Expand all Collapse all | Results 51 - 75 of 109 |
51. CMB 2009 (vol 52 pp. 245)
Involutions of RA Loops Let $L$ be an RA loop, that is, a loop whose loop ring
over any coefficient ring $R$
is an alternative, but not associative, ring. Let
$\ell\mapsto\ell^\theta$ denote an involution on $L$ and extend
it linearly to the loop ring $RL$. An element $\alpha\in RL$ is
\emph{symmetric} if $\alpha^\theta=\alpha$ and \emph{skew-symmetric}
if $\alpha^\theta=-\alpha$. In this paper, we show that
there exists an involution making
the symmetric elements of $RL$ commute if and only if
the characteristic of $R$ is $2$ or $\theta$ is the
canonical involution on $L$,
and an involution making the skew-symmetric elements of $RL$
commute if and only if
the characteristic of $R$ is $2$ or $4$.
Categories:20N05, 17D05 |
52. CMB 2009 (vol 52 pp. 9)
On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements
$[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the
symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{-1})}$ with
$0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in
\sn$. We give the spectrum of $R_n$ and show that the ratio of the
largest eigenvalue $\lambda_0$ to the second largest one (in absolute
value) increases as a positive power of $n$ as $n\rightarrow \infty$.
Keywords:symmetric group, representation theory, eigenvalue, statistical physics Categories:20B30, 20C30, 15A18, 82B20, 82B28 |
53. CMB 2008 (vol 51 pp. 584)
On Tensor Products of Polynomial Representations We determine the necessary and sufficient combinatorial
conditions for which the tensor product of two irreducible polynomial
representations of $\GL(n,\mathbb{C})$ is isomorphic to another.
As a consequence we discover families of Littlewood--Richardson
coefficients that are non-zero, and a condition on Schur non-negativity.
Keywords:polynomial representation, symmetric function, Littlewood--Richardson coefficient, Schur non-negative Categories:05E05, 05E10, 20C30 |
54. CMB 2008 (vol 51 pp. 114)
Zero Cycles on a Twisted Cayley Plane Let $k$ be a field of characteristic not $2,3$.
Let $G$ be an exceptional simple algebraic group over $k$
of type $\F$, $^1{\E_6}$ or $\E_7$ with trivial Tits algebras.
Let $X$ be a projective $G$-homogeneous variety.
If $G$ is of type $\E_7$, we assume in addition
that the respective
parabolic subgroup is of type $P_7$.
The main result of the paper says that
the degree map on the group of zero cycles of $X$
is injective.
Categories:20G15, 14C15 |
55. CMB 2008 (vol 51 pp. 81)
Homotopy Formulas for Cyclic Groups Acting on Rings The positive cohomology groups of a finite group acting on a ring
vanish when the ring has a norm one element. In this note we give
explicit homotopies on the level of cochains when the group is cyclic,
which allows us to express any cocycle of a cyclic group
as the coboundary of an explicit cochain.
The formulas in this note are closely related to the effective problems considered in previous joint work
with Eli Aljadeff.
Keywords:group cohomology, norm map, cyclic group, homotopy Categories:20J06, 20K01, 16W22, 18G35 |
56. CMB 2008 (vol 51 pp. 134)
Numerical Semigroups Having a Toms Decomposition We show that the class of system proportionally modular numerical semigroups
coincides with the class of numerical semigroups having a Toms
decomposition.
Categories:20M14, 11D75 |
57. CMB 2007 (vol 50 pp. 632)
Transformations and Colorings of Groups Let $G$ be a compact topological group and let $f\colon G\to G$ be a
continuous transformation of $G$. Define $f^*\colon G\to G$ by
$f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume
that $H=\Imag f^*$ is a subgroup of $G$ and for every
measurable $C\subseteq H$,
$\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable
$C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that
$f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.
Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10 |
58. CMB 2007 (vol 50 pp. 535)
Generalized Descent Algebras If $A$ is a subset of the set of reflections of a finite Coxeter
group $W$, we define a sub-$\ZM$-module $\DC_A(W)$ of the group
algebra $\ZM W$. We discuss cases where this submodule is a
subalgebra. This family of subalgebras includes strictly the
Solomon descent algebra, the group algebra and, if $W$ is of type
$B$, the Mantaci--Reutenauer algebra.
Keywords:Coxeter group, Solomon descent algebra, descent set Categories:20F55, 05E15 |
59. CMB 2007 (vol 50 pp. 268)
On the Lack of Inverses to $C^*$-Extensions Related to Property T Groups Using ideas of S. Wassermann on non-exact $C^*$-algebras and
property T groups, we show that one of his examples of non-invertible
$C^*$-extensions is not semi-invertible. To prove this, we
show that a certain element vanishes in the asymptotic tensor
product. We also show that a modification of the example gives
a $C^*$-extension which is not even invertible up to homotopy.
Keywords:$C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopy Categories:19K33, 46L06, 46L80, 20F99 |
60. CMB 2007 (vol 50 pp. 206)
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ |
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times
\SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional
$CW$-complex of the homotopy type of an $n$-sphere. We study the
automorphism group $\Aut (G)$ in order to compute the number of
distinct homotopy types of spherical space forms with respect to free
and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with
free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as
well.
Keywords:automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space form Categories:55M35, 55P15, 20E22, 20F28, 57S17 |
61. CMB 2006 (vol 49 pp. 347)
Affine Completeness of Generalised Dihedral Groups In this paper we study affine completeness of generalised dihedral
groups. We give a formula for the number of unary compatible
functions on these groups, and we characterise for every $k \in~\N$
the $k$-affine complete generalised dihedral groups. We find that
the direct product of a $1$-affine complete group with itself need not
be $1$-affine complete. Finally, we give an example of a nonabelian
solvable affine complete group. For nilpotent groups we find a
strong necessary condition for $2$-affine completeness.
Categories:08A40, 16Y30, 20F05 |
62. CMB 2006 (vol 49 pp. 285)
Orbits and Stabilizers for Solvable Linear Groups We extend a result of Noritzsch,
which describes the orbit sizes in the action of a
Frobenius group $G$ on a finite vector space $V$ under
certain conditions, to a more general class of finite
solvable groups $G$.
This result has applications in computing
irreducible character degrees of finite groups.
Another application, proved here, is a result
concerning the structure of certain groups with
few complex irreducible character degrees.
Categories:20B99, 20C15, 20C20 |
63. CMB 2006 (vol 49 pp. 296)
On the Modularity of Three Calabi--Yau Threefolds With Bad Reduction at 11 This paper investigates the modularity of three
non-rigid Calabi--Yau threefolds with bad reduction at 11. They are
constructed as fibre products of rational elliptic surfaces,
involving the modular elliptic surface of level 5. Their middle
$\ell$-adic cohomology groups are shown to split into
two-dimensional pieces, all but one of which can be interpreted in
terms of elliptic curves. The remaining pieces are associated to
newforms of weight 4 and level 22 or 55, respectively. For this
purpose, we develop a method by Serre to compare the corresponding
two-dimensional 2-adic Galois representations with uneven trace.
Eventually this method is also applied to a self fibre product of
the Hesse-pencil, relating it to a newform of weight 4 and level
27.
Categories:14J32, 11F11, 11F23, 20C12 |
64. CMB 2006 (vol 49 pp. 196)
Another Proof of Totaro's Theorem on $E_8$-Torsors We give a short proof of Totaro's theorem that every$E_8$-torsor over
a field $k$ becomes trivial over a finiteseparable extension of $k$of
degree dividing $d(E_8)=2^63^25$.
Categories:11E72, 14M17, 20G15 |
65. CMB 2006 (vol 49 pp. 127)
Character Degree Graphs of Solvable Groups of Fitting Height $2$ Given a finite group $G$, we attach to the character degrees
of $G$ a graph whose vertex set is the set of primes dividing the degrees of
irreducible characters of $G$, and with an edge between $p$ and $q$ if
$pq$ divides the degree of some irreducible character of $G$.
In this paper, we describe which graphs occur when $G$ is
a solvable group of Fitting height $2$.
Category:20C15 |
66. CMB 2006 (vol 49 pp. 96)
Roots of Simple Modules We introduce roots of indecomposable modules over group algebras of finite groups,
and we investigate some of their properties. This allows us to correct an error
in Landrock's book which has to do with roots of simple modules.
Categories:20C20, 20C05 |
67. CMB 2005 (vol 48 pp. 460)
$B$-Stable Ideals in the Nilradical of a Borel Subalgebra We count the number of strictly positive $B$-stable ideals in the
nilradical of a Borel subalgebra and prove that
the minimal roots of any $B$-stable ideal are conjugate
by an element of the Weyl group to a subset of the simple roots.
We also count the number of ideals whose minimal roots are conjugate
to a fixed subset of simple roots.
Categories:20F55, 17B20, 05E99 |
68. CMB 2005 (vol 48 pp. 211)
The Distribution of Totatives The integers coprime to $n$ are called the {\it totatives} \rm of $n$.
D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when
the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th of
the total number of totatives up to $n$. They provided criteria in
various cases. Here we give an ``if and only if'' criterion which
allows us to recover most of the previous results in this literature
and to go beyond, as well to reformulate the problem in terms of
combinatorial group theory. Our criterion is that the above holds if
and only if for every odd character $\chi \pmod \kappa$ (where
$\kappa:=k/\gcd(k,n/\prod_{p|n} p)$) there exists a prime $p=p_\chi$
dividing $n$ for which $\chi(p)=1$.
Categories:11A05, 11A07, 11A25, 20C99 |
69. CMB 2005 (vol 48 pp. 32)
Non-Left-Orderable 3-Manifold Groups We show that several torsion free 3-manifold groups
are not left-orderable.
Our examples are groups of cyclic branched coverings of $S^3$
branched along links.
The figure eight knot provides simple
nontrivial examples. The groups arising in these examples are known
as Fibonacci groups which we show not to be left-orderable.
Many other examples of non-orderable groups are obtained by taking
3-fold branched covers of $S^3$ branched along various hyperbolic
2-bridge knots.
%with various hyperbolic 2-bridge knots as branched sets.
The manifold obtained in such a way from the $5_2$ knot
is of special interest as it is conjectured to be the hyperbolic
3-manifold with the smallest volume.
Categories:57M25, 57M12, 20F60 |
70. CMB 2005 (vol 48 pp. 80)
Trivial Units for Group Rings with $G$-adapted Coefficient Rings For each finite group $G$ for which the integral group ring
$\mathbb{Z}G$ has only trivial units, we give ring-theoretic
conditions for a commutative ring $R$ under which the group ring
$RG$ has nontrivial units. Several examples of rings satisfying
the conditions and rings not satisfying the conditions are given.
In addition, we extend a well-known result for fields by showing
that if $R$ is a ring of finite characteristic and $RG$ has only
trivial units, then $G$ has order at most 3.
Categories:16S34, 16U60, 20C05 |
71. CMB 2005 (vol 48 pp. 41)
Degree Homogeneous Subgroups Let $G$ be a finite group and $H$ be a subgroup. We say that $H$
is \emph{degree homogeneous }if, for each $\chi\in \Irr(G)$, all
the irreducible constituents of the restriction $\chi_{H}$ have
the same degree. Subgroups which are either normal or abelian are
obvious examples of degree homogeneous subgroups. Following a
question by E.~M. Zhmud', we investigate general properties of
such subgroups. It appears unlikely that degree homogeneous
subgroups can be characterized entirely by abstract group
properties, but we provide mixed criteria (involving both group
structure and character properties) which are both necessary and
sufficient. For example, $H$ is degree homogeneous in $G$ if and
only if the derived subgroup $H^{\prime}$ is normal in $G$ and,
for every pair $\alpha,\beta$ of irreducible $G$-conjugate
characters of $H^{\prime}$, all irreducible constituents of
$\alpha^{H}$ and $\beta^{H}$ have the same degree.
Category:20C15 |
72. CMB 2004 (vol 47 pp. 530)
A Characterization of $ PSU_{11}(q)$ Order components of a finite simple group were introduced in [4].
It was proved that some non-abelian simple groups are uniquely determined
by their order components. As the main result of this paper, we
show that groups $PSU_{11}(q)$ are also uniquely determined by
their order components. As corollaries of this result, the
validity of a conjecture of J. G. Thompson and a conjecture of W.
Shi and J. Bi both on $PSU_{11}(q)$ are obtained.
Keywords:Prime graph, order component, finite group,simple group Categories:20D08, 20D05, 20D60 |
73. CMB 2004 (vol 47 pp. 439)
On the Stable Basin Theorem The stable basin theorem was introduced by Basmajian and Miner as a
key step in their necessary condition for the discreteness of a
non-elementary group of complex hyperbolic isometries. In this
paper we improve several of Basmajian and Miner's key estimates and
so give a substantial improvement on the main inequality in the
stable basin theorem.
Categories:22E40, 20H10, 57S30 |
74. CMB 2004 (vol 47 pp. 343)
Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras We construct new examples of non-nil algebras with any number of
generators, which are direct sums of two
locally nilpotent subalgebras. Like all previously known examples, our examples
are contracted semigroup algebras and the underlying semigroups are unions
of locally nilpotent subsemigroups.
In our constructions we make more
transparent
than in the past the close relationship between the considered problem
and combinatorics of words.
Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words Categories:16N40, 16S15, 20M05, 20M25, 68R15 |
75. CMB 2004 (vol 47 pp. 237)
Ramification des sÃ©ries formelles Let $p$ be a prime number. Let $k$ be a finite field of characteristic $p$.
The subset $X+X^2 k[[X]]$ of the ring $k[[X]]$ is a group under the substitution
law $\circ $ sometimes called the Nottingham group of $k$; it is denoted by
$\mathcal{R}_k$. The ramification of one series $\gamma\in\mathcal{R}_k$ is
caracterized by its lower ramification numbers: $i_m(\gamma)=\ord_X
\bigl(\gamma^{p^m} (X)/X - 1\bigr)$, as well as its upper ramification numbers:
$$
u_m (\gamma) = i_0 (\gamma) + \frac{i_1 (\gamma) - i_0(\gamma)}{p} +
\cdots + \frac{i_m (\gamma) - i_{m-1} (\gamma)}{p^m} , \quad (m \in
\mathbb{N}).
$$
By Sen's theorem, the $u_m(\gamma)$ are integers. In this paper, we determine
the sequences of integers $(u_m)$ for which there exists $\gamma\in\mathcal{R}_k$
such that $u_m(\gamma)=u_m$ for all integer $m \geq 0$.
Keywords:ramification, Nottingham group Categories:11S15, 20E18 |