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51. CJM 2003 (vol 55 pp. 1080)

Kellerhals, Ruth
Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds
We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$.

Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40

52. CJM 2003 (vol 55 pp. 750)

Göbel, Rüdiger; Shelah, Saharon; Strüngmann, Lutz
Almost-Free $E$-Rings of Cardinality $\aleph_1$
An $E$-ring is a unital ring $R$ such that every endomorphism of the underlying abelian group $R^+$ is multiplication by some ring element. The existence of almost-free $E$-rings of cardinality greater than $2^{\aleph_0}$ is undecidable in $\ZFC$. While they exist in G\"odel's universe, they do not exist in other models of set theory. For a regular cardinal $\aleph_1 \leq \lambda \leq 2^{\aleph_0}$ we construct $E$-rings of cardinality $\lambda$ in $\ZFC$ which have $\aleph_1$-free additive structure. For $\lambda=\aleph_1$ we therefore obtain the existence of almost-free $E$-rings of cardinality $\aleph_1$ in $\ZFC$.

Keywords:$E$-rings, almost-free modules
Categories:20K20, 20K30, 13B10, 13B25

53. CJM 2003 (vol 55 pp. 822)

Kim, Djun Maximilian; Rolfsen, Dale
An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements
We define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic. By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.

Category:20F36

54. CJM 2002 (vol 54 pp. 1229)

Gow, Roderick; Szechtman, Fernando
The Weil Character of the Unitary Group Associated to a Finite Local Ring
Let $\mathbf{R}/R$ be a quadratic extension of finite, commutative, local and principal rings of odd characteristic. Denote by $\mathbf{U}_n (\mathbf{R})$ the unitary group of rank $n$ associated to $\mathbf{R}/R$. The Weil representation of $\mathbf{U}_n (\mathbf{R})$ is defined and its character is explicitly computed.

Category:20G05

55. CJM 2002 (vol 54 pp. 970)

Cegarra, A. M.; García-Calcines, J. M.; Ortega, J. A.
On Graded Categorical Groups and Equivariant Group Extensions
In this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

Categories:18D10, 18D30, 20E22, 20F29

56. CJM 2002 (vol 54 pp. 795)

Möller, Rögnvaldur G.
Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations
Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results.

Keywords:totally disconnected locally compact groups, scale function, permutation groups, groups acting on graphs
Categories:22D05, 20B07, 20B27, 05C25

57. CJM 2001 (vol 53 pp. 1121)

Athanasiadis, Christos A.; Santos, Francisco
Monotone Paths on Zonotopes and Oriented Matroids
Monotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip. It is proved that any monotone path on a $d$-dimensional zonotope with $n$ generators admits at least $\lceil 2n/(n-d+2) \rceil-1$ flips for all $n \ge d+2 \ge 4$ and that for any fixed value of $n-d$, this lower bound is sharp for infinitely many values of $n$. In particular, monotone paths on zonotopes which admit only three flips are constructed in each dimension $d \ge 3$. Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented matroid. An application in the context of Coxeter groups of a result known to be valid for monotone paths on simple zonotopes is included.

Categories:52C35, 52B12, 52C40, 20F55

58. CJM 2000 (vol 52 pp. 1310)

Yagunov, Serge
On the Homology of $\GL_n$ and Higher Pre-Bloch Groups
For every integer $n>1$ and infinite field $F$ we construct a spectral sequence converging to the homology of $\GL_n(F)$ relative to the group of monomial matrices $\GM_n(F)$. Some entries in $E^2$-terms of these spectral sequences may be interpreted as a natural generalization of the Bloch group to higher dimensions. These groups may be characterized as homology of $\GL_n$ relatively to $\GL_{n-1}$ and $\GM_n$. We apply the machinery developed to the investigation of stabilization maps in homology of General Linear Groups.

Categories:19D55, 20J06, 18G60

59. CJM 2000 (vol 52 pp. 1018)

Reichstein, Zinovy; Youssin, Boris
Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-Varieties
Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety. We show that $X$ can be transformed, by a sequence of blowups with smooth $G$-equivariant centers, into a $G$-variety $X'$ with the following property the stabilizer of every point of $X'$ is isomorphic to a semidirect product $U \sdp A$ of a unipotent group $U$ and a diagonalizable group $A$. As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.

Categories:14L30, 14E15, 14E05, 12E05, 20G10

60. CJM 2000 (vol 52 pp. 449)

Adler, Jeffrey D.; Roche, Alan
An Intertwining Result for $p$-adic Groups
For a reductive $p$-adic group $G$, we compute the supports of the Hecke algebras for the $K$-types for $G$ lying in a certain frequently-occurring class. When $G$ is classical, we compute the intertwining between any two such $K$-types.

Categories:22E50, 20G05

61. CJM 2000 (vol 52 pp. 265)

Brion, Michel; Helminck, Aloysius G.
On Orbit Closures of Symmetric Subgroups in Flag Varieties
We study $K$-orbits in $G/P$ where $G$ is a complex connected reductive group, $P \subseteq G$ is a parabolic subgroup, and $K \subseteq G$ is the fixed point subgroup of an involutive automorphism $\theta$. Generalizing work of Springer, we parametrize the (finite) orbit set $K \setminus G \slash P$ and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of $\theta$-stable (resp. $\theta$-split) parabolic subgroups. We also describe the decomposition of any $(K,P)$-double coset in $G$ into $(K,B)$-double cosets, where $B \subseteq P$ is a Borel subgroup. Finally, for certain $K$-orbit closures $X \subseteq G/B$, and for any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero global sections, we show that the restriction map $\res_X \colon H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we describe the $K$-module $H^0 (X, \mathcal{L})$. This gives information on the restriction to $K$ of the simple $G$-module $H^0 (G/B, \mathcal{L})$. Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal $K$-types.

Keywords:flag variety, symmetric subgroup
Categories:14M15, 20G05

62. CJM 2000 (vol 52 pp. 438)

Wallach, N. R.; Willenbring, J.
On Some $q$-Analogs of a Theorem of Kostant-Rallis
In the first part of this paper generalizations of Hesselink's $q$-analog of Kostant's multiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a $q$-analog of the Kostant-Rallis theorem is given for the real group $\SL(4,\mathbb{R})$ (that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

Categories:22E47, 20G05

63. CJM 2000 (vol 52 pp. 197)

Radjavi, Heydar
Sublinearity and Other Spectral Conditions on a Semigroup
Subadditivity, sublinearity, submultiplicativity, and other conditions are considered for spectra of pairs of operators on a Hilbert space. Sublinearity, for example, is a weakening of the well-known property~$L$ and means $\sigma(A+\lambda B) \subseteq \sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The effect of these conditions is examined on commutativity, reducibility, and triangularizability of multiplicative semigroups of operators. A sample result is that sublinearity of spectra implies simultaneous triangularizability for a semigroup of compact operators.

Categories:47A15, 47D03, 15A30, 20A20, 47A10, 47B10

64. CJM 1999 (vol 51 pp. 1149)

Cohen, A. M.; Cuypers, H.; Sterk, H.
Linear Groups Generated by Reflection Tori
A reflection is an invertible linear transformation of a vector space fixing a given hyperplane, its axis, vectorwise and a given complement to this hyperplane, its center, setwise. A reflection torus is a one-dimensional group generated by all reflections with fixed axis and center. In this paper we classify subgroups of general linear groups (in arbitrary dimension and defined over arbitrary fields) generated by reflection tori.

Categories:20Hxx, 20Gxx, 51A50

65. CJM 1999 (vol 51 pp. 1307)

Johnson, Norman W.; Weiss, Asia Ivić
Quadratic Integers and Coxeter Groups
Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)$-space $\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of $\mbox{H}^{n+1}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\PSL_2 (\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.

Categories:11F06, 20F55, 20G20, 20H10, 22E40

66. CJM 1999 (vol 51 pp. 1240)

Monson, B.; Weiss, A. Ivić
Realizations of Regular Toroidal Maps
We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types $\{3,6\}$ and $\{6,3\}$.

Keywords:regular maps, realizations of polytopes
Categories:51M20, 20F55

67. CJM 1999 (vol 51 pp. 1226)

McKay, John
Semi-Affine Coxeter-Dynkin Graphs and $G \subseteq \SU_2(C)$
The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types.

Categories:20C99, 05C25, 14B05

68. CJM 1999 (vol 51 pp. 1194)

Lusztig, G.
Subregular Nilpotent Elements and Bases in $K$-Theory
In this paper we describe a canonical basis for the equivariant $K$-theory (with respect to a $\bc^*$-action) of the variety of Borel subalgebras containing a subregular nilpotent element of a simple complex Lie algebra of type $D$ or~$E$.

Category:20G99

69. CJM 1999 (vol 51 pp. 1175)

Lehrer, G. I.; Springer, T. A.
Reflection Subquotients of Unitary Reflection Groups
Let $G$ be a finite group generated by (pseudo-) reflections in a complex vector space and let $g$ be any linear transformation which normalises $G$. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset $gG$, a subquotient of $G$ which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in $G$ of certain elements of the coset. A criterion is also given in terms of the invariant degrees of $G$ for an integer to be regular for $G$. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

Categories:51F15, 20H15, 20G40, 20F55, 14C17

70. CJM 1999 (vol 51 pp. 881)

Witherspoon, Sarah J.
The Representation Ring and the Centre of a Hopf Algebra
When $H$ is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of $H$ which generalizes the conjugacy class sums of a finite group, and express the class equation of $H$ in terms of this basis. We show that the representation ring and the centre of $H$ are dual character algebras (or signed hypergroups).

Categories:16W30, 20N20

71. CJM 1999 (vol 51 pp. 658)

Shumyatsky, Pavel
Nilpotency of Some Lie Algebras Associated with $p$-Groups
Let $ L=L_0+L_1$ be a $\mathbb{Z}_2$-graded Lie algebra over a commutative ring with unity in which $2$ is invertible. Suppose that $L_0$ is abelian and $L$ is generated by finitely many homogeneous elements $a_1,\dots,a_k$ such that every commutator in $a_1,\dots,a_k$ is ad-nilpotent. We prove that $L$ is nilpotent. This implies that any periodic residually finite $2'$-group $G$ admitting an involutory automorphism $\phi$ with $C_G(\phi)$ abelian is locally finite.

Categories:17B70, 20F50

72. CJM 1998 (vol 50 pp. 1176)

Dobson, Edward
Isomorphism problem for metacirculant graphs of order a product of distinct primes
In this paper, we solve the isomorphism problem for metacirculant graphs of order $pq$ that are not circulant. To solve this problem, we first extend Babai's characterization of the CI-property to non-Cayley vertex-transitive hypergraphs. Additionally, we find a simple characterization of metacirculant Cayley graphs of order $pq$, and exactly determine the full isomorphism classes of circulant graphs of order $pq$.

Categories:05, 20

73. CJM 1998 (vol 50 pp. 1007)

Elder, G. Griffith
Galois module structure of ambiguous ideals in biquadratic extensions
Let $N/K$ be a biquadratic extension of algebraic number fields, and $G=\Gal (N/K)$. Under a weak restriction on the ramification filtration associated with each prime of $K$ above $2$, we explicitly describe the $\bZ[G]$-module structure of each ambiguous ideal of $N$. We find under this restriction that in the representation of each ambiguous ideal as a $\bZ[G]$-module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone. For a given group, $G$, define ${\cal S}_G$ to be the set of indecomposable $\bZ[G]$-modules, ${\cal M}$, such that there is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$ is a $\bZ[G]$-module summand of an ambiguous ideal of $N$. Can ${\cal S}_G$ ever be infinite? In this paper we answer this question of Chinburg in the affirmative.

Keywords:Galois module structure, wild ramification
Categories:11R33, 11S15, 20C32

74. CJM 1998 (vol 50 pp. 829)

Putcha, Mohan S.
Conjugacy classes and nilpotent variety of a reductive monoid
We continue in this paper our study of conjugacy classes of a reductive monoid $M$. The main theorems establish a strong connection with the Bruhat-Renner decomposition of $M$. We use our results to decompose the variety $M_{\nil}$ of nilpotent elements of $M$ into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.

Categories:20G99, 20M10, 14M99, 20F55

75. CJM 1998 (vol 50 pp. 719)

Göbel, Rüdiger; Shelah, Saharon
Indecomposable almost free modules---the local case
Let $R$ be a countable, principal ideal domain which is not a field and $A$ be a countable $R$-algebra which is free as an $R$-module. Then we will construct an $\aleph_1$-free $R$-module $G$ of rank $\aleph_1$ with endomorphism algebra End$_RG = A$. Clearly the result does not hold for fields. Recall that an $R$-module is $\aleph_1$-free if all its countable submodules are free, a condition closely related to Pontryagin's theorem. This result has many consequences, depending on the algebra $A$ in use. For instance, if we choose $A = R$, then clearly $G$ is an indecomposable `almost free' module. The existence of such modules was unknown for rings with only finitely many primes like $R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime $p$. The result complements a classical realization theorem of Corner's showing that any such algebra is an endomorphism algebra of some torsion-free, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorial-algebraic techniques related with what we call {\it rigid tree-elements\/} coming from a module generated over a forest of trees.

Keywords:indecomposable modules of local rings, $\aleph_1$-free modules of rank $\aleph_1$, realizing rings as endomorphism rings
Categories:20K20, 20K26, 20K30, 13C10
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