Faithful Irreducible Representation Research Articles

A massless neutrino and lepton mixing patterns from finite discrete subgroups of U(3)

Finite discrete subgroups of $U(3)$ as possible flavour symmetries $G_f$ for a massless neutrinos with predictive mixing angles are studied. This is done by assuming that a residual symmetry $S_\nu$ appropriate for describing a massless neutrino is contained in $G_f$. It is shown that all the groups $G_f$ admitting three dimensional faithful irreducible representation and generated from a specific set of $3\times 3$ matrices imply only one of the three flavour compositions for the massless state namely, unmixed, maximally mixed with equal probabilities and bimaximally mixed with probabilities (0,1/2,1/2) and their permutations. This result holds irrespective of the order of $G_f$ and the choice of $S_{\nu}$ within it. All of these lead to unfavourable leading order prediction for the solar mixing angle. Neutrino mixing pattern is then numerically investigated in case of subgroups of $U(3)$ with order less than 512 and it is found that only one of these can lead to a massless neutrino and leading order predictions for all the mixing angles close to their experimental values. Ways to correct for the solar angle prediction are proposed and two concrete examples giving the observed mixing pattern are discussed.

Sporadic simple groups and quotient singularities

We show that if a faithful irreducible representation of a central extension of a sporadic simple group with centre contained in the commutator subgroup gives rise to an exceptional (resp. weakly exceptional but not exceptional) quotient singularity, then that simple group is the Hall–Janko group (resp. the Suzuki group).

ON THE IRREDUCIBLE REPRESENTATIONS OF SOLUBLE GROUPS OF FINITE RANK

We obtained some sufficient and necessary conditions of existence of faithful irreducible representations of a soluble group G of finite rank over a field k. It was shown that the existence of such representations strongly depends on construction of the socle of the group G. The situation is especially complicated in the case where the field k is locally finite.

The full group C$^{*}$-algebra of the modular group is primitive

We show that the full group C*-algebra of PSL(n,Z) is primitive when n = 2 and not primitive when n >= 3. Moreover, we show that there exists an uncountable family of pairwise inequivalent, faithful irreducible representations of C*(PSL(2,Z)).

Groups with faithful irreducible projective unitary representations

For a countable group Γ and a multiplier , we study the property of Γ having a unitary projective -representation which is both irreducible and projectively faithful. We show that this property is equivalent to Γ being the quotient of an appropriate group by its centre. Moreover, we give a criterion in terms of the minisocle of Γ. Several examples are described to show the existence of various behaviours.

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.

Primitivity of some full group C*-algebras

We show that the full group C -algebra of the free product of two nontrivial countable amenable discrete groups, where at least one of them has more than two elements, is primitive. We also show that in many cases, this C - algebra is antiliminary and has an uncountable family of pairwise inequivalent, faithful irreducible representations.

On the finite subgroups of U(3) of order smaller than 512

We use the SmallGroups Library to find the finite subgroups of U(3) of order smaller than 512 which possess a faithful three-dimensional irreducible representation. From the resulting list of groups we extract those groups that cannot be written as direct products with cyclic groups. These groups are the basic building blocks for models based on finite subgroups of U(3). All resulting finite subgroups of SU(3) can be identified using the well-known list of finite subgroups of SU(3) derived by Miller, Blichfeldt and Dickson at the beginning of the 20th century. Furthermore, we prove a theorem which allows us to construct infinite series of finite subgroups of U(3) from a special type of finite subgroups of U(3). This theorem is used to construct some new series of finite subgroups of U(3). The first members of these series can be found in the derived list of finite subgroups of U(3) of order smaller than 512. In the last part of this work we analyze some interesting finite subgroups of U(3), especially the group , which is closely related to the important SU(3)-subgroup S4.

C*-Homomorphisms and duality of C*-discrete quantum groups

Let Open image in new window be a C*-discrete quantum group and let Open image in new window be the discrete quantum group associated with Open image in new window . Suppose that there exists a continuous action of Open image in new window on a unital C*-algebra Open image in new window so that Open image in new window becomes a Open image in new window -algebra. If there is a faithful irreducible vacuum representation π of Open image in new window on a Hilbert space H = Open image in new window with a vacuum vector Ω, which gives rise to a Open image in new window -invariant state, then there is a unique C*-representation (θ, H) of Open image in new window supplemented by the action. The fixed point subspace of Open image in new window under the action of Open image in new window is exactly the commutant of θ( Open image in new window ).

Irreducibly represented groups

A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented. We indicate some consequences of this for operator algebras. In particular, we characterise up to isomorphism the countable subgroups Δ of the unitary group of a separable infinite dimensional Hilbert space \mathcal H of which the bicommutants Δ'' (in the sense of the theory of von Neumann algebras) coincide with the algebra of all bounded linear operators on \mathcal H .

On faithful irreducible projective representations of finite groups over a field of characteristic p

Let K be a field of finite characteristic p and G a finite group with a normal Sylow p-subgroup. We give necessary and sufficient conditions for G to have a faithful irreducible projective representation over K. In the case when G is an abelian p-group and K is not a perfect field we find also all dimensions of faithful irreducible projective representations of G over K and show that the group G has infinitely many isomorphism classes of faithful irreducible projective representations of each possible dimension d ≠ 1 .

NEST REPRESENTATIONS OF DIRECTED GRAPH ALGEBRAS

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra ${\mathfrak{L}}_G$ of a countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra ${\mathcal{T}}^{+}(G)$.We prove that the finite-dimensional nest representations separate the points in ${\mathfrak{L}}_G$, and a fortiori, in ${\mathcal{T}}^{+}(G)$. The irreducible finite-dimensional representations separate the points in ${\mathfrak{L}}_G$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every vertex $x \in {\mathcal{T}}(G)$ supporting a cycle, $x$ also supports at least one loop edge.We also study faithful nest representations. We prove that ${\mathfrak{L}}_G$ (or ${\mathcal{T}}^{+}(G)$) admits a faithful irreducible representation if and only if $G$ is strongly transitive as a directed graph. More generally, we obtain a condition on $G$ which is equivalent to the existence of a faithful nest representation. We also give a condition that determines the existence of a faithful nest representation for a maximal type ${\mathbb{N}}$ nest.

FOCK REPRESENTATIONS AND QUANTUM MATRICES

In this paper we study the Fock representation of a certain *-algebra which appears naturally in the framework of quantum group theory. It is also a generalization of the twisted CCR-algebra introduced by Pusz and Woronowicz. We prove that the Fock representation is a faithful irreducible representation of the algebra by bounded operators in a Hilbert space, and, moreover, it is the only (up to unitary equivalence) representation possessing these properties.

Parametrization of Torsion Units in U1(ZS3)

In this study, a representation of the group ring ZS3 is obtained by using a faithful irreducible representation of S3 of degree 2. In ZS3, all torsion units are expressed in terms of parameters by means of this representation. It is shown that any torsion unit in U1 (ZS3) can be expressed in terms of two parameters.

Classical and quantum function reconstruction via character evaluation

We consider two natural instances of the problem of determining a function f: G→ G defined on a group G by making repeated queries to the oracle O(x)=χ( f(x)) , where χ is a known character of the group G. In particular, we consider the problem of recovering a hidden monic polynomial f( X) of degree d⩾1 over a finite field F p of p elements given a black box which, for any x∈ F p , evaluates the quadratic character of f( x). We design a classical algorithm of complexity O( d 2 p d+ ε ) and also show that the quantum query complexity of this problem is O( d). Some of our results extend those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a linear polynomial f( X)= X+ s (with unknown s); some are new even in this case. We then consider the related problem of determining an element s of a finite group G given the oracle O(x)=χ(sx) , where χ is the character of a (known) faithful irreducible representation ρ of G. We show that if d is the dimension of ρ, then O(d 2 log|G|) classical queries suffice to determine s. The proof involves development of a new “uncertainty principle” for the Fourier transform over finite non-Abelian groups.

Twisted group rings of metacyclic groups

Given a finite metacyclic group G G , a central extension F F having the projective lifting property over all fields is constructed. This extension and its group rings are used to investigate the faithful irreducible projective representations of G G and the fields over which they can be realized. A full description of the finite metacyclic groups having central simple twisted group rings over fields is given.

FINITE METACYCLIC GROUPS WITH FAITHFUL IRREDUCIBLE REPRESENTATIONS

A metacyclic group is a group that has a cyclic normal subgroup with cyclic quotient. We characterize finite metacyclic groups that have faithful irreducible representations over a given field.

On modular projective representations of finite nilpotent groups

Our aim is to determine necessary and sufficient conditions for a finite nilpotent group to have a faithful irreducible projective representation over a field of characteristic $p\geq 0$.

On primitive representations of soluble groups of finite rank

In the paper it is proved, in particular, that a group is polycyclic if and only if it is soluble of finite rank, satisfies the ascending chain condition for normal subgroups and admits a faithful irreducible primitive representation over a field of characteristic zero. Methods are developed that enable one to study induced representations of nilpotent and soluble groups of finite rank.

Complete positivity of elementary operators

In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([i ] + 1)-reflexive, where [n ] denotes the greatest integer not n larger than '. By the result, we show that if 1?( ) = E Ai( )Bi is an 2=1 elementary operator on a C*-algebra A, then 'D is completely positive if and only if 'D is ([n1 ] + 1)-positive. In this paper, let H denote a complex Hilbert space. Let H(') denote the direct sum of n copies of H. For T E L(H), we write T(') for the operator on H(') which is the direct sum of n copies of T; the notation is extended to a subset of L(H) by S(n) = {T(n) E L(H(n)): T E S}. If S is a subspace of L(H), S is called n-reflexive if S(n) = ref (S(n)) =_ {T(n) E L(H(n)): T(n)X E [S(n)X], for all x E H (n)}, where [] denotes norm closed linear span. By the definiton, we have that if S is m-reflexive, then S is n-reflexive for n > m. A separating vector for a subspace S of L(H) is a vector x E H such that T 4 Tx, T E S, is an invective map. For x, y E H, let x 0 y denote the rank-one operator u | 4 (u, x)y. Let A denote a C*-algebra. Then A is called primitive, if A has a faithful irreducible representation on some Hilbert space. An elementary operator AP on A n is a linear mapping of the form AP: T F4 AiTBi, where {Ai},nL1 and {Bi},nU1 are i=l subsets of A. In this paper, we assume that all elementary operators are nonzero. A linear map 4J on A is positive (resp. hermitian-preserving) if 4)(T) is positive (resp. hermitian) for all positive (resp. hermitian) T in A. We define 4'n = 4{I 0 In: Mn(A) -4 Mn(A) by 4) 0 In((Tij)nxn) = (4)(Ti ))nxn. 4) is said to be n-positive if 4J 0 In is positive. If 4J is n-positive for all n, then 4J is said to be completely positive. In [4], Magajna states the following problem: For each positive integer r determine the smallest k = k(r) such that all rdimensional subspaces of L(H) are k-reflexive. In [4], Magajna proves k < r. In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([n] + 1)-reflexive. Also by this result, we study complete positivity of elementary operators on a C*-algebra A. We prove that if n 4 ) = Ai( )Bi is an elementary operator on a C*-algebra A, then 4) is i=l1 Received by the editors July 8, 1996 and, in revised form, May 14, 1997. 1991 Mathematics Subject Classification. Primary 47B47, 47B49; Secondary 46L05.