Group Theoretical Interpretation Research Articles

Gaussian Binomial Coefficients in Group Theory, Field Theory, and Topology

In this article we offer group-theoretic, field-theoretic, and topological interpretations of the Gaussian binomial coefficients and their sum. For a finite p-group G of rank n, we show that the Gaussian binomial coefficient is the number of subgroups of G that are minimally expressible as an intersection of n – k maximal subgroups of G, and their sum is precisely the number of subgroups that are either G or an intersection of maximal subgroups of G. We provide a field-theoretic interpretation of these quantities through the lens of Galois theory and a topological interpretation involving covering spaces.

Group Theoretical Description of the Periodic System

The group theoretical description of the periodic system of elements in the framework of the Rumer–Fet model is considered. We introduce the concept of a single quantum system, the generating core of which is an abstract C*-algebra. It is shown that various concrete implementations of the operator algebra depend on the structure of the generators of the fundamental symmetry group attached to the energy operator. In the case of the generators of the complex shell of a group algebra of a conformal group, the spectrum of states of a single quantum system is given in the framework of the basic representation of the Rumer–Fet group, which leads to a group-theoretic interpretation of the Mendeleev’s periodic system of elements. A mass formula is introduced that allows giving the termwise mass splitting for the main multiplet of the Rumer–Fet group. The masses of elements of the Seaborg table (eight-periodic extension of the Mendeleev table) are calculated starting from the atomic number Z=3 and going to Z=220. The continuation of the Seaborg homology between lanthanides and actinides is established with the group of superactinides. A 10-periodic extension of the periodic table is introduced in the framework of the group-theoretic approach. The multiplet structure of the extended table’s periods is considered in detail. It is shown that the period lengths of the system of elements are determined by the structure of the basic representation of the Rumer–Fet group. The theoretical masses of the elements of 10th and 11th periods are calculated starting from Z=221 and going to to Z=364. The concept of hypertwistor is introduced.

The problem of motion in gauge theories of gravity

In this article we consider the problem to what extent the motion of gauge-charged matter that generates the gravitational field can be arbitrary, as well as what equations are superimposed on the gauge field due to conditions of compatibility of gravitational field equations. Considered problem is analyzed from the point of view symmetry of the theory with respect to the generalized gauge deformed groups without specification of Lagrangians. In particular it is shown, that the motion of uncharged particles along geodesics of Riemannian space is inherent in an extremely wide range of theories of gravity and is a consequence of the gauge translational invariance of these theories under the condition of fulfilling equations of gravitational field. In the cause of gauge-charged particles, the Lorentz force, generalized for gauge-charged matter, appears in equations of motion as a consequence of the gauge symmetry of the theory under the condition of fulfilling equations of gravitational and gauge fields. In addition, we found relationships of equations for some fields that follow from the assumption about fulfilling of equations for other fields. At the end of the article there is an Appendix, which briefly describes the main provisions and facts from the theory of generalized gauge deformed groups and presents the main ideas of a single group-theoretical interpretation of gauge fields of both external (space-time) and internal symmetry, which is an alternative to their geometric interpretation.

𝑞-Tricomi functions and quantum algebra representations

Abstract The quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of q-special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the q-Tricomi functions and 2D q-Tricomi functions through the generating function and series expansion and for the first time establishing a connecting relation between the q-Tricomi and q-Bessel functions. The behavior of these functions is described through shapes, and the contrast between them is observed using mathematical software. Further, the problem of framing the q-Tricomi and 2D q-Tricomi functions in the context of the irreducible representation ( ω ) {(\omega)} of the two-dimensional quantum algebra ℰ q ⁢ ( 2 ) {\mathcal{E}_{q}(2)} is addressed, and certain relations involving these functions are obtained. 2-Variable 1-parameter q-Tricomi functions and their relationship with the 2-variable 1-parameter q-Bessel functions are also explored.

De Sitter field equations from quadratic curvature gravity: A group theoretical approach

In this paper, the linearized field equations related to the quadratic curvature gravity theory have been obtained in the four-dimensional de Sitter (dS) space–time. The massless spin-2 field equations have been written in terms of the Casimir operators of dS group making use of the ambient space notations. By imposing some simple constraints, arisen from group theoretical interpretation of the field equations, a new four-dimensional Gauss–Bonnet (GB-)like action has been introduced with the related field equations transforming according to the unitary irreducible representations (UIRs) of dS group. Since, the field equations transform according to the UIRs of dS group, the GB-like action, we just obtained, is expected to be a successful model of modified gravity. For more clarity, the gauge invariant field equations have been solved in terms of a gauge-fixing parameter [Formula: see text]. It has been shown that the solution can be written as the multiplication of a symmetric rank-2 polarization tensor and a massless minimally coupled scalar field on dS space. The Krein–Gupta–Bleuler quantization method has been utilized and the covariant two-point function has been calculated in terms of the massless minimally coupled scalar two-point function, using the ambient space notations. It has been written in terms of dS intrinsic coordinates from the ambient space counterpart. The two-point functions are dS invariant and free of any theoretical problems. It means that the proposed model is a successful model of modified gravity and it can produce significant results in the contexts of classical theory of gravity and quantum gravity toy models.

A group-theoretical interpretation of the word problem for free idempotent generated semigroups

The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup IG(E) – the ‘free-est’ semigroup with a given biordered set E of idempotents. We show that when E is finite, the word problem for IG(E) is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of maximal subgroups of IG(E). As an application, we obtain decidability of the word problem for an important class of examples. Also, we prove that for finite E, IG(E) is always a weakly abundant semigroup satisfying the congruence condition.

Interactions between Ehrenfest’s urns arising from group actions

The Ehrenfest diffusion model is a well-known classical physical model consisting of two urns and n balls. There is a group theoretical interpretation of the model by using the Gelfand pair $$({\mathbb {Z}}/2{\mathbb {Z}}\wr S_{n},S_{n})$$ by Diaconis and Shahshahani (Z Wahrsch Verw Gebiete 57(2):159–179, 1981). This interpretation is still valid for an r-urns generalization. Then the corresponding Gelfand pair is $$(S_{r}\wr S_{n},S_{r-1}\wr S_{n})$$ . However, in these models, there are no restrictions for ball movements, i.e., each ball can freely move to any urns. In this paper, interactions between urns arising from actions of finite groups are introduced. Degree of freedom of ball movements are restricted by finite group actions. We then show that the cutoff phenomenon occurs in some particular (yet significant and interesting) cases.

Affine Deligne-Lusztig varieties and the action of 𝐽

We propose a new stratification of the reduced subschemes of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties that highlights the relation between the geometry of these spaces and the action of the associated automorphism group. We show that this provides a joint group-theoretic interpretation of well-known stratifications which only exist for special cases such as the Bruhat-Tits stratification of Vollaard and Wedhorn, the semi-module stratification of de Jong and Oort, and the locus where theaa-invariant is equal to11.

Group theoretical interpretation of the modified gravity in de Sitter space

A frame work has been presented for theoretical interpretation of various modified gravitational models which is based on the group theoretical approach and unitary irreducible representations (UIR's) of de Sitter (dS) group. In order to illustrate the application of the proposed method, a model of modified gravity has been investigated. The background field method has been utilized and the linearized modified gravitational field equation has been obtained in the 4-dimensional dS space-time as the background. The field equation has been written as the eigne-value equation of the Casimir operators of dS space using the flat 5-dimensional ambient space notations. The Minkowskian correspondence of the theory has been obtained by taking the zero curvature limit. It has been shown that under some simple conditions, the linearized modified field equation transforms according to two of the UIR's of dS group labeled by $\Pi^\pm_{2,1}$ and $\Pi^\pm_{2,2}$ in the discrete series. It means that the proposed modified gravitational theory can be a suitable one to describe the quantum gravitational effects in its linear approximation on dS space. The field equation has been solved and the solution has been written as the multiplication of a symmetric rank-2 polarization tensor and a massless scalar field using the ambient space notations. Also the two-point function has been calculated in the ambient space formalism. It is dS invariant and free of any theoretical problem.

Selected group-theoretic aspects of confirmation measure symmetries

The paper considers symmetry properties of rule interestingness measures (in particular: measures of confirmation). Many authors have studied various symmetries, however the discussion on which sets of symmetries should be taken into account, let alone why the particular symmetries are desirable or not, has still not produced a generally recognized consensus. Furthermore, the results published so far neglect the fact that symmetries can be the subject of group theory-based considerations. This paper aims at solving those problems by introducing group-theoretic interpretations of symmetries, indicating that all symmetries can be treated as permutations, and compositions of symmetries as compositions of permutations. Such an interpretation allows us to apply the well-known group-theoretic results to symmetries. In particular, using this approach, we reveal the phenomenon of incompleteness occurring in sets of symmetries considered by different authors, and propose an effective way of controlling it. Moreover, we demonstrate that assessing the symmetries as either desirable or undesirable, as introduced by these authors, brings in inconsistencies, which become evident under the permutation-based interpretation. Finally, we present group-theoretic guidelines to the design of such symmetry sets that are free of the incompleteness and inconsistency phenomena but remain meaningful in the context of rule evaluation.

Generalizations of the Szemerédi–Trotter Theorem

We generalize the Szemeredi---Trotter incidence theorem, to bound the number of complete flags in higher dimensions. Specifically, for each $$i=0,1,\ldots ,d-1$$i=0,1,?,d-1, we are given a finite set $$S_i$$Si of i-flats in $${\mathbb {R}}^d$$Rd or in $${\mathbb {C}}^d$$Cd, and a (complete) flag is a tuple $$(f_0,f_1,\ldots ,f_{d-1})$$(f0,f1,?,fd-1), where $$f_i\in S_i$$fi?Si for each i and $$f_i\subset f_{i+1}$$fi?fi+1 for each $$i=0,1,\ldots ,d-2$$i=0,1,?,d-2. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in $${\mathbb {R}}^3$$R3 such that among the lines incident to a point, at most O(1) of them can be coplanar, (ii) incidences with Legendrian lines in $${\mathbb {R}}^3$$R3, a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in $${\mathbb {R}}^3$$R3 (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case. Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems.

On the singlet projector and the monodromy relation for psu(2, 2|4) spin chains and reduction to subsectors

As a step toward uncovering the relation between the weak and the strong coupling regimes of the $\mathcal{N}=4$ super Yang-Mills theory beyond the specral level, we have developed in a previous paper [arXiv:1410.8533] a novel group theoretic interpretation of the Wick contraction of the fields, which allowed us to compute a much more general class of three-point functions in the SU(2) sector, as in the case of strong coupling [arXiv:1312.3727], directly in terms of the determinant representation of the partial domain wall partition funciton. Furthermore, we derived a non-trivial identity for the three point functions with monodromy operators inserted, being the discrete counterpart of the global monodromy condition which played such a crucial role in the computation at strong coupling. In this companion paper, we shall extend our study to the entire ${\rm psu}(2,2|4)$ sector and obtain several important generalizations. They include in particular (i) the manifestly conformally covariant construction, from the basic principle, of the singlet-projection operator for performing the Wick contraction and (ii) the derivation of the monodromy relation for the case of the so-called "harmonic R-matrix", as well as for the usual fundamental R-matrtix. The former case, which is new and has features rather different from the latter, is expected to have important applications. We also describe how the form of the monodromy relation is modified as ${\rm psu}(2,2|4)$ is reduced to its subsectors.

Intertwining Operator Realization of ANTI-DE sitter holography

We give a group-theoretic interpretation of relativistic holography as equivalence between representations of the anti de Sitter algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators for arbitrary integer spin the framework of representation theory. Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the de Sitter case, we show that each bulk field has two boundary (shadow) fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary.

Duality between the trigonometricBCnSutherland system and a completed rational Ruijsenaars–Schneider–van Diejen system

We present a new case of duality between integrable many-body systems, where two systems live on the action-angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one. Our investigation utilizes an idea that was exploited previously to provide group-theoretic interpretation for several dualities discovered originally by Ruijsenaars. In the group-theoretic framework, one applies Hamiltonian reduction to two Abelian Poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space. Taking the cotangent bundle of U(2n) as the upstairs space, we demonstrate how this mechanism leads to a new dual pair involving the BCn trigonometric Sutherland system. Thereby, we generalize earlier results pertaining to the An trigonometric Sutherland system as well as a recent work by Pusztai on the hyperbolic BCn Sutherland system.

Unitarity of the SoV Transform for the Toda Chain

The quantum separation of variables method consists in mapping the original Hilbert space where a spectral problem is formulated onto one where the spectral problem takes a simpler "separated" form. In order to realise such a program, one should construct the map explicitly and then show that it is unitary. In the present paper, we develop a technique which allows one to prove the unitarity of this map in the case of the quantum Toda chain. Our proof solely builds on objects and relations naturally arising in the framework of the so-called quantum inverse scattering method. Hence, with minor modifications, it should be readily transposable to other quantum integrable models solvable by the quantum separation of variables method. As such, it provides an important alternative to the proof of the map's unitarity based on the group theoretical interpretation of the quantum Toda chain, which is absent for more complex quantum integrable models.

On the Relation Between Gauge and Phase Symmetries

We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state (e.g. from \(q\)and\(p\) to \(q\)or\(p\) in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau’s moment map, the Mardsen–Weinstein symplectic reduction, the symplectic “category” introduced by Weinstein, and the conjecture (proposed by Guillemin and Sternberg) according to which “quantization commutes with reduction”. By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit (the extreme cases of) this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points.

Riemann–Hilbert Problems, Toeplitz Operators and $${\mathfrak{Q}}$$ Q -Classes

We generalize the notion of $${\mathfrak{Q}}$$ -classes $${C_{{Q_1} {Q_2}}}$$ , which was introduced in the context of Wiener–Hopf factorization, by considering very general 2 × 2 matrix functions Q 1, Q 2. This allows us to use a mainly algebraic approach to obtain several equivalent representations for each class, to study the intersections of $${\mathfrak{Q}}$$ -classes and to explore their close connection with certain non-linear scalar equations. The results are applied to various factorization problems and to the study of Toeplitz operators with symbol in a $${\mathfrak{Q}}$$ -class. We conclude with a group theoretic interpretation of some of the main results.

Schrödinger algebra and non-relativistic holography

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. The main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov-Witten we give the conditions when both boundary fields are physical.

Yukawas, G-flux, and spectral covers from resolved Calabi-Yau’s

We use the resolution procedure of Esole and Yau [1] to study Yukawa couplings, G-flux, and the emergence of spectral covers from elliptically fibered Calabi-Yau’s with a surface of A 4 singularities. We provide a global description of the Esole-Yau resolution and use it to explicitly compute Chern classes of the resolved 4-fold, proving the conjecture of [2] for the Euler character in the process. We comment on the physical implications of the surprising singular fibers in codimension 2 and 3 in [1] and emphasize a group theoretic interpretation based on the A 4 weight lattice. We then construct explicit G-fluxes by brute force in one of the 6 birationally equivalent Esole-Yau resolutions, quantize them explicitly using our result for the second Chern class, and compute the spectrum and flux-induced 3-brane charges, finding agreement with results and conjectures of local models in all cases. Finally, we provide a precise description of the spectral divisor formalism in this setting and sharpen the procedure described in [3] in order to explicitly demonstrate how the Higgs bundle spectral cover of the local model emerges from the resolved Calabi-Yau geometry. Along the way, we demonstrate explicitly how the quantization rules for fluxes in the local and global models are related.

Application of the [formula omitted]-symmetries approach and time independent integral of the modified Emden equation

In this paper we derive the time-independent integral for a nonlinear dissipative system, namely the modified Emden equation, from Lie point symmetries. We employ the recently introduced λ -symmetries method [C. Muriel, J.L. Romero, First integrals, integrating factors and λ -symmetries of second-order differential equations, J. Phys. A: Math. Theor. 42 (2009) 365207–365217] to complete this task. To begin with we recall Lie point symmetries of this system and derive λ -symmetries from the vector fields. The knowledge of λ -symmetries enables us to obtain integrating factors, integrals and the general solution for the linearizable case. While determining the integrating factor from the λ -symmetry for the integrable case we find that this case splits up into three sub-cases. We then obtain the integrating factor and integral for these three sub-cases. The results agree with the ones reported in the literature and thereby give a group theoretical interpretation for the nonstandard time independent integrals exhibited by the system.