Expand all Collapse all | Results 26 - 50 of 71 |
26. CJM 2007 (vol 59 pp. 418)
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)
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 nonassociative Category:20N05 |
28. CJM 2006 (vol 58 pp. 1144)
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)
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 distributions Categories:22E35, 20G40 |
30. CJM 2006 (vol 58 pp. 23)
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 representations Categories:20C40, 20C15 |
31. CJM 2005 (vol 57 pp. 1056)
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)
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, $K$-types Categories:20G25, 22E35, 20H25 |
33. CJM 2005 (vol 57 pp. 416)
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 planes Categories:20F67, 20F06 |
34. CJM 2004 (vol 56 pp. 945)
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)
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 product Categories:17B01, 05E10, 20C30, 16W30 |
36. CJM 2004 (vol 56 pp. 246)
ÃlÃ©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)
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)
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)
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 |
40. CJM 2002 (vol 54 pp. 1229)
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)
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)
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 |
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)
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)
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)
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 |
48. CJM 2000 (vol 52 pp. 438)
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 |
50. CJM 1999 (vol 51 pp. 1175)
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 |