Direct Product Representations Research Articles

On representations of direct products and the bounded generation property of branch groups

We prove that the minimal representation dimension of a direct product G of non-abelian groups G_1,ldots ,G_n is bounded below by n+1 and thereby answer a question of Abért. If each G_i is moreover non-solvable, then this lower bound can be improved to be 2n. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups, we show that branch groups cannot be boundedly generated.

Про матричнi зображення наднапiвгруп напiвгрупи, породженої двома взаємно анульовними iдемпотентами

Matrix representations of finite semigroups over fields are studied not so well as for finite groups. Representations of finite groups over fields are studied sufficiently well; in particular, the criterions of representation type are fully defined for an arbitrary field. If the characteristic $p$ of a field $K$ does not divide the order of a group (classical case), then the group has, up to equivalence, finite number of non-decomposable representations; such group is called a group of finite representation type. If the characteristic $p$ divides the order of a group (modular case), then the group has finite representation type only if its Sylow $p$-subgroup is cyclic. In this case for most finite groups the problems of describing their representations includes the problem on classification, up to similarity, of the pairs of matrices. Such groups are called wild, and groups that allow explicit descriptions of representations are called tame. The tame and wild groups in modular case are fully described by the first author together with Yu. A. Drozd. In the theory of representations of semigroups, the largest number of works are devoted to irreducible representations. Among the old results, there are only some results on semigroups of finite representation type, namely, for a finite quite simple semigroup (I. S. Ponizovsky) and some semigroups of all transformations of a finite set (I. S. Ponizovsky, C. Ringel). In cases, when the numbers of non-decomposable representations is infinite, the most famous are the results from the theory of representations of algebras that can be reformulated in terms of representations of semigroups: the description of representations of the algebra $ $ (I. M. Gelfand, V. A. Ponomarev and L. O. Nazarova, A. V. Roiter, V. V. Sergeichuk, V. M. Bondarenko) and the algebra $ $ (V. M. Bondarenko and C. Ringel). If we are not talking about individual semigroups, but about semigroup classes, then it should be noted works about on representations of the semigroups generated by idempotents with partial zero multiplication (V. M. Bondarenko, O. M. Tertychna), representations of the Rees semigroups (S. M. Dyachenko), semigroups generated by the potential elements (V. M. Bondarenko, O. V. Zubaruk) and representations of direct products of the symmetric second-order semigroup (V. M. Bondarenko, E. M. Kostyshyn). Such semigroups can have both a finite and infinite representation type. V. M. Bondarenko and Ja. V. Zatsikha described representation types of the third-order semigroups over a field, and indicate the canonical form of the matrix representations for any semigroup of finite representation type. This article is devoted to the study of similar problems for oversemigroups of commutative semigroups.

Raman Scattering Polarization and Single Spinon Identification in Two-Dimensional Kitaev Quantum Spin Liquids

Perfect deporalization of the Loudon-Fleury inelastic visible-light scattering in the Kitaev honeycomb model is well known. Though it happens in Heisenberg Kagome and triangular antiferromagnets as well, yet we prove it to be of geometric origin rather than peculiar to quantum spin liquids. A Kitaev spin liquid in the square planar geometry indeed exhibits polarized Raman spectra containing defferent symmetry species, each brought by symmetry-compatible spinon geminate excitations, i.e. arising from symmetry-compatible direct-product representations made of double-valued irreducible representations of mediating-spinon-belonging gauged $\bm{k}$-point symmetry groups. We combine a standard point-symmetry-group analysis of the Raman vertex in the real space and an elaborate projective-symmetry-group analysis of Raman-scattering-mediating Majorana spinons in the reciprocal space to identify emergent spinons singly.

Degrees of irreducible representations of direct products of nonabelian simple groups

For a finite group G, let the character degree set of G, denoted by $$\mathrm{cd}(G)$$, be the set of the degrees of all irreducible complex representations of G. In the present paper, the structure of finite groups whose character degree sets coincide with the character degree sets of direct product of nonabelian simple groups will be studied. For a finite group S and a positive integer n, let $$S^{n}$$ be the direct product of n copies of S. We prove that if G is a finite group with $$\mathrm{cd}(G)=\mathrm{cd}(H)$$, where $$\begin{aligned}&H\in \{\mathrm{PSL}_{3}(q)^{n}(q=2^{\alpha }\geqslant 4, 3\not \mid q-1), \mathrm{PSU}_{3}(q)^{n}(q=2^{\alpha }\geqslant 4, 3\not \mid q+1), \\&\quad M_{11}^{n} (1\leqslant n\leqslant 2), M_{23}^{n}, J_{1}^{n}, J_{2}^{n}, J_{3}^{n}, J_{4}^{n}\}, \end{aligned}$$then G is a quasi perfect group. This extends the first step of Huppert’s Conjecture to the direct product of simple groups. This conjecture states that the nonabelian simple groups are uniquely determined up to an abelian direct factor by the set of character degrees.

Про матричнi зображення наднапiвгру пнапiвгрупи, породженої взаємно анульовними 2-потентним i2-нiльпотентним елементами

Among the old results, there are only some results on the representation type of semigroups, namely, for a finite quite simple semigroup (I. S. Ponizovsky) and some semigroups of all transformations of a finite set (I. S. Ponizovsky, C. Ringel); these papers were discussed on finite representation type. If we talk about new results, and even for semigroup classes, then it should be noted works on representations of the semigroups generated by idempotents with partial zero multiplication (V. M. Bondarenko, O. M. Tertychna), semigroups generated by the potential elements (V. M. Bondarenko, O. V. Zubaruk) and representations of direct products of the symmetric second-order semigroup (V. M. Bondarenko, E. M. Kostyshyn). Such semigroups can have both a finite and infinite representation type. V. M. Bondarenko and Ja. V. Zatsikha described representation types of the third-order semigroups over a field, and indicate the canonical form of the matrix representations for any semigroup of finite representation type. This article is devoted to the study of similar problems for oversemigroups of commutative semigroups.

Study of MV-algebras via derivations

Abstract The main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

Efficient generation of sum-of-products representations of high-dimensional potential energy surfaces based on multimode expansions.

The transformation of multi-dimensional potential energy surfaces (PESs) from a grid-based multimode representation to an analytical one is a standard procedure in quantum chemical programs. Within the framework of linear least squares fitting, a simple and highly efficient algorithm is presented, which relies on a direct product representation of the PES and a repeated use of Kronecker products. It shows the same scalings in computational cost and memory requirements as the potfit approach. In comparison to customary linear least squares fitting algorithms, this corresponds to a speed-up and memory saving by several orders of magnitude. Different fitting bases are tested, namely, polynomials, B-splines, and distributed Gaussians. Benchmark calculations are provided for the PESs of a set of small molecules.

Direct Product Representation of Labelled Free Choice Nets

Hack’s theorem [7] shows that every live and bounded free choice net is covered by S-components [5, 2]. If the net is labelled, with a distribution of the alphabet into possibly overlapping subalphabets for components, the question arose whether the net can be decomposed into a product of automata, with several possible definitions of product and of equivalence [1, 13, 11, 4]. Zielonka showed [15] that there is a live and 1bounded net which is not direct product representable. We give a straightforward example of a live and 1-bounded labelled free choice net which is not direct product representable, we do not know of any earlier such example. We give two sufficient conditions for 1-bounded labelled free choice nets to be direct product representable. In the other direction, we give two sufficient conditions on products of automata using which we can construct labelled free choice nets. In [14] expressions corresponding to such products has been recently given.

Indecomposable Projective Representations of Direct Products of Finite Groups Over a Field of Characteristic p

Let K be a field of characteristic p > 0, K* the multiplicative group of K and G = G p × B a finite group, where G p is a p-group and B is a p′-group. Denote by K λ G a twisted group algebra of G over K with a 2-cocycle λ ∈Z 2(G, K*). In this article, we give necessary and sufficient conditions for K λ G to be of OTP representation type, in the sense that every indecomposable K λ G-module is isomorphic to the outer tensor product V#W of an indecomposable K λ G p -module V and an irreducible K λ B-module W.

Representation of direct product of permutation groups as symmetry groups of boolean functions

We consider boolean functions invariance groups S(f) and some special kinds of boolean functions. We construct a k valued boolean function f which represent a direct product of two permutation groups and give an upper bound of k. Moreover we show that in some cases we can construct 2 valued boolean function which represent this product.

BOOLEAN FACTOR CONGRUENCES AND PROPERTY (*)

A variety [Formula: see text] has Boolean factor congruences (BFC) if the set of factor congruences of any algebra in [Formula: see text] is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniqueness of direct product representations of algebras, since it is a strengthening of the refinement property. We provide an explicit Mal'cev condition for BFC. With the aid of this condition, it is shown that BFC is equivalent to a variant of the definability property (*), an open problem in Willard's work[9].

Group theory of non-abelian vortices

We investigate the structure of the moduli space of multiple BPS non-Abelian vortices in U(N) gauge theory with N fundamental Higgs fields, focusing our attention on the action of the exact global (color-flavor diagonal) SU(N) symmetry on it. The moduli space of a single non-Abelian vortex, \( \mathbb{C}{P^{N - 1}} \), is spanned by a vector in the fundamental representation of the global SU(N) symmetry. The moduli space of winding-number k vortices is instead spanned by vectors in the direct-product representation: they decompose into the sum of irreducible representations each of which is associated with a Young tableau made of k boxes, in a way somewhat similar to the standard group composition rule of SU(N) multiplets. The Kähler potential is exactly determined in each moduli subspace, corresponding to an irreducible SU(N) orbit of the highest-weight configuration.

Pointed semi-quantales and lattice-valued topological spaces

In a way more general than variable-basis approach to lattice-valued topological spaces, the present paper introduces an alternative approach to lattice-valued topological spaces-direct product representation spaces extending the notion of quantal spaces in the sense of Mulvey and Pelletier to semi-quantales recently proposed by Rodabaugh. This paper aims to give an answer to the main question whether there exists a categorical connection, possibly a categorical equivalence, between direct product representation spaces and variable-basis lattice-valued topological spaces. Small sources in the category of semi-quantales which are called pointed semi-quantales can be identified with direct product representation spaces. For this reason, the main problem will be handled in terms of pointed semi-quantales. Generalized quasi-lattice-valued topological spaces extending variable-basis quasi-topological spaces into the present setting are introduced to be a suitable topological counterpart of pointed semi-quantales. To formalize and to solve the main problem, categories of pointed semi-quantales and of generalized quasi-lattice-valued topological spaces are constructed, and the relations between these categories, providing a satisfactory answer to the main problem, are established in this paper.

Varieties with definable factor congruences

We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first-order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner that it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.

A complete list of symmetry adapted expressions to the fourth power for compact bending potentials in molecules with and symmetry from a general symbolic algebra program

An algorithm is proposed to perform the reduction of direct product representations of finite groups and, in particular, to perform the complete symmetry adaption of power representations, which may be used to obtain compact analytical representations of potential energy surfaces following an idea described in Marquardt and Quack [J. Chem. Phys. 109, 10628 (1998)]. The algorithm is general in the sense that it can be applied to any finite group being characterized by its set of irreducible representations. It is automatic in the sense that, in case the reduction yields multiple degenerate irreducible subspaces of the same species, all degenerate irreducible subspaces are obtained with a coherent phase relation. The algorithm is based on the standard reduction rule of traditional representation theory. A symbolic algebra computer program based on MAPLE is presented and applied here to obtain the complete list of symmetry adapted expressions of bond angle coordinates up to the fourth power in all irreducible representations of the C3v and Td point groups.

Decomposition of tensor product representations of the unitary matrix quantum group SUq(2)

Representations of SUq(2) are labeled by a phase playing the role of a Casimir operator. The coproduct SUq(2) is used to form direct product representations which can be reduced to irreducible ones. Direct product of two representations with given phases can be reduced to a representation with any given phase. Coefficients necessary for this procedure are calculated. The calculation can also be performed in the Fourier transformed space which replaces the phase by an integer.

Modular projective representations of direct products of finite groups

Modular projective representations of direct products of finite groups

Stereographic projections and direct products

The concept of stereographic projections of point groups is reviewed. If the focus of attention is moved from the symmetry-equivalence of points to the symmetry operations inter-relating points, it is possible to give simple diagrams of irreducible representations, at least for many axial groups. This approach may be elaborated by the use of trigonometric basis functions, when these may be shown on similar stereographic-related diagrams. These two simple steps enable some insights into direct products; in particular, it is possible to give a diagrammatic representation of an antisymmetric direct product.

A grid representation for spherical angles: Decoupling of the angular momentum operator

The angular momentum operator which is a function of the orientational angle θ and the azimuthal angle φ may be split into the φ-dependent and φ-independent parts so that the split exponential operator method can be exactly implemented (with orthogonal transformations) in a direct product discrete variable representation of θ and φ. Although one loses the exact representation of the angular momentum in the spherical harmonic basis, the direct product representations have been proved to converge and to be stable and efficient. An advantage is that computational time for a wave-packet propagation (for a matrix-vector product) is reduced for split exponential propagators since a direct product representation is preserved for all the angles.

Unified effective theory for light mesons and heavy mesons

We present a unified effective field theory for the light mesons and heavy mesons which integrates many features from the quark model, vector dominance, chiral dynamics, Skyrmion physics, and the heavy quark effective theory into a single universal theory. By identifying the meson fields as the Bargmann-Wigner superfields with the direct product representation of a quark and antiquark and using the Bargmann-Wigner equation and Lagrangian, we can readily match the meson Lagrangian with the quark Lagrangian. Thus we not only incorporate various aspects of the quark contents into the meson fields but we can also transcribe the implementation of the chiral symmetry and spontaneous symmetry breakings from the quark Lagrangian to the meson Lagrangian. The physical light meson fields can be constructed nonlinearly from the quark and antiquark direct product representation, with the ${0}^{\ensuremath{-}}$ pseudoscalar mesons as the Goldstone mesons and the ${1}^{\ensuremath{-}}$ vector mesons as the non-Abelian gauge fields of the hidden local SU(3) symmetry. Phenomenological consequences and theoretical implications of this theory will be discussed. Various aspects of the heavy quark effective theory are automatically integrated into this formulation.