Small Representations Research Articles

Small representations of the relation algebra ?n+1(1, 2, 3)

Applying combinatorial methods, we prove that the symmetric relation algebra ɛn+1(1, 2, 3) ofn+1 atoms is finitely representable for alln ≳ 1, on at most (2+o(1))n2 elements asn → ∞. We explicitly construct a representation of size 4.5n2, for every n >1.

Small representations of finite distributive lattices as congruence lattices

A recent result of G. Grätzer, H. Lakser, and E. T. Schmidt states that for any distributive lattice D with n join-irreducible elements, there exists a lattice L with O ( n 2 ) O({n^2}) elements, whose congruence lattice ConL is isomorphic to D. We show that this result is best possible.

Reproducing Kernels and Composition Series for Spaces of Vector-Valued Holomorphic Functions on Tube Domains

We study Hilbert spaces of vector-valued holomorphic sections on tube domains of type II and III. We consider some small representations of the maximal compact subgroups and calculate the norm of each K-type in the Hilbert spaces. As a consequence we get composition series of the analytic continuations of certain holomorphic discrete series and expansions of matrix-valued reproducing kernels.

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U q ( $$\widehat{sl}$$ n+1) (whereq ∈ ℂ× is not a root of unity) which are irreducible as representations of U q (sl n+1). We call such representations ‘small’. In 1986, Jimbo defined a family of homomorphismsev a from U q (sl n+1) to (an enlargement of) U q (sl,n+1), depending on a parametera ∈ ℂ·. A second family,ev a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U q ( $$\widehat{sl}$$ n+1) is obtained by pulling back an irreducible representation of U q (sl n+1) byev a orev a for somea ∈ ℂ·.

BRST quantization of 2D Yang-Mills theory with anomalies

In this paper we discuss the BRST quantization of anomalous 2D Yang-Mills (YM) theory. Since we use an oscillator basis for the YM Fock space the anomaly appears already for a pure YM system and the constraints form a Kac-Moody algebra with negative central charge. We also discuss the coupling of chiral fermions and find that the BRST cohomology for systems with chiral fermions in a sufficiently large representation of the gauge group is completely equivalent to the cohomology of the finite-dimensional gauge group. For pure YM theory of YM theory coupled to chiral fermions in small representations there exists an infinite number of inequivalent cohomology classes. This is discussed in some detail for example of SU(2).

Examples of tensor categories

Let G be a tensor category over a field k of characteristic 0. In his paper [-i], w 7, Delinge had established rather general conditions for G to be the category of representations of an algebraic group. He has shown that if any object neC satisfies the condition A " n = 0 for a sufficiently large n, then 0 is equivalent to the category Rep(G) of representations of a proalgebraic group scheme G over k. For tensor categories over fields of positive characteristic this result is no longer true. In the first part of this paper we present a construction of a class of tensor categories over fields of positive characteristic for which an analogue of Deligne's theorem does not hold. To describe these categories, denote by Gz an algebraic Chevalley group over Z and by G the corresponding group over Fp (for a sufficiently large p). Our category C will be a certain "subquotient" of the category R(G) of finite-dimensional representations of G. More precisely, let S be the full subcategory of R(G) whose objects (called small representations) are direct sums of the irreducible representations of G with highest weights inside the fundamental alcove. Let also SL 2 c G be the principal SLz-subgroup in G and (SLz)Im c SL2 be the infinitesimal neighbourhood of the Cartan subgroup H in SL2. Denote by O the full subcategory of R(G) consisting of representations M of G (called quasiprojective representations) such that the restriction of M to (SL2)m) is projective. The category 0 is a quotient of the subcategory S+PcR(G). In the last section of the paper we show that the same approach can be applied to representations of quantum groups. Let U be the quantum group at a root of prime order p of 1 (which is a Hopf algebra over the ring of integers in a cyclotomic field, see [2, 3]). Starting from the category of representations of U, we construct a semisimple braided category ~(G, p) with the finite number of simple objects. On the other hand, in Wess-Zumino-Witten conformal field theory one encounters the category O(G, m) of integrable representations of the correspond-

A fast image simulation technique for high resolution electron microscopy with multicomponent atomic species

A new method has been developed for simulating high resolution electron microscope images of weak phase objects via a digital television frame store system and fast Fourier transforms of a graphical representation of structures containing several atomic species. Here masks are constructed which consist of circular disk regions whose areas are proportional to the scattering power of the different atom types. These masks represent the object transmission function. The method extends the previous work on image computations of monatomic species objects using small point-like model representations of atom positions. Several examples will be given to verify the relative scattering power from masks corresponding to different atomic species such as Y, Ba, Cu, and O, as well as the limitations of this method for representing objects. Image simulations, which are in agreement with experiments, will also be presented for superconducting oxide materials of the form YBa2Cu3O7 using the circular disk method.

Thalamic connections of three representations of the body surface in somatosensory cortex of gray squirrels

The anatomical tracer, wheat germ agglutinin, was used to determine the connections of electrophysiologically identified locations in three architectonically distinct representations of the body surface in the somatosensory cortex of gray squirrels. Injections in the first somatosensory area, S-I, revealed reciprocal connections with the ventroposterior nucleus (VP), a portion of the thalamus just dorsomedial to VP, the posterior medial nucleus, Pom, and sometimes the ventroposterior inferior nucleus (VPI). As expected, injections in the representation of the face in S-I resulted in label in ventroposterior medial (VPM), the medial subnucleus of VP, whereas injections in the representation of the body labeled ventroposterior lateral (VPL), the lateral subnucleus of VP. Furthermore, there was evidence from connections that the caudal face and head are represented dorsolaterally in VPM, and the forelimb is represented centrally and medially in VPL. The results also support the conclusion that a representation paralleling that in VP exists in Pom, so that the ventrolateral part of Pom represents the face and the dorsomedial part of Pom is devoted to the body. Because connections with VPI were not consistently revealed, the possibility exists that only some parts or functional modules of S-I are interconnected with VPI. Two separate small representations of the body surface adjoin the caudoventral border of S-I. Both resemble the second somatosensory area, S-II, enough to be identified as S-II in the absence of evidence for the other. We term the more dorsal of the two fields S-II because it was previously defined as S-II in squirrels (Nelson et al., '79), and because it more closely resembles the S-II identified in most other mammals. We refer to the other field as the parietal ventral area, PV (Krubitzer et al, '86). Injections in S-II revealed reciprocal connections with VP, Pom, and a thalamic region lateral and caudal to Pom and dorsal to VP, the posterior lateral nucleus, Pol. Whereas major interconnections between S-II and VPI have been reported for cats, raccoons, and monkeys, no such interconnections were found for S-II in squirrels. The parietal ventral area, PV, was found to have prominent reciprocal interconnections with VP, VPI, and the internal (magnocellular) division of the medial geniculate complex (MGi). The pattern of connections conforms to the established somatotopic organization of VP and suggests a crude parallel somatotopic organization in VPI. Less prominent interconnections were with Pol.(ABSTRACT TRUNCATED AT 400 WORDS)

Minimal parabolic systems and small representations

Minimal parabolic systems and small representations

Symmetry of Bloch functions in the space groupD4h6of perfect antiferromagnetic chromium

Compatibility relations are determined between the space group of paramagnetic chromium ${O}_{h}^{9}$, and the nonsymmorphic space group ${D}_{4h}^{6}$ of the perfect antiferromagnetic phase. These relations give, for each paramagnetic symmetry label, the corresponding small representations which label the Bloch functions in the antiferromagnetic space group. Using these relations one can determine linear combinations of paramagnetic Bloch functions which are symmetry-adapted to the antiferromagnetic structure. Further, one can determine those band degeneracies in the reduced Brillouin zone which split in the antiferromagnetic phase. Results are given for the paramagnetic bands of chromium.

Full representations of the double space groups based on the simple cubic lattice

The generating matrices of the induced (full) representations for the space groups Pm3m(=O 1 h), Pn3n(=O 2 h), Pm3n(=O 3 h) and Pn3m(=O 4 h) have been calculated. We have considered the representations corresponding to four high-symmetry points Γ, R, X and M in the Brillouin zone. We give explicitly the generators and generating relations for the full irreducible representations induced from the small irreducible representations of Miller and Love (M-L). Using these generators and generating relations and the program written in ALGOL, for the computer ODRA 1204 we have calculated the matrices of the full irreducible representations of the full space groups G k Γ , G k R , G ⋇ k M and G ⋇ k X .

Induced representations of the full holosymmetric double space groups based on the body-centred cubic bravais lattice

We introduce a modified formula for the element matrices of the induced representations of any crystallographic space group. With the help of this formula we can obtain all the induced representations of any space group. The generating matrices of the induced representations for the groups Im3m ( O 9 h) and Ia3d ( O 10 h) have been calculated. In particular, we have considered the representations corresponding to high-symmetry points in the Brillouin zone. For these two space groups we give explicitly the generators and generating relations for the full irreducible representations induced from the small irreducible representations of Miller and Love. A program, written in ALGOL for the computer ODRA 1204, computes from the generating relations and from the generators the induced unitary irreducible representations of the little and full wave vector groups at high symmetry points. The matrices of the full irreducible representations of the full space groups are needed for the construction of the invariants involved in the theory of second-order phase transitions.

Nonferroic phase transitions

Nonferroic phase transitions are defined as the structural transitions occurring with a breaking of translational symmetry within the same crystal class. They involve no new macroscopic tensor components below the transition point, and they are generally identified experimentally through the onset of superlattice reflections denoting the multiplication of the crystal's unit cell. A theoretical analysis of these transitions is presented, based on Landau's symmetry criteria for continuous transitons, of the order-parameter symmetries, space-group changes, and free-energy expansions. We establish that the order parameters of such transitions are necessarily related to one-dimensional (real or complex) small representations of the group of the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ vector. Their symmetry characteristics are, in general, simpler than those of other types of structural transitions. Most of them possess a one-component order parameter inducing a doubling of the unit cell. The remaining ones are associated with order-parameter dimensions as high as six, and unit-cell multiplications up to thirty-two. The coupling of the order parameter to macroscopic quantities, illustrated by the example of dielectric ones, is shown to belong to two possible schemes. The relation between nonferroics and antiferroelectrics is discussed. The theoretical results are compared to the available experimental data which pertain mainly to organic compounds and metallic alloys.

Small rank permutation representations of finite Chevalley groups

Small rank permutation representations of finite Chevalley groups

Computers in Lie Algebras. II: Calculation of Outer Multiplicities

The tensor product of two irreducible representations $\rho _1 $ and $\rho _2 $ of a complex simple Lie algebra L is a completely reducible representation of L, but it is often not irreducible. Associated with each irreducible representation $\rho $ of L is its outer multiplicity with respect to $\rho _1 \otimes \rho _2 $. The outer multiplicity of $\rho $ is the number of times it appears in the decomposition of $\rho _1 \otimes \rho _2 $ into its sum of irreducible components.This paper starts with a survey of basic results about outer multiplicities and the structure of the tensor product of two representations. Then many of the methods of computing the outer multiplicities of a given tensor product are discussed. Finally computer implementations of the Racah formula and the method of successive subtractions are described. The Racah formula is very fast for small dimensioned representations and ranks less than 5. The method of successive subtractions will handle larger ranks and very large dimensioned ...

Stimulus characteristics and object naming in aphasic patients

The influence of two stimulus characteristics (three-dimensional versus two-dimensional representations and large versus small two-dimensional representations) on visual object naming in aphasic patients was assessed. The patients produced a significantly higher number of correct naming responses to the three-dimensional than to the two-dimensional representations. The size of the two-dimensional representations did not influence naming performances significantly. The correlations between performance levels on the three forms of stimulus presentations were extremely high. The theoretical and practical implications of the findings are considered.

A General Method for Obtaining the Characters of Space Groups

A general method is developed for obtaining the characters of irreducible and small repre­ sentations of space groups. This method is applied especially when a given space group is nonsymmorphic and small representations are specified by reduced wave vectors lying on the boundary of the first Brillouin zone. It is shown that for most of the nonsymmorphic space groups this method reduces the calculations of the characters of such small repre sentations to those within the framework of point groups.

Some comments on the theory of space-group representations. II. Basic domain and representation domain

For pt. I see ibid., vol. 3, no.3, 610 (1970). Tables of space-group representations now available are incomplete for those space groups which do not have the full symmetry of the Bravais lattice on which they are based. It is shown that this defect can be overcome without recourse to further tabulation for all space groups except Pa3(Th6). This particular space group is considered separately, and is shown to be exceptional in that alone out of all the 230 space groups it possesses two k-vectors, which are related by an operation of the holosymmetric point group, but whose groups of k possess small representations of different dimensionalities. Nor does the addition of time reversal remove this anomaly.

Some comments on the theory of space-group representations: I

It is suggested that the Brillouin zones conventionally used for triclinic, monoclinic and, perhaps, trigonal crystals are unnecessarily complicated and can usefully be simplified. Secondly, the use of induced representations of a space group G (the full-group method) and the use of small representations of the little groups, Gk, of the wave vector k (the subgroup method) are compared. Thirdly, the authors discuss the various possible kinds of extra degeneracy which may be produced among the eigenfunctions of a particle or quasi- particle in a crystal as a result of the existence of time-reversal symmetry.

Selection Rules for Scattering Processes in Crystals

Different approaches in finding selection rules for scattering processes in crystals are discussed. It is shown that both the subgroup and the full-group procedures lead, not only to the same selection rules, but also to the same formula for finding them. The final result is expressed in terms of characters of small representations of the groups of k, and it is a matter of taste which procedure to use in deriving the formula for selection rules.