Nonequivalent Representations Research Articles

Generic length functions on countable groups

Let L(G) denote the space of integer-valued length functions on a countable group G endowed with the topology of pointwise convergence. Assuming that G does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on G is a word length and the associated Cayley graph is isomorphic to a certain universal graph U independent of G. On the other hand, we show that every comeager subset of L(G) contains 2ℵ0 “asymptotically incomparable” length functions. A combination of these results yields 2ℵ0 pairwise non-equivalent regular representations G→Aut(U). We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of G on L(G) by conjugation plays a crucial role in the proof of the latter result.

On Cayley representations of central Cayley graphs over almost simple groups

A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. We prove that every central Cayley graph over a simple group G has at most two pairwise nonequivalent Cayley representations over G associated with the subgroups of $${{\,\mathrm{Sym}\,}}(G)$$ induced by left and right multiplications of G. We also provide an algorithm which, given a central Cayley graph $$\Gamma $$ over an almost simple group G whose socle is of a bounded index, finds the full set of pairwise nonequivalent Cayley representations of $$\Gamma $$ over G in time polynomial in size of G.

On Cayley representations of finite graphs over Abelian $p$-groups

We construct a polynomial-time algorithm which given a graph $\Gamma$ finds the full set of non-equivalent Cayley representations of $\Gamma$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies that the recognition and the isomorphism problems for Cayley graphs over $D$ can be solved in polynomial time.

On a class of unitary representations of the braid groups B3 and B4

We describe a class of irreducible non-equivalent unitary representations of the braid group B3 in every dimension n≥6 which depends continuously on n2/6+1 real parameters. We show that the upper bound on the number of the parameters of which the class of irreducible non-equivalent unitary representations of B3 depends smoothly is equal to n2/4+2. The proof is achieved by a construction of such a class. We also prove that the tensor product of the Burau unitarisable representation of B4 and the irreducible unitary representation of B4 that coincide on commuting standard generators always forms irreducible unitary representations for the braid group B4. This gives a new class of unitary representations for the braid group B4 in 3n dimensions.

Unraveling complex relations between forest-cover change and conflicts through spatial and relational analyses

Despite the increasing prevalence of forest-cover change and conflicts, most studies have been unable to unravel the complex relations between the two processes. We attribute this failure to methodological limitations. We put forward an alternative approach that combines different datasets (remote sensing, GIS, local narratives, official censuses, newspaper articles), methods (spatial and relational analyses), and scales (subregions, economic sectors, land-based activities) to create a robust explanation of the relations between different intensities of forest-cover change and conflict in the Meseta Purepecha region, central Mexico. This is an important forest region, inhabited by indigenous and mestizo peasants; it has a worldwide reputation for community forestry and is also the epicenter of international avocado production. Forest-cover change is intense and there are recurrent episodes of conflict. We clustered communities in three subregions according to their patterns of forest-cover change. We analyzed the spatial patterns of forest-cover change and conflicts and we characterized the structure and function of the different economic sectors to unravel the nonlinear, interdependent (and sometimes contradictory) relations among these processes. We found that avocado production has differentially shaped the composition and working of society within each subregion, leading to three diverging patterns. Avocado production has provoked conflicts over landownership and over illegal logging in nearby areas. In some areas, a low incidence of conflicts over forest clearance might be explained by high profits, coercion, and violence. We suggest that, by combining spatial and relational analyses, we can integrate and check the congruence of nonequivalent representations from quantitative sources and observant participation at different scales and explain the heterogeneity that processes display across space. Our methodological approach can thus improve our understanding of similar and other complex and uncertain environmental problems elsewhere, especially when accurate or appropriate data are missing.

Formula omitted]-character varieties are not varieties of characters

We prove that for n≥2 and all groups Γ of corank ≥2 the coordinate rings of SO(2n,C)-character varieties of Γ are not generated by trace functions nor generalized trace functions τϕ,γ, sending ρ:Γ→SO(2n,C) to trϕρ(γ), for an arbitrary representation ϕ of SO(2n,C) and any γ∈Γ.Furthermore, we give examples of non-conjugate completely reducible representations undistinguishable by generalized trace functions. Hence, SO(2n,C)-character varieties are not varieties of characters! However, we also prove that any generic SO(2n,C)-representation of a free group can be distinguished from all non-equivalent representations by trace functions and by a single generalized trace function.

On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum

We consider the problem of classification of nonequivalent representations of a scalar operator λI in the form of a sum of k self-adjoint operators with at most n 1 , . . . ,n k points in their spectra, respectively. It is shown that this problem is *-wild for some sets of spectra if (n 1 , . . . ,n k ) coincides with one of the following k -tuples: (2, . . . , 2) for k ≥ 5, (2, 2, 2, 3), (2, 11, 11), (5, 5, 5), or (4, 6, 6). It is demonstrated that, for the operators with points 0 and 1 in the spectra and k ≥ 5, the classification problems are *-wild for every rational λϵ 2 [2, 3].

On the arithmetic and geometry of binary Hamiltonian forms

Given an indefinite binary quaternionic Hermitian form f with coefficients in a maximal order of a definite quaternion algebra over Q, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most s by f , as s tends to +∞. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad’s general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes

For Klein-Gordon fields, it is well known that there exist an infinite number of nonequivalent Fock representations of the canonical commutation relations and, therefore, of inequivalent quantum theories. A context in which this kind of ambiguities arises and prevents the derivation of robust results is, e.g., in the quantum analysis of cosmological perturbations. In these situations, typically, a suitable scaling of the field by a time dependent function leads to a description in an auxiliary static background, though the nonstationarity still shows up in a time dependent mass. For such a field description, and assuming the compactness of the spatial sections, we recently proved in three or less spatial dimensions that the criteria of a natural implementation of the spatial symmetries and of a unitary time evolution are able to select a unique class of unitarily equivalent vacua, and hence of Fock representations. In this work, we succeed to extend our uniqueness result to the consideration of all possible field descriptions that can be reached by a time dependent canonical transformation which, in particular, involves a scaling of the field by a function of time. This kind of canonical transformations modify the dynamics of the system and introduce a further ambiguity in its quantum description, exceeding the choice of a Fock representation. Remarkably, for any compact spatial manifold in less than four dimensions, we show that our criteria eliminate any possible nontrivial scaling of the field other than that leading to the description in an auxiliary static background. Besides, we show that either no time dependent redefinition of the field momentum is allowed or, if this may happen, the redefinition does not introduce any Fock representation that cannot be obtained by a unitary transformation.

Unitarizable Representations of Quivers

We investigate representations of *-algebras associated with posets. Unitarizable representations of the corresponding (bound) quivers (which are polystable representations for some appropriately chosen slope function) give rise to representations of these algebras. Considering posets which correspond to unbound quivers this leads to an ADE-classification which describes the unitarization behaviour of their representations. Considering posets which correspond to bound quivers, it is possible to construct unitarizable representations starting with polystable representations of related unbound quivers which can be glued together with a suitable direct sum of simple representations. Finally, we estimate the number of complex parameters parametrizing irreducible unitary non-equivalent representations of the corresponding algebras.

On the representation of integers by indefinite binary Hermitian forms

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

Quasi-invariant measures on sets of piecewise smooth homeomorphisms of closed intervals and circles and representations of diffeomorphism groups

Families of measures are constructed that are quasi-invariant with respect to the action of C1-diffeomorphisms, for a closed interval and a circle, and have bounded Borel measurable second derivatives. A series of pairwise nonequivalent irreducible representations of the group of C3-diffeomorphisms of the circle is introduced on the space of functions that are square integrable against these measures.

Theory and phenomenology of spin-1 chiral particles

We consider all possible finite representations of the Lorentz group and their association with the spin properties of the particles. It is shown that, for a given nontrivial spin, there exist several nonequivalent representations differing in chirality, which in the massless case correspond to different particles with different helicities. A spin-1 case, which includes the standard field description by the vector-potential and a non-standard one by the second rank antisymmetric tensor field, is considered in detail. The first field transforms under the real representation (1/2,1/2), while the second one does so under the chiral representations (1,0) and (0,1). By considering spin-1 hadron resonances as an example, it is shown that these two fields describe two different types of particles existing in nature. This idea is further applied to the construction of the Standard Model extension using a new type of spin-1 chiral particles. Its phenomenological consequences are studied in detail.

Integration of the thirring model equations

It is demonstrated that an exact solution for the Thirring massless model in the canonical quantization scheme follows from its Heisenberg equations in the method of dynamic mapping and the presence of unitary nonequivalent representations of the Dirac massless field.

ChemInform Abstract: REACCS in the Chemical Development Environment. Part 3. Graphically Nonequivalent Representations of Molecules and Reactions.

ChemInform Abstract: REACCS in the Chemical Development Environment. Part 3. Graphically Nonequivalent Representations of Molecules and Reactions.

On a Whitham-Type Equation

The Hunter-Saxton equation and the Gurevich-Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly.

Nonequivalent representations of canonical commutation relations in quantum mechanics: The case of the Aharonov-Bohm effect

We stress the role of nonequivalent representations of the canonical commutation relations in quantum mechanics. Such representations occur when the space accessible to the system is not simply connected. An example is the magnetic Aharonov-Bohm effect, which can be interpreted in terms of nonequivalent representations, without explicitly invoking the vector potential in the region accessible to the electrons.

Finite dimensional representations of the Euclidean algebra e(2) having two generators

The Euclidean algebra e(2) is the Lie algebra of the group E(2) of Euclidean transformations of the plane. This paper examines finite dimensional representations of e(2) having two generators. To each representation with two generators we associate a graph. In term of graphs, we give a criterion for the indecomposability of such representations and describe an invariant for indecomposable representations. We also classify the indecomposable representations of dimensions 5 and 6, regardless of the number of generators (dimensions less than 5 have been classified). In each case there are finitely many such representations. Next, we show that for each dimension ⩾8 there are infinitely many nonequivalent indecomposable representations.

Nonequivalent operator representations for Bose-condensed systems

The necessity of accurately taking into account the existence of nonequivalent operator representations associated with canonical transformations is discussed. It is demonstrated that Bose systems in the presence of the Bose-Einstein condensate and without it correspond to different Fock spaces, orthogonal to each other. A composite representation for the field operators is constructed allowing for a self-consistent description of Bose-condensed systems. Equations of motion are derived from the given Hamiltonian, which guarantees the validity of conservation laws and thermodynamic self-consistency. At the same time, the particle spectrum obtained either from diagonalizing this Hamiltonian or from linearizing the field-operator equations of motion has no gap. The condition of the condensate existence assures the absence of the gap in the spectrum, irrespectively to the approximation involved. The suggested self-consistent theory is both conserving and gapless.

A NAIADE based approach for sustainability benchmarking

Sustainability management and planning is mainly a social decision process about the desired future of a given city, region or country. A possible reduction of the unavoidable complexity, present in this type of problems, consists in considering explicitly non-equivalent representations which arise in the interaction between the various observer subjects and the different systemic levels considered. The reduction of the number of non-equivalent representations leads us to the use of descriptors such as indicators and indexes. Then a question arises: how could such indicators be used for policy making? This paper presents a multicriteria framework, aimed at helping real-world policy processes, for the benchmarking of sustainability indicators.