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26. CJM 2007 (vol 59 pp. 418)

Stoimenow, A.
 On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials It is known that the Brandt--Lickorish--Millett--Ho polynomial $Q$ contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from $Q$ is an open problem. We show that this is not so up to degree $9$. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree $d \le 10$ are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems. Categories:57M25, 57M27, 20F36, 57M50

27. CJM 2007 (vol 59 pp. 296)

Chein, Orin; Goodaire, Edgar G.
 Bol Loops of Nilpotence Class Two Call a non-Moufang Bol loop \emph{minimally non-Moufang} if every proper subloop is Moufang and \emph{minimally nonassociative} if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to $1$. In the process, we derive many commutator and associator identities which hold in such loops. Keywords:Bol loop, Moufang loop, nilpotent, commutator, associator, minimally nonassociativeCategory:20N05

28. CJM 2006 (vol 58 pp. 1144)

Hamana, Masamichi
 Partial $*$-Automorphisms, Normalizers, and Submodules in Monotone Complete $C^*$-Algebras For monotone complete $C^*$-algebras $A\subset B$ with $A$ contained in $B$ as a monotone closed $C^*$-subalgebra, the relation $X = AsA$ gives a bijection between the set of all monotone closed linear subspaces $X$ of $B$ such that $AX + XA \subset X$ and $XX^* + X^*X \subset A$ and a set of certain partial isometries $s$ in the normalizer" of $A$ in $B$, and similarly for the map $s \mapsto \Ad s$ between the latter set and a set of certain partial $*$-automorphisms" of $A$. We introduce natural inverse semigroup structures in the set of such $X$'s and the set of partial $*$-automorphisms of $A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough $B$ the homomorphism becomes surjective and all the partial $*$-automorphisms of $A$ are realized via partial isometries in $B$. In particular, the inverse semigroup associated with a type ${\rm II}_1$ von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the $C^*$-algebra version of these results. Categories:46L05, 46L08, 46L40, 20M18

29. CJM 2006 (vol 58 pp. 897)

Courtès, François
 Distributions invariantes sur les groupes rÃ©ductifs quasi-dÃ©ployÃ©s Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \a l'espace de leurs restrictions \a l'alg\ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis. Keywords:reductive $p$-adic groups, orbital integrals, invariant distributionsCategories:22E35, 20G40

30. CJM 2006 (vol 58 pp. 23)

Dabbaghian-Abdoly, Vahid
 Constructing Representations of Finite Simple Groups and Covers Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. An efficient and simple method to construct representations of finite groups is applicable whenever $G$ has a subgroup $H$ such that $\chi_H$ has a linear constituent with multiplicity $1$. In this paper we show (with a few exceptions) that if $G$ is a simple group or a covering group of a simple group and $\chi$ is an irreducible character of $G$ of degree less than 32, then there exists a subgroup $H$ (often a Sylow subgroup) of $G$ such that $\chi_H$ has a linear constituent with multiplicity $1$. Keywords:group representations, simple groups, central covers, irreducible representationsCategories:20C40, 20C15

31. CJM 2005 (vol 57 pp. 1056)

Ozawa, Narutaka; Rieffel, Marc A.
 Hyperbolic Group $C^*$-Algebras and Free-Product $C^*$-Algebras as Compact Quantum Metric Spaces Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a Dirac'' operator for $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is a word-length function on $G$, then the topology from this metric coincides with the weak-$*$ topology (our definition of a compact quantum metric space''). We show that a convenient framework is that of filtered $C^*$-algebras which satisfy a suitable Haagerup-type'' condition. We also use this framework to prove an analogous fact for certain reduced free products of $C^*$-algebras. Categories:46L87, 20F67, 46L09

32. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
 Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup. Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-typesCategories:20G25, 22E35, 20H25

33. CJM 2005 (vol 57 pp. 416)

Wise, Daniel T.
 Approximating Flats by Periodic Flats in \\CAT(0) Square Complexes We investigate the problem of whether every immersed flat plane in a nonpositively curved square complex is the limit of periodic flat planes. Using a branched cover, we reduce the problem to the case of $\V$-complexes. We solve the problem for malnormal and cyclonormal $\V$-complexes. We also solve the problem for complete square complexes using a different approach. We give an application towards deciding whether the elements of fundamental groups of the spaces we study have commuting powers. We note a connection between the flat approximation problem and subgroup separability. Keywords:CAT(0), periodic flat planesCategories:20F67, 20F06

34. CJM 2004 (vol 56 pp. 945)

Helminck, Aloysius G.; Schwarz, Gerald W.
 Smoothness of Quotients Associated \\With a Pair of Commuting Involutions Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$ and $K:=G^\theta$ be the fixed point groups. We have an action $(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson \cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$. Categories:20G15, 20G20, 22E15, 22E46

35. CJM 2004 (vol 56 pp. 871)

Schocker, Manfred
 Lie Elements and Knuth Relations A coplactic class in the symmetric group $\Sym_n$ consists of all permutations in $\Sym_n$ with a given Schensted $Q$-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of $\Sym_n$ which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras. Keywords:symmetric group, descent set, coplactic relation, Hopf algebra,, convolution productCategories:17B01, 05E10, 20C30, 16W30

36. CJM 2004 (vol 56 pp. 246)

Bonnafé, Cédric
 ÃlÃ©ments unipotents rÃ©guliers des sous-groupes de Levi We investigate the structure of the centralizer of a regular unipotent element of a Levi subgroup of a reductive group. We also investigate the structure of the group of components of this centralizer in relation with the notion of cuspidal local system defined by Lusztig. We determine its unipotent radical, we prove that it admits a Levi complement, and we get some properties on its Weyl group. As an application, we prove some results which were announced in previous paper on regular unipotent elements. Nous \'etudions la structure du centralisateur d'un \'el\'ement unipotent r\'egulier d'un sous-groupe de Levi d'un groupe r\'eductif, ainsi que la structure du groupe des composantes de ce centralisateur en relation avec la notion de syst\`eme local cuspidal d\'efinie par Lusztig. Nous d\'eterminons son radical unipotent, montrons l'existence d'un compl\'ement de Levi et \'etudions la structure de son groupe de Weyl. Comme application, nous d\'emontrons des r\'esultats qui \'etaient annonc\'es dans un pr\'ec\'edent article de l'auteur sur les \'el\'ements unipotents r\'eguliers. Category:20G

37. 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

38. 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

39. 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 modulesCategories:20K20, 20K30, 13B10, 13B25

40. 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

41. 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

42. 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 graphsCategories:22D05, 20B07, 20B27, 05C25

43. CJM 2001 (vol 53 pp. 1121)

 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

44. 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

45. 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

46. CJM 2000 (vol 52 pp. 449)

 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

47. 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 subgroupCategories:14M15, 20G05

48. 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

49. CJM 2000 (vol 52 pp. 197)

 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
 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