Search

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

Let G be a finite group. Let X 1(G) be the first column of the ordinary character table of G. We will show that if X 1(G) = X1(PGU3(q 2)), then G ≅ PGU3(q 2). As a consequence, we show that the projective general unitary groups PGU3(q 2) are uniquely determined by the structure of their complex group algebras.

It is well known that a Gabor representation on L 2 ( R d ) L^{2}(\mathbb {R}^{d}) admits a frame generator h ∈ L 2 ( R d ) h\in L^{2}(\mathbb {R}^{d}) if and only if the associated lattice satisfies the Beurling density condition, which in turn can be characterized as the “trace condition” for the associated von Neumann algebra. It happens that this trace condition is also necessary for any projective unitary representation of a countable group to admit a frame vector. However, it is no longer sufficient for general representations, and in particular not sufficient for Gabor representations when they are restricted to proper time-frequency invariant subspaces. In this short note we show that the condition is also sufficient for a large class of projective unitary representations, which implies that the Gabor density theorem is valid for subspace representations in the case of irrational types of lattices.

A procedure of calculation of two-valued space group representations and two-valued projective representations of point groups is considered. A method of construction of factor systems w2(r2, r1), which reflect the transformations of half-integer spin quantum wave functions and are required in order to find the two-valued irreducible projective representations of the point groups, is presented. This method is based on the introduction of an operation q, firstly used by Bethe, as an additional symmetry element. The pathway of introducing the relations, which permit to make a one-valued algebra of double groups and, particularly, their multiplication tables, is shown by the examples of the 222 (D2) and 32 (D3) groups. The construction of a standard factor-system w′(1)(r2, r1) of the projective class K1 for the group 222 on the base of the discussed relations is presented for the first time. The whole role and the possibilities of Bethe’s method and its modifications for the construction of two-valued representations of the point and space groups are discussed., Розглянуто методику побудови двозначних представлень просторових та двозначних проективних представлень точкових груп. Представлено метод побудови фактор-систем w2(r2, r1), якi вiдображають перетворення хвильових функцiй квантових систем з напiвцiлим спiном, i якi є необхiдними для знаходження двозначних незвiдних проективних представлень точкових груп. Цей метод ґрунтується на введеннi в ролi додаткового елемента симетрiї операцiї q, вперше використаної Бете. На прикладi груп 222 (D2) та 32 (D3) показано, яким чином вводяться спiввiдношення, що дозволяють зробити однозначними алгебру подвiйних груп та, зокрема, їх таблицi множення. Показано, яким чином на основi спiввiдношень, що обговорюються, будується вперше представлена для групи 222 стандартна фактор-система класу K1 – фактор-система w′(1)(r2, r1). Обговорюються також в цiлому роль та можливостi методу Бете та його модифiкацiй в побудовi двозначних представлень точкових та просторових груп.

We prove that a periodic group, saturated with projective linear groups of dimension 2 over finite fields, is isomorphic to the projective linear group of dimension 2 over a locally finite field.

In this paper, we construct a character theory for projective representations of finite groups. Consequently, we compute the number of distinct irreducible projective representations (up to isomorphism) of a finite group with a given associated Schur multiplier and deduce properties on the degrees of such projective representations.

In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of projective and symplectic groups of small degree.

In this paper, all the binary codes from the primitive permutation group P S U 2 ( 16 ) are constructed. It is shown that the groups P S U 2 ( 16 ) : 4 and S 17 are the automorphism groups of the constructed codes.

If m and n are non-negative integers, then three new classes of abelian p-groups are defined and studied: the m,n-simply presented groups, the m,n-balanced projective groups and the m,n-totally projec- tive groups. These notions combine and generalize both the theories of simply presented groups and p !+n -projective groups. If m,n = 0, these all agree with the class of totally projective groups, but when m+n � 1, they also include the p !+m+n -projective groups. These classes are related to the (strongly) n-simply presented and (strongly) n-balanced projective groups considered in (15) and the n-summable groups considered in (2). The groups in these classes whose lengths are less than ! 2 are character- ized, and if in addition we have n = 0, they are determined by isometries of their p m -socles.

We describe the theory and implementation in Magma of algorithms to compute the projective indecomposable KG-modules for finite groups G and finite fields K. We describe also how they may be used together with dimension-shifting techniques to compute cohomology groups Hn (G, M) for finite-dimensional KG-modules M and n⩾3.

We focus in this chapter on facts about group algebras that are not true for finite-dimensional algebras in general. The results are a mix of general statements and specific examples describing the representations of certain types of groups. At the beginning of the chapter, we summarize the properties of projective modules for p -groups and also the behavior of projective modules under induction and restriction. Toward the end, we show that the Cartan matrix is symmetric (then the field is algebraically closed) and also that projective modules are injective. In the middle, we describe quite explicitly the structure of projective modules formany semidirect products, and we do this by elementary arguments. It shows that the important general theorems are not always necessary to understand specific representations, and it also increases our stock of examples of groups and their representations. Because of the diversity of topics, it is possible to skip certain results in this chapter without affecting comprehension of what remains. For example, the reader who ismore interested in the general results could skip the description of representations of specific groups between Example 8.2.1 and Theorem 8.4.1. The Behavior of Projective Modules under Induction, Restriction, and Tensor Product We start with a basic fact about group algebras of p -groups in characteristic p . Theorem 8.1.1. Let k be a field of characteristic p and G a p-group. The regular representation is an indecomposable projective module that is the projective cover of the trivial representation. Every finitely generated projective module is free. The only idempotents in kG are 0 and 1 . Proof . We have seen in 6.12 that kG is indecomposable and it also follows from 7.14. By Nakayama's Lemma, kG is the projective cover of k . By 7.13 and 6.3, every indecomposable projective is isomorphic to kG . Every finitely generated projective is a direct sum of indecomposable projectives, and so is free. Finally, every idempotent e ∈ kG gives a module decomposition kG = kGe ⊕ kG (1 − e ). If e ≠ 0 then we must have kG = kGe , so kG (1 − e ) = 0 and e = 1.

We construct a projective variety with discrete, non-finitely generated automorphism group. As an application, we show that there exists a complex projective variety with infinitely many non-isomorphic real forms., Comment: 13 pages, 3 figures. Substantial revision, including new example of variety with infinitely many real forms. Comments appreciated!

For a smooth projective unitary representation (ρ,H) of a locally convex Lie group G, the projective space P(H∞) of smooth vectors is a locally convex Kahler manifold. We show that the action of G on P(H∞) is weakly Hamiltonian, and lifts to a Hamiltonian action of the central U(1)- extension G # obtained from the projective representation. We identify the non-equivariance cocycles obtained from the weakly Hamiltonian action with those obtained from the projective representation, and give some integrality conditions on the image of the momentum map.

We study the renormalization group action in a fermionic hierarchical model in the space of coefficients determining the Grassmann-valued density of the free measure. This space is interpreted as the two-dimensional projective space. The renormalization group map is a homogeneous quadratic map and has a special geometric property that allows describing invariant sets and the global dynamics in the whole space.

The semi-center of an artinian semisimple module-algebra over a finite group G can be described using the projective representations of G. In particular, the semi-center of the endomorphism ring of an irreducible projective representation over an algebraically closed field has a structure of a twisted group algebra. The following group-theoretic result is deduced: the center of a group of central type embeds into the group of its linear characters.

Let L2(q) be the projective special linear group, where q is a prime power. In the present paper, we prove that L2(q) is OD-characterizable by using the classification of finite simple groups. A new method is introduced in order to deal with the subtle changes of the prime graph of a group in the discussion of its OD-characterization. This not only generalizes a result of Moghaddamfar, Zokayi and Darafsheh, but also gives a positive answer to a conjecture put forward by Shi.

Let Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

We give some conditions under which no two non-conjugate projective representations, in an algebra, of a given group can become conjugate over a separable extension of the base field. In particular, we show this is always the case for groups with trivial abelianization.

We show by a combination of theoretical argument and computer search that if a projective (75, 4, 12, 5) set in PG(3, 7) exists then its automorphism group must be trivial. This corresponds to the smallest open case of a coding problem posed by H. Ward in 1998, concerning the possible existence of an infinite family of projective two-weight codes meeting the Griesmer bound.

In this study, we give one-to-one correspondence between the elements of the unit split three-sphere S ( 3 , 2 ) with the complex hyperbolic special unitary matrices SU ( 2 , 1 ) . Thus, we express spherical concepts such as meridians of longitude and parallels of latitude on SU ( 2 , 1 ) by using the method given in Toth [Toth G. Glimpses of algebra and geometry. Springer-Verlag; 1998] for S 3 . The relation among the special orthogonal group SO ( R 3 ) , the quotient group of unit quaternions S 3 / { ± 1 } and the projective space RP 3 given as SO ( R 3 ) ≅ S 3 / { ± 1 } = RP 3 is known as the Euclidean projective spaces [Toth G. Glimpses of algebra and geometry. Springer-Verlag; 1998]. This relation was generalized to the semi-Euclidean projective space and then, the expression SO ( 3 , 1 ) ≅ S ( 3 , 2 ) / { ± 1 } = RP 2 3 was acquired. Thus, it was found that Hopf fibriation map of S ( 2 , 1 ) can be used for Twistors (in not-null state) in quantum mechanics applications. In addition, the octonions and the split-octonions can be obtained from the Cayley-Dickson construction by defining a multiplication on pairs of quaternions or split quaternions. The automorphism group of the octonions is an exceptional Lie group. The split-octonions are used in the description of physical law. For example, the Dirac equation in physics (the equation of motion of a free spin 1/2 particle, like e.g. an electron or a proton) can be represented by a native split-octonion arithmetic.

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 .

By the infinite generalized symmetric group we mean the group , where is the group of all sequences in containing only finitely many non-zero elements and is the group of all finitely supported permutations of the positive integers. A complete description of the projective factor representations of of finite type is obtained.Bibliography: 18 titles.

Our aim is to characterize the projective extensions of bounded semilattices. As a consequence we get a new description of projective extensions of semilattices.

The "projective lightcone limit" has been proposed as an alternative holographic dual of an AdS space. It is a new type of group contraction for a coset G/H preserving the isometry group G but changing H. In contrast to the usual group contraction, which changes G preserving the spacetime dimension, it reduces the dimensions of the spacetime on which G is realized. The obtained space is a projective space on which the isometry is realized as a linear fractional transformation. We generalize and apply this limiting procedure to the "Hopf reduction" and obtain (n-1)-dimensional complex projective space from (2n-1)-dimensional sphere preserving SU(n) symmetry., 16 pages, v2: modified the section 2.1 clarifying the difference from IW contraction, added notes & references, version to appear in Nuclear Physics B

We settle the isomorphism problem for finite cyclic projective planes by proving the following analogue of the Bays–Lambossy theorem: two cyclic projective planes of order n are isomorphic if and only if they are multiplier equivalent, that is, if and only if the associated difference sets in $${\mathbb{Z}_{n^2+n+1}}$$ are equivalent. In fact, we establish a more general result for abelian groups, where multipliers are replaced by group automorphisms.

Given an arbitrary set π of primes, we complete the description of the finite linear and unitary groups having the D π-property.

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups.

We prove that if the p-reduced abelian group G is a special countable extension of its totally projective p-component of torsion Gp and R is a perfect commutative unitary ring of prime characteristic p, then the group S(G) of all normed p-units in the group algebra RG modulo Gp, that is, S(G)/Gp, is totally projective. Our result strengthens both classical results obtained by May and Hill–Ullery.

In case G is a finite group, there is a well-known criterion for projective modules: A ℤ G-module M is projective if and only if it is ℤ -free and has finite projective dimension. We first investigate whether only finite groups satisfy the above criterion. In the class of groups L H 𝔉, we conclude that this is true. Secondly, we consider the problem when a stably flat Γ-module is projective, where Γ is an arbitrary group. We show that if Γ is an L H 𝔉-group, then every stably flat cofibrant ℤ Γ-module is projective.

Conditions are given for a class 2 nilpotent group to have no central extensions of class 3. This is related to Betti numbers and to the problem of representing a class 2 nilpotent group as the fundamental group of a smooth projective variety.

Associated to each finite group $\Gamma$ in $SL_2(C)$ there is a family of noncommutative algebras which deforms the coordinate ring of the Kleinian singularity corresponding to that group. These algebras were defined by W. Crawley-Boevey and M. Holland, who also suggested a conjectural correspondence between the set of isomorphism classes of rank one projective modules over these algebras and associated Nakajima quiver varieties. In \cite{BGK}, V.Baranovski, V.Ginzburg and A.Kuznetsov proved the Crawley-Boevey-Holland conjecture using the methods of noncommutative projective geometry. In this paper we will state a refined ($G$-equivariant) version of this conjecture and, in the case of cyclic groups, give a new construction of this correspondence based on the notion of DG-model of a rank one projective module. This construction leads to a completely explicit description of ideals of the Crawley-Boevey-Holland algebras., Comment: 27 pages

The equations: $$ x_\nu ' = \frac{a_{1\nu }\,x_1+\cdots +a_{n\nu }\,x_n+a_{n+1,\nu }}{ a_{1,n+1}\,x_1+\cdots +a_{n,n+1}\,x_n+a_{n+1,n+1}}\qquad {\scriptstyle {(\nu \,=\,1\,\cdots \,n)}}\quad {(1)} $$ determine a group, as one easily convinces oneself, the so-called general projective group of the manifold \(x_1, \dots , x_n\). In the present chapter, we want to study more closely this important group, which is also called the group of all collineations [Collineationen] of the space \(x_1, \dots , x_n\), by focusing our attention on its subgroups.

In this paper, the classification of two dimension objects under the projective transformation group, interested group in computer vision, is presented. Results between the proposed invariants with previous invariant features are compared to show the performance of the method.

We prove that every non-finitely generated projective module over the integral group ring of a polycyclic-by-finite group G is free if and only if G is polycyclic., Comment: 15 pages, to appear in J. Algebra

All binary projective codes of dimension up to 6 are classified. Information about the number of the codes with different minimum distances and automorphism group orders is given.

Let be a twisted group algebra with basis and be a partition of G. A projective class algebra associated with P is a subalgebra of generated by all class sums . A main object of the paper is to find interrelationships of projective class algebras in and in for H functions as characters of projective class algebras.

Let X X be a set (with at least three different points) and let G G be a group of bijections of X X . If the action of G G on X X satisfies three natural conditions, then X X admits a canonical structure of a projective line over a commutative field, such that G G is the group of all projective transformations of X X .

The group consisting of the based homotopy classes of self-homotopy equivalences is called the self-equivalence group. We determine the group structures of self-equivalence groups, for the suspended real projective space whose dimension is less than or equal to six. The method is to study the multiplicative structure of self-homotopy set induced from the composition of maps. Finding out the invertible element of this monoid give almost all structures of self-equivalence groups. The group of the 1-fold suspension of the four-dimensional real projective space which is not determined similarly is obtained by the another method thought of from Rutter's paper.

Let Pn be the real n-dimensional projective space. We determine the group structure of the self-homotopy set of the double suspension of Pn where n is 3, 4, 5 and 6 using the ideas and methods of the second author (The suspension order of the real even dimensional projective space, J. Math. Kyoto Univ. 43(4) (2003), 755-769).

We will prove two results about the topology of complex projective surfaces. The first result says that if the Shafarevich Conjecture has an affirmative answer in dimension two then the second homotopy group of a smooth projective surface is a torsion-free abelian group. The second result is that for any 2-dimensional function field K/C there is a normal projective simply-connected surface with function field K.

The sequentially pure projective dimension of a global group with a decomposition basis is computed in terms of a structural property that generalizes the earlier treatment of p -local torsion groups by Fuchs and Hill. As a corollary, it is established that a global group with a decomposition basis and cardinality ℵ n , for some nonnegative integer n , has sequentially pure projective dimension not exceeding n .

This article is a contribution to the study of linear spaces admitting a line-transitive automorphism group. We classify such linear spaces where PSL(2,q), q>3 acts line transitively. We prove that the only cases which arise are projective planes, a Bose–Witt–Shrikhande linear space and one more space admitting PSL(2,26) as a line-transitive automorphism group.

Let V be a hyperbolic 2-torus bundle over the circle, or equivalently, a quotient of the three-dimensional group Sol by a lattice Γ. We study real projective structures on V. We prove that all such structures are left-invariant which means that they are all obtained by the following process: take a representation of Sol in PGL (4, R) with an open orbit Ω in RP3; it induces a projective structure on Sol and, hence, on the left-quotient Γ\Sol = V. We classify all the real projective structures on V and show that they are in fact all affine. The proof involves a study of codimension one projective subsurfaces, which are tori, and the dynamic of their holonomy group in the whole manifold.