Irreducible Representation Space Research Articles

Magnetic structure of the noncentrosymmetric magnet Sr2MnSi2O7 through irreducible representation and magnetic space group analyses.

Magnetic structures of the noncentrosymmetric magnet Sr2MnSi2O7 were examined through neutron diffraction for powder and single-crystalline samples, as well as magnetometry measurements. All allowed magnetic structures for space group P421m with the magnetic wavevector qm = (0, 0, ½) were refined via irreducible representation and magnetic space group analyses. The compound was refined to have in-plane magnetic moments within the magnetic space group Cmc21.1'c (No. 36.177) under zero field, which can be altered to P212121.1'c (No.19.28) above μ0H = 0.067 (5) T to align induced weak-ferromagnetic components within one layer on the ab plane. All refined parameters are provided following the recent framework based upon the magnetic space group, which better conveys when exchanging crystallographic information for commensurate magnetic structures.

Kazhdan constants and isomorphic graph pairs

Let G be a finite group, and let Γ be a subset of G. The Kazhdan constant of the pair (G,Γ) is defined to bethe maximum distance we can guarantee that an arbitrary unitvector in an arbitrary nontrivial irreducible unitary representation space of G can be moved by some element of Γ. The Kazhdanconstant relates to the expansion properties of the Cayley graph generated by G and Γ, and has been much studied in this context. Different pairs (G1,Γ1) and (G2,Γ2) may give rise to isomorphic Cayley graphs. In this paper, we investigate the question: To whatextent is the Kazhdan constant a graph invariant? In other words, if the pairs yield isomorphic Cayley graphs, must the corresponding Kazhdan constants be equal? In our main theorem, we constructan infinite family of such pairs where the Kazhdan constants areunequal. Other relevant results are presented as well.

Algebraic structure of Dirac Hamiltonians in non-commutative phase space

In this article we study two-dimensional Dirac Hamiltonians with non-commutativity both in coordinates and momenta from an algebraic perspective. In order to do so, we consider the graded Lie algebra generated by Hermitian bilinear forms in the non-commutative dynamical variables and the Dirac matrices in dimensions. By further defining a total angular momentum operator, we are able to express a class of Dirac Hamiltonians completely in terms of these operators. In this way, we analyze the energy spectrum of some simple models by constructing and studying the representation spaces of the unitary irreducible representations of the graded Lie algebra . As application of our results, we consider the Landau model and a fermion in a finite cylindrical well.

Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit

Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites with the carrier space of irreducible representations of U(N) described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M)×U(N−M). We embed this N-component fermion mixture in Fock space through a Schwinger–Jordan (boson and fermion) representation of U(N)-spin operators. We provide different realizations of basis vectors using Young diagrams, Gelfand–Tsetlin patterns and Fock states (for an electron/flux occupation number in the fermionic/bosonic representation). U(N)-spin operator matrix elements in the Gelfand–Tsetlin basis are explicitly given. Coherent state excitations above the ground state are computed and labeled by complex (N−M)×M matrix points Z on the Grassmannian phase space. They adopt the form of a U(N) displaced/rotated highest-weight vector, or a multinomial Bose–Einstein condensate in the flux occupation number representation. Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy (long wavelength) U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.

A Hamiltonian approach for obtaining irreducible projective representations and the k ⋅ p perturbation for anti-unitary symmetry groups

As is known, the irreducible projective representations (Reps) of anti-unitary groups contain three different situations, namely, the real, the complex and quaternionic types with torsion number 1, 2, 4 respectively. This subtlety increases the complexity in obtaining irreducible projective Reps of anti-unitary groups. In the present work, a physical approach is introduced to derive the condition of irreducibility for projective Reps of anti-unitary groups. Then a practical procedure is provided to reduce an arbitrary projective Rep into direct sum of irreducible ones. The central idea is to construct a Hermitian Hamiltonian matrix which commutes with the representation of every group element g ∈ G, such that each of its eigenspaces forms an irreducible representation space of the group G. Thus the Rep is completely reduced in the eigenspaces of the Hamiltonian. This approach is applied in the k ⋅ p effective theory at the high symmetry points (HSPs) of the Brillouin zone for quasi-particle excitations in magnetic materials. After giving the criterion to judge the power of single-particle dispersion around an HSP, we then provide a systematic procedure to construct the k ⋅ p effective model.

Matrix elements of unitary group generators in many-fermion correlation problem. II. Graphical methods of spin algebras

This second instalment continues our survey inter-relating various approaches to the evaluation of matrix elements (MEs) of the unitary group generators, their products, and spin-orbital U(2n) generators, in the basis of the electronic Gel’fand–Tsetlin (G–T) [or Gel’fand–Paldus (G–P) or P-] states spanning the carrier spaces of the two-column irreducible representations (irreps) of the unitary group approach (UGA) to the many-electron correlation problem. Exploiting an intimate relationship between the U(n), $${\mathcal {S}}_N$$ , and SU(2) groups we rely here on the SU(2) formalism within the confines of UGA rather than on an earlier approach relying on the permutation group $${\mathcal {S}}_N$$ . The key is a close relationship between the P-states of UGA and the Yamanouchi–Kotani (Y–K) states of SU(2) that differ only in a phase. This enables one to exploit the powerful graphical techniques of spin (or angular momentum) algebras as first developed by Jucys. We show the usefulness and power of these techniques in the problem of UGA ME evaluation which, moreover, leads automatically to their segmentation, first derived for single generators by Shavitt and for generator products by Boyle and Paldus. We also show how the same technique can be successfully employed for MEs of spin-dependent U(2n) generators in the UGA P-basis.

The Bernstein center of a p-adic unipotent group

François Rodier proved that smooth representations of a p-adic additive group are equivalent to sheaves of vector spaces on the Pontryagin dual group. For a general p-adic unipotent group, the category of smooth representations still shares certain formal properties with the category of sheaves on the space of irreducible representations. We give some representation-theoretic characterizations of the topology on this space, and deduce that the Bernstein center of the category of smooth representations is identified with the algebra of locally constant functions on irreducible representations.

The Mellin Transform and the Plancherel Theorem for the Discrete Heisenberg Group

In the classical representation theory of locally compact groups, there are well-known constructions of a unitary dual space of irreducible representations, the Fourier transform, and the Plancherel theorem. In this paper, we present analogs of these constructions for the discrete Heisenberg group and its irreducible infinite-dimensional representations in a vector space without topology.

Irreducible Representations of the Group of Unipotent Matrices of Order 4 over Integers

Abstract We study a coarse moduli space of irreducible representations of the group of unipotent matrices of order $4$ over the ring of integers that have finite weight. All such representations are known to be monomial (see [4]). To describe a coarse moduli space of such representations, we need to study pairs of subgroups and their characters, which induce non-isomorphic irreducible representations. We obtain a full classification of such pairs and, respectively, a coarse moduli space.

Generalization of Faddeev–Popov rules in Yang–Mills theories: N = 3,4 BRST symmetries

The Faddeev–Popov rules for a local and Poincaré-covariant Lagrangian quantization of a gauge theory with gauge group are generalized to the case of an invariance of the respective quantum actions, [Formula: see text], with respect to [Formula: see text]-parametric Abelian SUSY transformations with odd-valued parameters [Formula: see text], [Formula: see text] and generators [Formula: see text]: [Formula: see text], for [Formula: see text], implying the substitution of an [Formula: see text]-plet of ghost fields, [Formula: see text], instead of the parameter, [Formula: see text], of infinitesimal gauge transformations: [Formula: see text]. The total configuration spaces of fields for a quantum theory of the same classical model coincide in the [Formula: see text] and [Formula: see text] symmetric cases. The superspace of [Formula: see text] SUSY irreducible representation includes, in addition to Yang–Mills fields [Formula: see text], [Formula: see text] ghost odd-valued fields [Formula: see text], [Formula: see text] and [Formula: see text] even-valued [Formula: see text] for [Formula: see text], [Formula: see text]. To construct the quantum action, [Formula: see text], by adding to the classical action, [Formula: see text], of an [Formula: see text]-exact gauge-fixing term (with gauge fermion), a gauge-fixing procedure requires [Formula: see text] additional fields, [Formula: see text]: antighost [Formula: see text], [Formula: see text] even-valued [Formula: see text], 3 odd-valued [Formula: see text] and Nakanishi–Lautrup [Formula: see text] fields. The action of [Formula: see text] transformations on new fields as [Formula: see text]-irreducible representation space is realized. These transformations are the [Formula: see text] BRST symmetry transformations for the vacuum functional, [Formula: see text]. The space of all fields [Formula: see text] proves to be the space of an irreducible representation of the fields [Formula: see text] for [Formula: see text]-parametric SUSY transformations, which contains, in addition to [Formula: see text] the [Formula: see text] ghost–antighost, [Formula: see text], even-valued, [Formula: see text], odd-valued [Formula: see text] and [Formula: see text] fields. The quantum action is constructed by adding to [Formula: see text] an [Formula: see text]-exact gauge-fixing term with a gauge boson, [Formula: see text]. The [Formula: see text] SUSY transformations are by [Formula: see text] BRST transformations for the vacuum functional, [Formula: see text]. The procedures are valid for any admissible gauge. The equivalence with [Formula: see text] BRST-invariant quantization method is explicitly found. The finite [Formula: see text] BRST transformations are derived and the Jacobians for a change of variables related to them but with field-dependent parameters in the respective path integral are calculated. They imply the presence of a corresponding modified Ward identity related to a new form of the standard Ward identities and describe the problem of a gauge-dependence. An introduction into diagrammatic Feynman techniques for [Formula: see text] BRST invariant quantum actions for Yang–Mills theory is suggested.

Towards topological quantum computer

Quantum R-matrices, the entangling deformations of non-entangling (classical) permutations, provide a distinguished basis in the space of unitary evolutions and, consequently, a natural choice for a minimal set of basic operations (universal gates) for quantum computation. Yet they play a special role in group theory, integrable systems and modern theory of non-perturbative calculations in quantum field and string theory. Despite recent developments in those fields the idea of topological quantum computing and use of R-matrices, in particular, practically reduce to reinterpretation of standard sets of quantum gates, and subsequently algorithms, in terms of available topological ones. In this paper we summarize a modern view on quantum R-matrix calculus and propose to look at the R-matrices acting in the space of irreducible representations, which are unitary for the real-valued couplings in Chern–Simons theory, as the fundamental set of universal gates for topological quantum computer. Such an approach calls for a more thorough investigation of the relation between topological invariants of knots and quantum algorithms.

The local Langlands correspondence for inner forms of SL$$_{n}$$

This is a beamer presentation of MIMS eprint 2013.38 of the same title (joint authors are A-M. Aubert, P. Baum, R. Plymen, M. Solleveld). Let $F$ be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group $SL_n (F)$. It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for $SL_n (F)$ enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of $SL_n (F)$. An analogous result is shown in the archimedean case. To settle the case where $F$ has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of $GL_n (F)$, when the fields are close enough compared to the depth of the representations.

Second-order differential equations for bosons with spin $j \geq 1$ and in the bases of general tensor-spinors of rank 2j

A boson of spin-j>1 can be described in one of the possibilities within the Bargmann-Wigner framework by means of one sole differential equation of order twice the spin, which however is known to be inconsistent as it allows for non-local, ghost and acausally propagating solutions, all problems which are difficult to tackle. The other possibility is provided by the Fierz-Pauli framework which is based on the more comfortable to deal with second order Klein-Gordon equation, but it needs to be supplemented by an auxiliary condition. Although the latter formalism avoids some of the pathologies of the high-order equations, it still remains plagued by some inconsistencies such as the acausal propagation of the wave fronts of the (classical) solutions within an electromagnetic environment. We here suggest a method alternative to the above two that combines their advantages while avoiding the related difficulties. Namely, we suggest one sole strictly (j,0)+ (0,j) representation specific second order differential equation, which is derivable from a Lagrangian and whose solutions do not violate causality. The equation under discussion presents itself as the product of the Klein-Gordon operator with a momentum independent projector on Lorentz irreducible representation spaces constructed from one of the Casimir invariants of the spin-Lorentz group. The basis used is that of general tensor-spinors of rank-2j.

Optimal measurements for symmetric quantum states with applications to optical communication

The minimum probability of error (MPE) measurement discriminates between a set of candidate quantum states with the minimum average error probability allowed by quantum mechanics. Conditions for a measurement to be MPE were derived by Yuen, Kennedy and Lax (YKL). MPE measurements have been found for states that form a single orbit under a group action, i.e., there is a transitive group action on the states in the set. For such state sets, termed geometrically uniform (GU) by Forney, it was shown that the `pretty good measurement' (PGM) attains the MPE. Even so, evaluating the actual probability of error (and other performance metrics) attained by the PGM on a GU set involves inverting large matrices, and is not easy in general. Our first contribution is a formula for the MPE and conditional probabilities of GU sets, using group representation theory. Next, we consider sets of pure states that have multiple orbits under the group action. Such states are termed compound geometrically uniform (CGU). MPE measurements for general CGU sets are not known. In this paper, we show how our representation-theoretic description of optimal measurements for GU sets naturally generalizes to the CGU case. We show how to compute the MPE measurement for CGU sets by reducing the problem to solving a few simultaneous equations. The number of equations depends on the sizes of the multiplicity space of irreducible representations. For many common group representations (such as those of several practical good linear codes), this is much more tractable than solving large semi-definite programs---which is what is needed to solve the YKL conditions numerically for arbitrary state sets. We show how to evaluate MPE measurements for CGU states for some examples relevant to quantum-limited classical optical communication.

Single-shell states for multi-shell description of nuclei

The DUSM method is described for constructing the single-shell states and the corresponding matrix elements needed for describing the isotopes of a given nucleus within a major shell in the framework of the nuclear shell model. The advantage of this approach is that a streamline set of states are constructed, thereby reducing the calculation requirements. The idea is to truncate the full space of permutational irreducible representations with two columns to subspaces where the second column is restricted in its number of boxes – depending on the isospin value. The usefulness of this method for large-scale shell model calculations in the fp- and sdgp-shell is emphasized. An application to Sn isotopes in the sdgp-shell is presented.

On the orbit spaces of irreducible representations of simple compact Lie groups of types B, C, and D

It is proved that the orbit space of an irreducible representation of a simple connected compact Lie group of type B, C, or D can be a smooth manifold only in two cases.

Kravchuk polynomials and irreducible representations of the rotation group $$SO(3)$$ S O ( 3 )

We set forth that the Kravchuk polynomials of a discrete variable are actually “encoded” within appropriate finite-dimensional irreducible representations of the group of rotations of three-dimensional space \(SO(3)\). Hence, discrete irreducible representation spaces of the group \(SO(3)\) can be naturally interpreted as finite (discrete) versions of the linear quantum harmonic oscillator.

Oscillators in a (2+1)-dimensional noncommutative space

We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncommutative space where the noncommutativity is induced by the shift of the dynamical variables with generators of \documentclass[12pt]{minimal}\begin{document}$SL(2,\mathbb {R})$\end{document}SL(2,R) in a unitary irreducible representation considered in Falomir et al. [Phys. Rev. D 86, 105035 (2012)]. This redefinition is interpreted in the framework of the Levi's decomposition of the deformed algebra satisfied by the noncommutative variables. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, finding no constraints between coordinates and momenta noncommutativity parameters. Since the representation space of the unitary irreducible representations \documentclass[12pt]{minimal}\begin{document}$SL(2,\mathbb {R})$\end{document}SL(2,R) can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.

THE DUAL STRUCTURE OF CROSSED PRODUCT -ALGEBRAS WITH FINITE GROUPS

AbstractWe study the space of irreducible representations of a crossed product ${C}^{\ast } $-algebra ${\mathop{A\rtimes }\nolimits}_{\sigma } G$, where $G$ is a finite group. We construct a space $\widetilde {\Gamma } $ which consists of pairs of irreducible representations of $A$ and irreducible projective representations of subgroups of $G$. We show that there is a natural action of $G$ on $\widetilde {\Gamma } $ and that the orbit space $G\setminus \widetilde {\Gamma } $ corresponds bijectively to the dual of ${\mathop{A\rtimes }\nolimits}_{\sigma } G$.

Noncommutativity in (2+1)-dimensions and the Lorentz group

In this article we considered models of particles living in a three-dimensional space-time with a nonstandard noncommutativity induced by shifting canonical coordinates and momenta with generators of a unitary irreducible representation of the Lorentz group. The Hilbert space gets the structure of a direct product with the representation space, where we are able to construct operators which realize the algebra of Lorentz transformations. We study the modified Landau problem for both Schr\"odinger and Dirac particles, whose Hamiltonians are obtained through a kind of non-Abelian Bopp's shift of the dynamical variables from the ones of the usual problem in the normal space. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters. We find no constraint between the parameters referring to noncommutativity in coordinates and momenta but they rather play similar roles. Since the representation space of the unitary irreducible representations $SL(2,\mathbb{R})$ can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.