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51. CMB 2010 (vol 53 pp. 629)

Chinen, Naotsugu; Hosaka, Tetsuya
Asymptotic Dimension of Proper CAT(0) Spaces that are Homeomorphic to the Plane
In this paper, we investigate a proper CAT(0) space $(X,d)$ that is homeomorphic to $\mathbb R^2$ and we show that the asymptotic dimension $\operatorname{asdim} (X,d)$ is equal to $2$.

Keywords:asymptotic dimension, CAT(0) space, plane
Categories:20F69, 54F45, 20F65

52. CMB 2010 (vol 53 pp. 602)

Boij, Mats; Geramita, Anthony
Notes on Diagonal Coinvariants of the Dihedral Group
The bigraded Hilbert function and the minimal free resolutions for the diagonal coinvariants of the dihedral groups are exhibited, as well as for all their bigraded invariant Gorenstein quotients.

Categories:13D02, 20C33, 20F55

53. CMB 2009 (vol 52 pp. 598)

Moreno, M. A.; Nicola, J.; Pardo, E.; Thomas, H.
Numerical Semigroups That Are Not Intersections of $d$-Squashed Semigroups
We say that a numerical semigroup is \emph{$d$-squashed} if it can be written in the form $$ S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$ for $N,a_1,\dots,a_d$ positive integers with $\gcd(a_1,\dots, a_d)=1$. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular. Recent works by Rosales \emph{et al.} give a concrete example of a numerical semigroup that cannot be written as an intersection of $2$-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of $2$-squashed semigroups. We also will prove the same result for $3$-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of $d$-squashed semigroups for any fixed $d$, and we prove some partial results towards this conjecture.

Keywords:numerical semigroup, squashed semigroup, proportionally modular semigroup
Categories:20M14, 06F05, 46L80

54. CMB 2009 (vol 52 pp. 435)

Monson, B.; Schulte, Egon
Modular Reduction in Abstract Polytopes
The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind:\ first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in $\mathbb{Z}[\tau]$ (with $\tau$ the golden ratio), to construct new regular $4$-polytopes of hyperbolic types $\{3,5,3\}$ and $\{5,3,5\}$ with automorphism groups given by finite orthogonal groups.

Keywords:abstract polytopes, regular and chiral, Coxeter groups, modular reduction
Categories:51M20, 20F55

55. CMB 2009 (vol 52 pp. 273)

MacDonald, John; Scull, Laura
Amalgamations of Categories
We consider the pushout of embedding functors in $\Cat$, the category of small categories. We show that if the embedding functors satisfy a 3-for-2 property, then the induced functors to the pushout category are also embeddings. The result follows from the connectedness of certain associated slice categories. The condition is motivated by a similar result for maps of semigroups. We show that our theorem can be applied to groupoids and to inclusions of full subcategories. We also give an example to show that the theorem does not hold when the property only holds for one of the inclusion functors, or when it is weakened to a one-sided condition.

Keywords:category, pushout, amalgamation
Categories:18A30, 18B40, 20L17

56. CMB 2009 (vol 52 pp. 245)

Goodaire, Edgar G.; Milies, César Polcino
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

57. CMB 2009 (vol 52 pp. 9)

Chassé, Dominique; Saint-Aubin, Yvan
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

58. CMB 2008 (vol 51 pp. 584)

Purbhoo, Kevin; Willigenburg, Stephanie van
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

59. CMB 2008 (vol 51 pp. 134)

Rosales, J. C.; Garc\'{\i}a-Sánchez, P. A.
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

60. CMB 2008 (vol 51 pp. 114)

Petrov, V.; Semenov, N.; Zainoulline, K.
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

61. CMB 2008 (vol 51 pp. 81)

Kassel, Christian
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

62. CMB 2007 (vol 50 pp. 535)

Hohlweg, Christophe
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

63. CMB 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
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

64. CMB 2007 (vol 50 pp. 206)

Golasiński, Marek; Gonçalves, Daciberg Lima
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

65. CMB 2007 (vol 50 pp. 268)

Manuilov, V.; Thomsen, K.
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

66. CMB 2006 (vol 49 pp. 347)

Ecker, Jürgen
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

67. CMB 2006 (vol 49 pp. 285)

Riedl, Jeffrey M.
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

68. CMB 2006 (vol 49 pp. 296)

Sch"utt, Matthias
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

69. CMB 2006 (vol 49 pp. 196)

Chernousov, Vladimir
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

70. CMB 2006 (vol 49 pp. 127)

Lewis, Mark L.
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 $2$.


71. CMB 2006 (vol 49 pp. 96)

Külshammer, Burkhard
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

72. CMB 2005 (vol 48 pp. 460)

Sommers, Eric N.
$B$-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

73. CMB 2005 (vol 48 pp. 211)

Germain, Jam
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

74. CMB 2005 (vol 48 pp. 32)

Dąbkowski, Mieczysław K.; Przytycki, Józef H.; Togha, Amir A.
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

75. CMB 2005 (vol 48 pp. 80)

Herman, Allen; Li, Yuanlin; Parmenter, M. M.
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
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