Distinct Irreducible Representations Research Articles

First-principles investigation of the magnetoelectric properties of Ba7Mn4O15

Type-II multiferroics, in which the magnetic order breaks inversion symmetry, are appealing for both fundamental and applied research due their intrinsic coupling between magnetic and electrical orders. Using first-principles calculations we study the ground state magnetic behaviour of Ba7Mn4O15 which has been classified as a type-II multiferroic in recent experiments. Our constrained moment calculations with the proposed experimental magnetic structure shows the spontaneous emergence of a polar mode giving rise to an electrical polarisation comparable to other known type-II multiferroics. When the constraints on the magnetic moments are removed, the spins self-consistently relax into a canted antiferromagnetic ground state configuration where two magnetic modes transforming as distinct irreducible representations coexist. While the dominant magnetic mode matches well with the previous experimental observations, the second mode is found to possess a different character resulting in a non-polar ground state. Interestingly, the non-polar magnetic ground state exhibits a significantly strong linear magnetoelectric (ME) coupling comparable to the well-known multiferroic BiFeO3, suggesting strategies to design new linear MEs.

Comments of [formula omitted]-supersymmetry in superfield formalism

We investigate superfield formulation of the minimal Z22-supersymmetry. It is shown that the integrability on Z22-superspace guarantees the invariance of action. Then we present two superfields which carry distinct irreducible representations of the Z22-supersymmetry algebra. One of them gives integrable Lagrangian and the other does not. We also show that integrable superfields with different Z22-degree also carry irreducible representations and they give invariant actions. To perform this analysis, the representation theory of the minimal Z22-supersymmetry algebra is studied in some detail.

Quantum expanders and growth of group representations

Let π be a finite dimensional unitary representation of a group G with a generating symmetric n-element set S⊂G. Fix ε>0. Assume that the spectrum of |S| -1 ∑ s∈S π(s)⊗π(s) ¯ is included in [-1,1-ε] (so there is a spectral gap ≥ε). Let r N ' (π) be the number of distinct irreducible representations of dimension ≤N that appear in π. Then let R n,ε ' (N)=supr N ' (π) where the supremum runs over all π with n,ε fixed. We prove that there are positive constants δ ε and c ε such that, for all sufficiently large integer n (i.e. n≥n 0 with n 0 depending on ε) and for all N≥1, we have expδ ε nN 2 ≤R n,ε ' (N)≤expc ε nN 2 . The same bounds hold if, in r N ' (π), we count only the number of distinct irreducible representations of dimension exactly =N.

Lattice modes and the Jahn-Teller ferroelectric transition ofGaV4S8

Crystal of ${\mathrm{GaV}}_{4}{\mathrm{S}}_{8}$, a multiferroic system hosting a N\'eel-type skyrmion lattice phase, has been investigated by polarized Raman and IR spectroscopy above and below the ferroelectric phase transition. Counts of the observed IR and Raman-active modes belonging to distinct irreducible representations agree quite well with group-theory predictions. Phonon spectra are assigned and interpreted with the aid of ab initio calculations of the phonon spectra in the ferroelectric phase. Results allow appreciation of phonon frequencies of the modes involved in Jahn-Teller distortion and their contribution to the spontaneous polarization.

Coexistence of Ferroelectric Phases and Phonon Dynamics in Relaxor Ferroelectric Na0.5Bi0.5TiO3 Based Single Crystals

A combination of polarized Raman technique, infrared reflectance spectra, and first‐principles density‐functional theoretical calculations were used to investigate structure transformation and lattice vibrations of Na0.5Bi0.5TiO3, Na0.5Bi0.5TiO3–5%BaTiO3, and Na0.5Bi0.5TiO3–8%K0.5Bi0.5TiO3 single crystals. It was found that Na0.5Bi0.5TiO3 is of a two‐phase mixture with rhombohedral and monoclinic structures at room temperature. Correspondingly, three Raman‐active phonon modes located at 395, 790, and 868 cm−1, which were previously assumed as A1 modes of rhombohedral phase have been reassigned as A′′, A′, and A′ modes of monoclinic phase in the present work. In particular, a strong low‐frequency A′′ mode at 49 cm−1 was found and its temperature dependence was revealed. Two deviations from linearity for the abrupt frequency variation in the A′′ mode and Ti–O bond have been detected at temperatures of ferroelectric to antiferroelectric phase transition TF–AF and dielectric maximum temperature Tmax. The appearance of Na–O vibrations at 150 cm−1 was found below Tmax, indicating the existence of nanosized Na+TiO3 clusters. The observed Raman and infrared active modes belonging to distinct irreducible representations are in good agreement with group‐theory predictions, which suggests 9A1+9E and 36A′′+24A′ modes for the rhombohedral and monoclinic phases of Na0.5Bi0.5TiO3, respectively.

Schur asymptotics of Veronese syzygies

We study the asymptotic behavior of Veronese syzygies as representations of the general linear group. For a fixed homological degree p of the syzygies, we describe the exact asymptotic growth for the number of distinct irreducible representations and for the number of irreducible representations also counting multiplicities. This shows that asymptotically Veronese syzygies have a very rich algebraic and representation-theoretic structure as the degree of the embedding grows.

Multiple soft-mode vibrations of lead zirconate.

Polarized Raman, IR, and time-domain THz spectroscopy of orthorhombic lead zirconate single crystals have yielded a comprehensive picture of temperature-dependent quasiharmonic frequencies of its low-frequency phonon modes. It is argued that these modes primarily involve vibrations of Pb ions and librations of oxygen octahedra. Their relation to phonon modes of the parent cubic phase is proposed. Counts of the observed IR and Raman active modes belonging to distinct irreducible representations agree quite well with group-theory predictions. Analysis of the results yields insight into the phase transition mechanism, involving a soft ferroelectric branch coupled by a trilinear term to another two oxygen octahedra tilt modes.

Helical magnetic state in the distorted triangular lattice ofα-CaCr2O4

The magnetic properties of the high temperature alpha form of the CaCr2O4 compound have been investigated for the first time by magnetic susceptibility, specific heat and powder neutron diffraction. The system undergoes a unique magnetic phase transition at 43K to a long range order incommensurate helical phase with magnetic propagation vector k=(0,0.3317(2),0). The magnetic model proposed from neutron diffraction data shows that the plane of rotation of the spins is perpendicular to the wave-vector, and that the magnetic modulation is consistent with two modes belonging to distinct irreducible representations of the group. The magnetic point group 2221' is not compatible with ferroelectricity unlike the CuCrO2 delafossite [Kimura et al., Phys. Rev. B, 78 140401 (2008)] but predicts the existence of quadratic magnetoelectric effects, discussed based on a Landau analysis.

REPRESENTATIONS AND RIGIDITY OF Aut(F3)

We construct some new matrix representations of Aut (F3), the automorphism group of a free group of rank 3, and show that there are infinitely many distinct irreducible representations of Aut (F3) in dimensions at most 42.

On the weighted complexity of a regular covering of a graph

We consider the weighted complexity of a graph G, and present a generalization of Northshield's Theorem on the complexity of G. Furthermore, we give an explicit formula for the weighted complexity of a regular covering H of G in terms of that of G and a product of determinants over the all distinct irreducible representations of the covering transformation group of H.

Simple SCF method with spin-orbit interaction: SOSCF method

An equivalence restricted open-shell self-consistent field (SCF) method has been extended to include spin-orbit (SO) interaction for the variational calculation of individual SO sublevels both in the LS- and jj-coupling schemes. The method can be applied to simple atomic and molecular systems within an abelian point group with higher symmetry, e.g., C2v, D2, or D2h, in which the three rotational symmetries of Rx, Ry, and Rz belong to distinct irreducible representations. Test calculations with relativistic effective core potentials (RECPs) by Christiansen and Ermler et al. are performed for most main-group p-block elements except for group 15, and for diatomic hydrides of group 14 and 16 elements, and the results are compared with previous theoretical and experimental data. In spite of the simplicity, our results are in reasonable agreement with the experimental data for most group 13 and 17 elements and for lighter group 14 and 16 elements. For heavier group 14 and 16 elements, however, the results in LS coupling become poorer due to the significant second-order SO effect. In such cases, the jj-coupling scheme yields better results, and the turning point between LS and jj preference was found between the fourth- and fifth-row elements. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 515–530, 1999

A decomposition formula for representations

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

Composition series and colour symmetry point groups

The idea of composition series that exist among the 32 crystallographic point groups is explored in rederiving the 106 colour symmetry point groups (58 magnetic, 18 polychromatic, 30 multicolour groups) which are associated with the distinct irreducible representations of the crystallographic point groups employing the concept of colour generators. The colour symmetry group associated with an irreducible representation of Gi is generated either from the maximal normal subgroup Gi+1 or from an appropriate colour symmetry group associated with Gi+1. Simultaneous generation of various colour symmetry point groups that may exist for all the point groups contained in a chosen composition series is a feature of this present method. The elegance of this method and its physical significance are also briefly discussed.

0-Hecke algebras

AbstractThe structure of a 0-Hecke algebra H of type (W, R) over a field is examined. H has 2n distinct irreducible representations, where n = ∣R∣, all of which are one-dimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2n indecomposable left ideals, in a similar way to Solomon's (1968) decomposition of the underlying Coxeter group W.

Embedding numbers for finite groups

This note is concerned with the following problem. Let H denote a subgroup of a finite group G and let L denote a linear or one dimensional representation (i.e., a character) of H. We assume throughout that the field F is algebraically closed and is either of characteristic 0 or of prime characteristic which does not divide the order of any groups under consideration. Let GIL denote the corresponding induced representation of G. How many distinct (i.e., nonequivalent) irreducible representations appear in the decomposition of GI L into irreducible parts? (This number is just the central intertwining number of GI L, which is denoted by Ct(GI L). Cf. [1].) More specifically, we are interested in determining an upper bound on the number of distinct irreducible representations which will appear, purely in terms of the way H is embedded in G, and in terms which do not depend on the particular linear representation L of H. Two such bounds come quickly to mind. The number of classes (of conjugates) of the super group G, which we denote { G: e}, is clearly an upper bound. Dimension considerations also give [G: H] as an upper bound. We now introduce a new group theoretic invariant which heuristically is a measure of the manner in which the classes of G are distributed among the H-cosets of G. DEFINITION. Let H be a (not necessarily normal) subgroup of a finite group G. For each normal subset N of G, let +1(N) denote the number of classes (of conjugates) of G contained in N. Let +2(N) denote the number of right H-cosets of G which have nonzero intersection with N. Let +(N) = { G: e} -45(N) +42(N). We then define the embedding number of H in G, denoted by (G: H), to be the minimum of the +(N), as N is taken over all normal subsets of G. We remark that a definition of 42 using left cosets would yield the same value for (G: H) since N-' intersects the same number of left cosets as N does right cosets. Taking N= {e } where e is the identity element of the group we have (G: H) {G: e}. Taking N=G we have (G: H)? [G: H]. If H$ G, it is easy to verify that (G: H) > 1. If His a proper normal subgroup, then, taking N=H we have (G:H)<{G:e}. In the case where H is a normal subgroup of G, another number associated with the embedding of H in G is the number of classes in the factor group G/H. We call this the class number of H in G and denote it by { G: H}.