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A top-down approach to the flavor problem motivated from string theory leads to the concept of eclectic flavor groups that combine traditional and modular flavor symmetries. To make contact with models constructed in the bottom-up approach, we analyze a specific example based on the eclectic flavor group Omega(1) (a nontrivial combination of the traditional flavor group Delta(54) and the finite modular group T') in order to extract general lessons from the eclectic scheme. We observe that this scheme is highly predictive since it severely restricts the possible group representations and modular weights of matter fields. Thereby, it controls the structure of the Kaehler potential and the superpotential, which we discuss explicitly. In particular, both Kaehler potential and superpotential are shown to transform nontrivially, but combine to an invariant action. Finally, we find that discrete R-symmetries are intrinsic to eclectic flavor groups., 30 pages, 2 tables, 1 figure; v2: assumptions clarified, references added and updated, matches version published in NPB

We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into $\SL(n,\R)$ that satisfy partial hyperconvexity properties inspired from Labourie's work. This is the case for several open sets of Anosov representations not contained in higher rank Teichm\"uller spaces, as well as for $\Theta$-positive representations into $\SO(p,q)$ if $p\geq 4$. We moreover show that 'positivity properties' known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations., Comment: 40 pages, 9 figures. Comments welcome! V2: small corrections in sec. 4.4; V3: final version, incorporating referee comments, to appear in Trans. AMS

We explicitly construct every kinematically allowed three particle graviton-graviton-$P$ and photon-photon-$P$ S-matrix in every dimension and for every choice of the little group representation of the massive particle $P$. We also explicitly construct the spacetime Lagrangian that generates each of these couplings. In the case of gravitons we demonstrate that this Lagrangian always involves (derivatives of) two factors of the Riemann tensor, and so is always of fourth or higher order in derivatives. This result verifies one of the assumptions made in the recent preprint \cite{Chowdhury:2019kaq} while attempting to establish the rigidity of the Einstein tree level S-matrix within the space of local classical theories coupled to a collection of particles of bounded spin., Comment: added references

A systematic design procedure for placing ports with low correlation on an asymmetric antenna geometry is presented. By applying the mathematical description of symmetry based on group theory and group representations to the theory of characteristic modes, it is shown that it is in general not possible to realize uncorrelated antenna ports on an asymmetric antenna. Therefore, a port placement procedure based on characteristic modes is proposed for such geometries in order to realize ports with low correlation. The design procedure is based on modal parameters alone. Thus, only one full simulation run is needed in order to perform the modal analysis. The proposed procedure is illustrated by means of numerical examples.

A coarse graining operation of spatially homogeneous quantum states based on an SU(1,1) Lie group structure has recently been proposed in [1] and used in [2] to compute an explicit renormalisation group flow in the context of loop quantum cosmology. In this note, we explain the group theoretical origin of this procedure and generalise previous results based on these insights. We also highlight how the group theoretical origin of these techniques implies their immediate generalisation to other Lie groups., Comment: 11 pages

Semi-Standard Young Tableaux includes many applications in combinatorics, group representations, invariant theory, symmetric functions, and the theory of algorithms. Out of the many combinatorial objects Semi-Standard Young Tableau (SSYT) is the one on k-symbols. This paper aims to present how to find hook lengths, formulae for evaluation of total number of standard and semi standard young tableaux.The Semi-Standard Young Tableaux are used to label the basis vectors of the standard or Young Yamanouchi basis of the symmetric group sn. This papergives a new graphical method to find ranking and unranking of the semi-standard Young tableau for a partition of degree n in sn. Graphical method of approach is explained forthe symmetric group s6 using a partition of degree 6 as an example.

We study the balance of $G$-gain graphs, where $G$ is an arbitrary group, by investigating their adjacency matrices and their spectra. As a first step, we characterize switching equivalence and balance of gain graphs in terms of their adjacency matrices in $M_n(\mathbb C G)$. Then we introduce a represented adjacency matrix, associated with a gain graph and a group representation, by extending the theory of Fourier transforms from the group algebra $\mathbb C G$ to the algebra $M_n(\mathbb C G)$. We prove that a gain graph is balanced if and only if the spectrum of the represented adjacency matrix associated with any (or equivalently all) faithful unitary representation of $G$ coincides with the spectrum of the underlying graph, with multiplicity given by the degree of the representation. We show that the complex adjacency matrix of unit gain graphs and the adjacency matrix of a cover graph are indeed particular cases of our construction. This enables us to recover some classical results and prove some new characterizations of balance in terms of spectrum, index or structure of these graphs., Comment: 27 pages, 3 tables, 5 figures. In this second version, the word "balancedness" (with the meaning of "property of being balanced") has been replaced by the word "balance" both in the title and in the body of the article

The research of this paper has been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union, by the grant PGC2018-095140-B-I00 from the Ministerio de Ciencia, Innovacion y Universidades and the Agencia Estatal de Investigacion, Spain, and FEDER, European Union, and by the grant PROMETEO/2017/057 from the Generalitat, Valencian Community, Spain. The first author is supported by the predoctoral grant 201606890006 from the China Scholarship Council. The second author is supported by the grant 11401597 from the National Science Foundation of Chin

In this paper we construct an example of two representations \((V_{1},G_{1})\) and \((V_{2},G_{2})\) which are action type geometrically equivalent and groups \(G_{1}\) and \(G_{2}\) are geometrically equivalent, but the representations \((V_{1},G_{1})\) and \((V_{2},G_{2})\) are not geometrically equivalent.

In this paper, we propose a novel approach to predict group activities given the beginning frames with incomplete activity executions. Existing action (We define action as the behavior performed by a single person, and define activity as the behavior performed by a group of people) prediction approaches learn to enhance the representation power of the partial observation (We define partial observation as the beginning frames with incomplete activity execution, and full observation as the one with complete activity execution). However, for group activity prediction, the relation evolution of people’s activity and their positions over time is an important cue for predicting group activity. To this end, we propose a sequential relational anticipation model (SRAM) that summarizes the relational dynamics in the partial observation and progressively anticipates the group representations with rich discriminative information. Our model explicitly anticipates both activity features and positions by two graph auto-encoders, aiming to learn a discriminative group representation for group activity prediction. Experimental results on two popularly used datasets demonstrate that our approach significantly outperforms the state-of-the-art activity prediction methods.

Learning group representation is a commonly concerned issue in tasks where the basic unit is a group, set, or sequence. Previously, the research community tries to tackle it by aggregating the elements in a group based on an indicator either defined by humans such as the quality and saliency, or generated by a black box such as the attention score. This article provides a more essential and explicable view. We claim the most significant indicator to show whether the group representation can be benefited from one of its element is not the quality or an inexplicable score, but the discriminability w.r.t. the model. We explicitly design the discrimiability using embedded class centroids on a proxy set. We show the discrimiability knowledge has good properties that can be distilled by a light-weight distillation network and can be generalized on the unseen target set. The whole procedure is denoted as discriminability distillation learning (DDL). The proposed DDL can be flexibly plugged into many group-based recognition tasks without influencing the original training procedures. Comprehensive experiments on various tasks have proven the effectiveness of DDL for both accuracy and efficiency. Moreover, it pushes forward the state-of-the-art results on these tasks by an impressive margin.

In this paper, we describe properties of the Dedekind η-function, constructions arising from it, and their applications in various topics of the number theory and algebra. We discuss connections with the group representation theory and the study of the structure of spaces of modular forms. Special attention is paid to the special class of modular forms, namely, η-functions with multiplicative coefficients.

It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space H ${\mathcal H}$ correspond to direct product decompositions H ≃ A × A ⊥ $\mathcal{H}\simeq A\times A^\perp$ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.

The Fourier method on commutative groups is used in many fields of mathematics, physics, and engineering. Nowadays, this method finds increasingly wide application to non-commutative groups. Along with the one-sided convolution operators and the corresponding convolution equations, two-sided convolution operators on noncommutative groups are studied. Two-sided convolution operators have a number of applications in complex analysis and are used in quantum mechanics. In this paper, two-sided convolutions on arbitrary finite noncommutative groups are considered. A criterion for the inversibility of the two-sided convolution operator is obtained. An algorithm for solving the two-sided convolution equation on an arbitrary finite noncommutative group, using the Fourier transform, is developed. Estimates of the computational complexity of the algorithm developed are given. It is shown that the complexity of solving the two-sided convolution equation depends both on the type of the group representation and on the computational complexity of the Fourier transform. The algorithm is considered in detail on the example of the finite dihedral group $${{\mathbb{D}}_{m}}$$ and the Heisenberg group $$\mathbb{H}({{\mathbb{F}}_{p}})$$ over a simple Galois field, and the results of numerical experiments are presented.

The calculation of evolutionary distance via models of genome rearrangement has an inherent combinatorial complexity. Various algorithms and estimators have been used to address this; however, many of these set quite specific conditions for the underlying model. A recently proposed technique, applying representation theory to calculate evolutionary distance between circular genomes as a maximum likelihood estimate, reduces the computational load by converting the combinatorial problem into a numerical one. We show that the technique may be applied to models with any choice of rearrangements and relative probabilities thereof; we then investigate the symmetry of circular genome rearrangement models in general. We discuss the practical implementation of the technique and, without introducing any bona fide numerical approximations, give the results of some initial calculations for genomes with up to 11 regions.

Crystalline solids consisting of three-dimensional networks of interconnected rigid units are ubiquitous amongst functional materials. In many cases, application-critical properties are sensitive to rigid-unit rotations at low temperature, high pressure or specific stoichiometry. The shared atoms that connect rigid units impose severe constraints on any rotational degrees of freedom, which must then be cooperative throughout the entire network. Successful efforts to identify cooperative-rotational rigid-unit modes (RUMs) in crystals have employed split-atom harmonic potentials, exhaustive testing of the rotational symmetry modes allowed by group representation theory, and even simple geometric considerations. This article presents a purely algebraic approach to RUM identification wherein the conditions of connectedness are used to construct a linear system of equations in the rotational symmetry-mode amplitudes.

Our study examines the extent to which parliamentary candidates believe that membership of a certain social group allows Members of Parliament (MPs) to more effectively represent that group. Using the 2009 German Candidate Survey, we look at four social groups: women, immigrants, religious people and East Germans. The descriptive results indicate that support for group representation is highest for women and lowest for East Germans. The explanatory analyses reveal that women are more likely than men to believe that women are better at representing the interests of women. The same holds for immigrants, religious people, and East Germans. Candidates’ belief that MPs from a certain social group are better at representing that group tends to be limited to their own social group. Our results thus indicate that the belief that one should belong to a social group in order to effectively represent the interests of that group is mainly based on identity, rather than an overall belief in the link between descriptive and substantive representation. They also show that attitudes towards representation change significantly as a result of parliamentary socialisation, with candidates with parliamentary experience being significantly less likely to support the idea of group representation compared with those without such experience.

‎In the present paper‎, ‎we are going to use geometric and topological concepts‎, ‎entities and properties of the‎ ‎integral curves of linear vector fields‎, ‎and the theory of differential equations‎, ‎to establish a representation for some groups on $R^{n} (ngeq 1)$‎. ‎Among other things‎, ‎we investigate the surjectivity and faithfulness of the representation‎. At the end‎, ‎we give some applications‎. .

We describe representations of groupoid C*-algebras on Hilbert modules over arbitrary C*-algebras by a universal property. For Hilbert space representations, our universal property is equivalent to Renault's Integration-Disintegration Theorem. For a locally compact group, it is related to the automatic continuity of measurable group representations. It implies known descriptions of groupoid C*-algebras as crossed products for \'etale groupoids and transformation groupoids of group actions on spaces.

We carry out a systematic study of primary operators in the conformal field theory of a free Weyl fermion. Using SO(4,2) characters we develop counting formulas for primaries constructed using a fixed number of fermion fields. By specializing to particular classes of primaries, we derive very explicit formulas for the generating functions for the number of primaries in these classes. We present a duality map between primary operators in the fermion field theory and polynomial functions. This allows us to construct the primaries that were counted. Next we show that these classes of primary fields correspond to polynomial functions on certain permutation orbifolds. These orbifolds have palindromic Hilbert series., v2: matches published version

Let $$\mathrm {Sp}(2,1)$$ be the isometry group of the quaternionic hyperbolic plane $$\mathbf{H}_{\mathbb {H}}^2$$ . An element g in $$\mathrm {Sp}(2,1)$$ is hyperbolic if it fixes exactly two points on the boundary of $$\mathbf{H}_{\mathbb {H}}^2$$ . We classify pairs of hyperbolic elements in $$\mathrm {Sp}(2,1)$$ up to conjugation. A hyperbolic element of $$\mathrm {Sp}(2,1)$$ is called loxodromic if it has no real eigenvalue. We show that the set of $$\mathrm {Sp}(2,1)$$ conjugation orbits of irreducible loxodromic pairs is a $$(\mathbb {C}\mathbb {P}^1)^4$$ bundle over a topological space that is locally a semi-analytic subspace of $$\mathbb {R}^{13}$$ . We use the above classification to show that conjugation orbits of ‘geometric’ representations of a closed surface group (of genus $$g \ge 2$$ ) into $$\mathrm {Sp}(2,1)$$ can be determined by a system of $$42g-42$$ real parameters. Further, we consider the groups $$\mathrm {Sp}(1,1)$$ and $$\mathrm {GL}(2, \mathbb {H})$$ . These groups also act by the orientation-preserving isometries of the four and five dimensional real hyperbolic spaces respectively. We classify conjugation orbits of pairs of hyperbolic elements in these groups. These classifications determine conjugation orbits of ‘geometric’ surface group representations into these groups.

Let G be a topological group. We investigate relations between two classes of “polynomial like” continuous functions on G defined, respectively, by the conditions 1) Δ h n + 1 f = 0 for every h ∈ G , and 2) Δ h n + 1 Δ h n ⋯ Δ h 1 f = 0 for every h 1 , ⋯ , h n + 1 ∈ G . It is shown that for many (but not all) groups these classes coincide. We consider also Montel type versions of the above conditions – when 1) and 2) hold only for h in a generating subset of G. Our approach is based on the study of the counterparts of the discussed classes for general representations of groups (instead of the regular representation).

We study colored generalizations of the symmetric algebra and its Koszul dual, the exterior algebra. The symmetric group $\mathfrak{S}_n$ acts on the multilinear components of these algebras. While $\mathfrak{S}_n$ acts trivially on the multilinear components of the colored symmetric algebra, we use poset topology techniques to understand the representation on its Koszul dual. We introduce an $\mathfrak{S}_n$-poset of weighted subsets that we call the weighted boolean algebra and we prove that the multilinear components of the colored exterior algebra are $\mathfrak{S}_n$-isomorphic to the top cohomology modules of its maximal intervals. We use a technique of Sundaram to compute group representations on Cohen-macaulay posets to give a generating formula for the Frobenius series of the colored exterior algebra. We exploit that formula to find an explicit expression for the expansion of the corresponding representations in terms of irreducible $\mathfrak{S}_n$-representations. We show that the two colored Koszul dual algebras are Koszul in the sense of Priddy., Comment: 30 pages, 5 figures, full version of extended abstract published in proceedings FPSAC2016, related to arXiv:1309.5527 and arXiv:1408.5415, some of the results were announced in arXiv:1506.01628. To appear in Journal of Algebra

Presence, of bodies and ideas, is often taken as the primary indicator of political equality and, hence, democratic health. Intersectionality and constructivism question the validity of measuring presence. Turning theory into practice, we propose a comparative reflexive design guided by two research questions: (1) Who are the groups? and (2) What are their problems? This reveals both prototypical and non-prototypical groups and interests, from the perspectives of politicians (from above) and citizens (from below). We suggest concrete qualitative and quantitative methodological strategies to study these questions empirically.

Tensegrities consisting of axially loaded strings and bars have various technologically important applications, for which symmetric configurations are preferred as prototypes. In this study, we investigate the self-equilibrated states of rhombic truncated regular tetrahedral and cubic tensegrities by combining group representation theory and force-density matrix method. The force-density matrices of these tensegrities are analytically block-diagonalized using the irreducible representation matrices of tetrahedral and cubic groups. By making the symmetry-adapted force-density matrix satisfy the necessary number of nullity and positive semi-definiteness, we derive the analytical expressions for self-equilibrium and super-stability of rhombic truncated regular tetrahedral and cubic tensegrities. This work helps to design highly symmetric tensegrities for developing biomechanical models, mechanical metamaterials, and flexible robotics.

The elementary particles of relativistic quantum field theory are not simple field quanta, as has long been assumed. Rather, they supplement quantum fields, on which they depend on but to which they are not reducible, as shown here with particles defined instead as a unified collection of properties that appear in both physical symmetry group representations and field propagators. This notion of particle provides consistency between the practice of particle physics and its basis in quantum field theory.