Inequivalent Irreducible Representations Research Articles

Groups with three projective characters

Working over the field of complex numbers, the inequivalent irreducible projective representations of a finite group G with 2-cocycle α are considered. The greatest common divisor of the degrees of the α-characters of these representations is used to obtain information about the degrees of the irreducible αS -characters of a Sylow subgroup S of G. Suppose now that G has exactly three irreducible α-characters. Then the potential form that the three degrees can take is found if G has a nontrivial cyclic Sylow subgroup. However, the main result is to show that G is solvable when the degrees are of the form a, ra and ka with certain restrictions on r and k, most notably that r = 1 or k. This involves detailed analysis of the possible actions of G on the irreducible αN -characters of a normal subgroup N of G.

Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups.

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument's evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

Can you take Akemann–Weaver's [formula omitted] away?

By Glimm's dichotomy, a separable, simple C⁎-algebra has continuum many unitarily inequivalent irreducible representations if, and only if, it is non-type I while all of its irreducible representations are unitarily equivalent if, and only if, it is type I. Naimark asked whether the latter equivalence holds for all C⁎-algebras.In 2004, Akemann and Weaver gave a negative answer to Naimark's problem using Jensen's Diamond Principle ⋄ℵ1, a powerful diagonalization principle that implies the Continuum Hypothesis (CH). By a result of Rosenberg, a separably represented, simple C⁎-algebra with a unique irreducible representation is necessarily of type I. We show that this result is sharp by constructing an example of a separably represented, simple C⁎-algebra that has exactly two inequivalent irreducible representations, and therefore does not satisfy the conclusion of Glimm's dichotomy. Our construction uses a weakening of Jensen's ⋄ℵ1, denoted ⋄Cohen, that holds in the original Cohen's model for the negation of CH. We also prove that ⋄Cohen suffices to give a negative answer to Naimark's problem. Our main technical tool is a forcing notion that generically adds an automorphism of a given C⁎-algebra with a prescribed action on its space of pure states.

Topological Effects in a Fermionic Condensate Induced by a Cosmic String and Compactification on the AdS Bulk

In this paper, we analyzed the fermionic condensate (FC) associated with a massive fermionic field on a five-dimensional anti-de Sitter (AdS) spacetime in the presence of a cosmic string taking into account a magnetic flux running along its core. In addition, a compactified dimension was considered. Due to this compactification, the FC is expressed in terms of two distinct contributions: the first one corresponds to the geometry without compactification, and the second one is induced by the compactification. Depending on the values of the physical parameters, the total FC can be positive or negative. As a limiting case, the expression for the FC on locally Minkowski spacetime was derived. It vanishes for a massless fermionic field, and the nonzero FC on the AdS background space in the massless case is an effect induced by gravitation. This shows that the gravitational field may essentially influence the parameter space for phase transitions. For a massive field, the FC diverges on the string as the inverse cube of the proper distance from the string. In the case of a massless field, depending on the magnetic flux along the string and planar angle deficit, the limiting value of the FC on the string can be either finite or infinite. At large distances, the decay of the FC as a function of the distance from the string is a power law for both cases of massive and massless fields. For a cosmic string on the Minkowski bulk and for a massive field, the decay is exponential. The topological part in the FC vanishes on the AdS boundary. We show that the FCs coincide for the fields realizing two inequivalent irreducible representations of the Clifford algebra. In the special case of the zero planar angle deficit, the results presented in this paper describe Aharonov–Bohm-type effects induced by magnetic fluxes in curved spacetime.

Representations of special Jordan triple systems of all symmetric and hermitian n by n matrices

We study the representations of two special Jordan triple systems (with respect to the product xyz + zyx): The first is the Jordan triple system of all symmetric by matrices over a field of characteristic zero, the second is the Jordan triple system of all Hermitian by matrices over the complex numbers . We show that the universal (associative) envelope of is isomorphic to , while the universal (associative) envelope of is isomorphic to . As corollaries, the Jordan triple system has two nontrivial finite-dimensional inequivalent irreducible representations, while the Jordan triple system has four nontrivial inequivalent finite-dimensional irreducible representations.

Double covering of the modular A5 group and lepton flavor mixing in the minimal seesaw model

In this paper, we investigate the double covering of modular $\Gamma^{}_5 \simeq A^{}_5$ group and derive all the modular forms of weight one for the first time. The modular forms of higher weights are also explicitly given by decomposing the direct products of weight-one forms. For the double covering group $\Gamma^\prime_5 \simeq A^\prime_5$, there exist two inequivalent two-dimensional irreducible representations, into which we can assign two right-handed neutrino singlets in the minimal seesaw model. Two concrete models with such a salient feature have been constructed to successfully explain lepton mass spectra and flavor mixing pattern. The allowed parameter space for these two minimal scenarios has been numerically explored, and analytically studied with some reasonable assumptions.

Casimir Effect for Fermion Condensate in Conical Rings

The fermion condensate (FC) is investigated for a (2+1)-dimensional massive fermionic field confined on a truncated cone with an arbitrary planar angle deficit and threaded by a magnetic flux. Different combinations of the boundary conditions are imposed on the edges of the cone. They include the bag boundary condition as a special case. By using the generalized Abel-Plana-type summation formula for the series over the eigenvalues of the radial quantum number, the edge-induced contributions in the FC are explicitly extracted. The FC is an even periodic function of the magnetic flux with the period equal to the flux quantum. Depending on the boundary conditions, the condensate can be either positive or negative. For a massless field the FC in the boundary-free conical geometry vanishes and the nonzero contributions are purely edge-induced effects. This provides a mechanism for time-reversal symmetry breaking in the absence of magnetic fields. Combining the results for the fields corresponding to two inequivalent irreducible representations of the Clifford algebra, the FC is investigated in the parity and time-reversal symmetric fermionic models and applications are discussed for graphitic cones.

On the universal envelope of a Jordan triple system of n × n matrices

We study the universal (associative) envelope of the Jordan triple system of all [Formula: see text] [Formula: see text] matrices with the triple product [Formula: see text] over a field of characteristic 0. We use the theory of non-commutative Gröbner–Shirshov bases to obtain the monomial basis and the center of the universal envelope. We also determine the decomposition of the universal envelope and show that there exist only five finite-dimensional inequivalent irreducible representations of the Jordan triple system of all [Formula: see text] matrices.

Representations and classification of the compact quantum groups Uq(2) for complex deformation parameters

In this paper, we obtain a complete list of inequivalent irreducible representations of the compact quantum group [Formula: see text] for nonzero complex deformation parameters [Formula: see text], which are not roots of unity. The matrix coefficients of these representations are described in terms of the little [Formula: see text]-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of [Formula: see text] is obtained. Thus, we have an explicit description of the Peter–Weyl decomposition of [Formula: see text]. As an application, we discuss the Fourier transform and establish the Plancherel formula. We also describe the decomposition of the tensor product of two irreducible representations into irreducible components. Finally, we classify the compact quantum group [Formula: see text].

Vacuum fermionic currents in braneworld models on AdS bulk with a cosmic string

We investigate the effects of a brane and magnetic-flux-carrying cosmic string on the vacuum expectation value (VEV) of the current density for a charged fermionic field in the background geometry of (4+1)-dimensional anti-de Sitter (AdS) spacetime. The brane is parallel to the AdS boundary and the cosmic string is orthogonal to the brane. Two types of boundary conditions are considered on the brane that include the MIT bag boundary condition and the boundary conditions in Z2-symmetric braneworld models. The brane divides the space into two regions with different properties of the vacuum state. The only nonzero component of the current density is along the azimuthal direction and in both the regions the corresponding VEV is decomposed into the brane- free and brane-induced contributions. The latter vanishes on the string and near the string the total current is dominated by the brane-free part. At large distances from the string and in the region between the brane and AdS horizon the decay of the brane-induced current density, as a function of the proper distance, is power-law for both massless and massive fields. For a massive field this behavior is essentially different from that in the Minkowski bulk. In the region between the brane and AdS boundary the large-distance decay of the current density is exponential. Depending on the boundary condition on the brane, the brane-induced contribution is dominant or subdominant in the total current density at large distances from the string. By using the results for fields realizing two inequivalent irreducible representations of the Clifford algebra, the vacuum current density is investigated in C - and P -symmetric fermionic models. Applications are given for a cosmic string in the Randall-Sundrum-type braneworld model with a single brane.

New R-matrices with non-additive spectral parameters and integrable models of strongly correlated fermions

We present a general formula for constructing R-matrices with non-additive spectral parameters associated with a type-I quantum superalgebra. The spectral parameters originate from two one-parameter families of inequivalent finite-dimensional irreducible representations of the quantum superalgebra upon which the R-matrix acts. Applying to the quantum superalgebra Uq(gl(2|1)), we obtain the explicit expression for the Uq(gl(2|1))− invariant R-matrix which is of non-difference form in spectral parameters. Using this R­ matrix we derive a new two-parameter integrable model of strongly correlated electrons with pure imaginary pair hopping terms.

The Casimir effect for fermionic currents in conical rings with applications to graphene ribbons

We investigate the combined effects of boundaries and topology on the vacuum expectation values (VEVs) of the charge and current densities for a massive 2D fermionic field confined on a conical ring threaded by a magnetic flux. Different types of boundary conditions on the ring edges are considered for fields realizing two inequivalent irreducible representations of the Clifford algebra. The related bound states and zero energy fermionic modes are discussed. The edge contributions to the VEVs of the charge and azimuthal current densities are explicitly extracted and their behavior in various asymptotic limits is considered. On the ring edges the azimuthal current density is equal to the charge density or has an opposite sign. We show that the absolute values of the charge and current densities increase with increasing planar angle deficit. Depending on the boundary conditions, the VEVs are continuous or discontinuous at half-integer values of the ratio of the effective magnetic flux to the flux quantum. The discontinuity is related to the presence of the zero energy mode. By combining the results for the fields realizing the irreducible representations of the Clifford algebra, the charge and current densities are studied in parity and time-reversal symmetric fermionic models. If the boundary conditions and the phases in quasiperiodicity conditions for separate fields are the same the total charge density vanishes. Applications are given to graphitic cones with edges (conical ribbons).

Conjugacy classes of Renner monoids

In this paper we describe conjugacy classes of a Renner monoid R with unit group W, the Weyl group. We show that every element in R is conjugate to an element ue where u∈W and e is an idempotent in a cross section lattice. Denote by W(e) and W⁎(e) the centralizer and stabilizer of e∈Λ in W, respectively. Let W(e) act by conjugation on the set of left cosets of W⁎(e) in W. We find that ue and ve (u,v∈W) are conjugate if and only if uW⁎(e) and vW⁎(e) are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugacy classes of R and the orbits of this action. We then obtain a formula for calculating the number of conjugacy classes of R, and describe in detail the conjugacy classes of the Renner monoid of some J-irreducible monoids.We then generalize Munn conjugacy on a rook monoid to any Renner monoid and show that Munn conjugacy coincides with semigroup conjugacy, action conjugacy, and character conjugacy. We also show that the number of inequivalent irreducible representations of R over an algebraically closed field of characteristic zero equals the number of Munn conjugacy classes in R.

Finite Groups with Irreducible Projective Representations of Large Degree

In this article, we will investigate finite groups G that possess a 2-cocycle for which all the complex irreducible projective representations have the same degree. We will show that, provided the number of such inequivalent irreducible projective representations is relatively small compared to the order of G, then the degree equality property is inherited by all the Sylow subgroups of G.

Representations of certain generalized Schur algebras

This paper studies semigroup algebras A over a domain R for a certain family of semigroups which includes the symmetric groups, the full-transformation semigroups, and the rook semigroups. For each algebra A an A-module M on which A acts is introduced and an algebra B is defined as the commuting algebra of A acting on M. It is shown that A is also the commuting algebra of B acting on M. When A is a symmetric group algebra R [ S r ] , B is the corresponding Schur algebra S R ( r , r ) . The representation theory of the semigroup algebras A is well known. This paper analyzes the irreducible representations of the B algebras. In many cases, including the symmetric group, full-transformation semigroup, and rook semigroup cases, a complete set of inequivalent irreducible representations of the B algebra is obtained. In particular, this gives what appears to be a new description of a complete set of irreducible representations for the Schur algebra S R ( r , r ) .

Symmetry based analysis of the Kohn anomaly and electron-phonon interaction in graphene and carbon nanotubes

Symmetry based analysis of the electron-phonon coupling in graphene and carbon nanotubes is performed and Kohn anomalies, their Brillouin-zone positions together with the complete set of good quantum numbers are predicted. Interestingly, graphene dynamical representation is found to contain only a small portion of quite a large set of inequivalent irreducible representations of the relevant full symmetry group. Besides, vanishing of the electron-phonon interaction for majority of the normal displacements is also shown to be a consequence of the symmetry. The results are further numerically confirmed within full and tight-binding density-functional calculations and force constants model and enhanced coupling to the Fermi level electrons of the Dirac point ${A}_{1g}$ mode with respect to the $\ensuremath{\Gamma}$ point ${E}_{2g}$ mode is confirmed. Finally, energy dispersion of the kink phonon spectrum is analytically evaluated and compared to the classical phonon spectrum.

MATRIX UNITS FOR THE GROUP ALGEBRA kGf = k((ℤ2 × ℤ2) ≀ Sf)

The irreducible representations of the group Gf := (ℤ2 × ℤ2) ≀ Sf are indexed by 4-partitions of f, i.e., by the set {[α]3[β]2[γ]1[δ]0|α ⊢ u3, β ⊢ u2, γ ⊢ u1, δ ⊢ u0, u0 + u1 + u2 + u3 = f}. This set is in 1 - 1 correspondence with partitions of 4f whose 4-core is empty. In this paper we construct the inequivalent irreducible representations of Gf. We also compute a complete set of seminormal matrix units for the group algebra kGf.

Charged Representations of the Infinite Fermi and Clifford Algebras

The real and quaternionic charge conjugation operators invariant under the infinite-dimensional Clifford algebra, or compatible with the Fermi algebra, are determined. There results a maze of inequivalent irreducible charged representations, all of which are non-Fock. The representation vectors and their charges admit two interpretations besides those of spinors or states of quantum fields: as wavelets on the circle, with charge conjugations acting via ordinary complex conjugation; and as infinite-dimensional numbers, with charge conjugations acting by automorphisms.

Generating fast Fourier transforms of solvable groups

This paper presents a new algorithm for constructing a complete list of pairwise inequivalent ordinary irreducible representations of a finite solvable group G. The input of the algorithm is a pc presentation corresponding to a composition series refining a chief series of G. Modifying the Baum–Clausen algorithm for supersolvable groups and combining this with an idea of Plesken for constructing intertwining spaces, we derive a worst-case upper complexity bound O( p·| G| 2log(| G|)), where p is the largest prime divisor of | G|. The output of the algorithm is well suited to performing a fast Fourier transform of G. For supersolvable groups there are composition series which are already chief series. In this case the generation of discrete Fourier transforms can be done more efficiently than in the solvable case. We report on a recent implementation for the class of supersolvable groups.

Two-Dimensional Electron Gas in a Periodic Potential and External Magnetic Field: States of Pairs and Three-Particle Systems

The group-theoretical classification of states of identical particle pairs is presented. Then obtained states are coupled with those of an antiparticle to construct states of a three-particle system. Investigations are performed using products of irreducible projective representations of the 2D translation group. For a given BvK period N degeneracy of pair states is N, whereas three-particle states are N 2 -fold degenerated. It has to be underlined that the case of even N is more complicated since pair states are labelled by four inequivalent irreducible projective representations. The problem of symmetry properties with respect to particles transposition is briefly discussed