containing in particular all locally (soluble-by-finite) groups. On the other hand, an example shows that there exist periodic metabelian minimal non-V groups.

We give an introduction to non-abelian Hodge theory for curves with the aim of stating the \(P = W\) conjecture both in its original cohomological version and in the more recent geometric one, and proposing a strategy to relate the two conjectures.

We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus Conjecture for the group $\operatorname{PSL}(2,19)$. We also prove the Zassenhaus Conjecture for $\operatorname{PSL}(2,23)$. In a second application we show that there are no normalized units of order $6$ in the integral group rings of $M_{10}$ and $\operatorname{PGL}(2,9)$. This completes the proof of a theorem of W. Kimmerle and A. Konovalov that the Prime Graph Question has an affirmative answer for all groups having an order divisible by at most three different primes., [v3]: 22 pages, extended outline of the new method, typos corrected

In the present paper we introduce a notion of the \(\mathbb {K}\)-Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the \(\mathbb {K}\)-Riemann integral and the convexity notion is replaced by \(\mathbb {K}\)-convexity.

Recently the method based on irreducible representations of finite groups has been proposed as a tool for investigating the more sophisticated versions of Bell inequalities (V. Ugǔr Gűney, M. Hillery, Phys. Rev. A90, 062121 ([2014]) and Phys. Rev. A91, 052110 ([2015])). In the present paper an example based on the symmetry group S 4 is considered. The Bell inequality violation due to the symmetry properties of regular tetrahedron is described. A nonlocal game based on the inequalities derived is described and it is shown that the violation of Bell inequality implies that the quantum strategies outperform their classical counterparts.