26. CMB 2012 (vol 56 pp. 630)
 Sundar, S.

Inverse Semigroups and Sheu's Groupoid for the Odd Dimensional Quantum Spheres
In this paper, we give a different proof of the fact that the odd dimensional
quantum spheres are groupoid $C^{*}$algebras. We show that the $C^{*}$algebra
$C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial
isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the
inverse semigroup $T$ by Exel is exactly the same as the groupoid
considered by Sheu.
Keywords:inverse semigroups, groupoids, odd dimensional quantum spheres Categories:46L99, 20M18 

27. CMB 2011 (vol 56 pp. 395)
 Oancea, D.

Coessential Abelianization Morphisms in the Category of Groups
An epimorphism $\phi\colon G\to H$ of groups, where $G$ has rank $n$, is called
coessential if every (ordered) generating $n$tuple of $H$ can be
lifted along $\phi$ to a generating $n$tuple for $G$. We discuss this
property in the context of the category of groups, and establish a criterion
for such a group $G$ to have the property that its abelianization
epimorphism $G\to G/[G,G]$, where $[G,G]$ is the commutator subgroup, is
coessential. We give an example of a family of 2generator groups whose
abelianization epimorphism is not coessential.
This family also provides counterexamples to the generalized AndrewsCurtis conjecture.
Keywords:coessential epimorphism, Nielsen transformations, AndrewCurtis transformations Categories:20F05, 20F99, 20J15 

28. CMB 2011 (vol 55 pp. 783)
 Motallebi, M. R.; Saiflu, H.

Products and Direct Sums in Locally Convex Cones
In this paper we define lower, upper, and symmetric completeness and
discuss closure of the sets in product and direct sums. In particular,
we introduce suitable bases for these topologies, which leads us to
investigate completeness of the direct sum and its components. Some
results obtained about $X$topologies and polars of the neighborhoods.
Keywords:product and direct sum, duality, locally convex cone Categories:20K25, 46A30, 46A20 

29. CMB 2011 (vol 56 pp. 272)
 Cheng, Lixin; Luo, Zhenghua; Zhou, Yu

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate
In this note, we first give a characterization of super weakly
compact convex sets of a Banach space $X$:
a closed bounded convex set $K\subset X$ is
super weakly compact if and only if there exists a $w^*$ lower
semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in
K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly FrÃ©chet
differentiable on each bounded set of $X^*$. Then we present a
representation theorem for the dual of the semigroup $\textrm{swcc}(X)$
consisting of all the nonempty super weakly compact convex sets of the
space $X$.
Keywords:super weakly compact set, dual of normed semigroup, uniform FrÃ©chet differentiability, representation Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50 

30. CMB 2011 (vol 56 pp. 13)
 Alon, Gil; Kozma, Gady

Ordering the Representations of $S_n$ Using the Interchange Process
Inspired by Aldous' conjecture for
the spectral gap of the interchange process and its recent
resolution by Caputo, Liggett, and Richthammer, we define
an associated order $\prec$ on the irreducible representations of $S_n$. Aldous'
conjecture is equivalent to certain representations being comparable
in this order, and hence determining the ``Aldous order'' completely is a
generalized question. We show a few additional entries for this order.
Keywords:Aldous' conjecture, interchange process, symmetric group, representations Categories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35 

31. CMB 2011 (vol 55 pp. 673)
 Aizenbud, Avraham; Gourevitch, Dmitry

Multiplicity Free Jacquet Modules
Let $F$ be a nonArchimedean local field or a finite field.
Let $n$ be a natural number and $k$ be $1$ or $2$.
Consider $G:=\operatorname{GL}_{n+k}(F)$ and let
$M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup.
Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.
Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$.
In this paper we prove that $J$ is a multiplicity free functor, i.e.,
$\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$,
for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$.
We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors.
At the end we discuss whether other Jacquet functors are multiplicity free.
Keywords:multiplicity one, Gelfand pair, invariant distribution, finite group Categories:20G05, 20C30, 20C33, 46F10, 47A67 

32. CMB 2011 (vol 54 pp. 654)
 Forrest, Brian E.; Runde, Volker

Norm One Idempotent $cb$Multipliers with Applications to the Fourier Algebra in the $cb$Multiplier Norm
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)
Keywords:amenability, bounded approximate identity, $cb$multiplier norm, Fourier algebra, norm one idempotent Categories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25 

33. CMB 2011 (vol 55 pp. 48)
 Chebolu, Sunil K.; Christensen, J. Daniel; Mináč, Ján

Freyd's Generating Hypothesis for Groups with Periodic Cohomology
Let $G$ be a finite group, and let $k$ be a field whose characteristic $p$
divides
the order of $G$.
Freyd's generating hypothesis for the stable module category of
$G$ is the statement that a map between finitedimensional
$kG$modules in the thick subcategory generated by $k$ factors through a
projective if the induced map on Tate cohomology is trivial. We show that if
$G$
has periodic cohomology, then the generating hypothesis holds if and only if
the Sylow
$p$subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions
that are equivalent to the GH
for groups with periodic cohomology.
Keywords:Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomology Categories:20C20, 20J06, 55P42 

34. CMB 2011 (vol 55 pp. 390)
 Riedl, Jeffrey M.

Automorphisms of Iterated Wreath Product $p$Groups
We determine the order of
the automorphism group
$\operatorname{Aut}(W)$ for each member
$W$ of an important family
of finite $p$groups that
may be constructed as
iterated regular wreath
products of cyclic groups.
We use a method based on
representation theory.
Categories:20D45, 20D15, 20E22 

35. CMB 2011 (vol 54 pp. 663)
 Haas, Ruth; G. Helminck, Aloysius

Admissible Sequences for Twisted Involutions in Weyl Groups
Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$
related to a basis $\Delta$ for the root system $\Phi$ associated with
$W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We
show that the set of $\theta$twisted involutions in $W$,
$\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{1}\}$ is in one
to one correspondence with the set of regular involutions
$\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are
characterized by sequences in $\Sigma$ which induce an ordering called
the RichardsonSpringer Poset. In particular, for $\Phi$ irreducible,
the ascending RichardsonSpringer Poset of $\mathcal{I}_{\theta}$,
for nontrivial $\theta$ is identical to the descending
RichardsonSpringer Poset of $\mathcal{I}_{\operatorname{Id}}$.
Categories:20G15, 20G20, 22E15, 22E46, 43A85 

36. CMB 2011 (vol 55 pp. 98)
 Glied, Svenja

Similarity and Coincidence Isometries for Modules
The groups of (linear) similarity and coincidence isometries of
certain modules $\varGamma$ in $d$dimensional Euclidean space, which
naturally occur in quasicrystallography, are considered. It is shown
that the structure of the factor group of similarity modulo
coincidence isometries is the direct sum of cyclic groups of prime
power orders that divide $d$. In particular, if the dimension $d$ is a
prime number $p$, the factor group is an elementary abelian
$p$group. This generalizes previous results obtained for lattices to
situations relevant in quasicrystallography.
Categories:20H15, 82D25, 52C23 

37. CMB 2011 (vol 55 pp. 38)
 Butske, William

Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras
Zarhin proves that if $C$ is the curve $y^2=f(x)$ where
$\textrm{Gal}_{\mathbb{Q}}(f(x))=S_n$ or $A_n$, then
${\textrm{End}}_{\overline{\mathbb{Q}}}(J)=\mathbb{Z}$. In seeking to examine his
result in the genus $g=2$ case supposing other Galois groups, we
calculate
$\textrm{End}_{\overline{\mathbb{Q}}}(J)\otimes_{\mathbb{Z}} \mathbb{F}_2$
for a genus $2$ curve where $f(x)$ is irreducible.
In particular, we show that unless the Galois group is $S_5$ or
$A_5$, the Galois group does not determine ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$.
Categories:11G10, 20C20 

38. CMB 2011 (vol 55 pp. 188)
39. CMB 2011 (vol 54 pp. 487)
 Kong, Xiangjun

Some Properties Associated with Adequate Transversals
In this paper, another relationship between the quasiideal adequate transversals
of an abundant semigroup is given. We introduce the concept of a weakly multiplicative
adequate transversal and the classic result that an adequate transversal is multiplicative
if and only if it is weakly multiplicative and a quasiideal is obtained.
Also, we give two equivalent conditions for an adequate transversal to be weakly
multiplicative. We then consider the case when $I$ and $\Lambda$ (defined below) are
bands. This is analogous to the inverse transversal if the regularity condition is adjoined.
Keywords:abundant semigroup, adequate transversal, Green's $*$relations, quasiideal Category:20M10 

40. CMB 2011 (vol 54 pp. 255)
 Dehaye, PaulOlivier

On an Identity due to Bump and Diaconis, and Tracy and Widom
A classical question for a Toeplitz matrix with given symbol is how to
compute asymptotics for the determinants of its reductions to finite
rank. One can also consider how those asymptotics are affected when
shifting an initial set of rows and columns (or, equivalently,
asymptotics of their minors). Bump and Diaconis
obtained a formula for such shifts involving Laguerre polynomials and
sums over symmetric groups. They also showed how the Heine identity
extends for such minors, which makes this question relevant to Random
Matrix Theory. Independently, Tracy and Widom
used the WienerHopf factorization to
express those shifts in terms of products of infinite matrices. We
show directly why those two expressions are equal and uncover some
structure in both formulas that was unknown to their authors. We
introduce a mysterious differential operator on symmetric functions
that is very similar to vertex operators. We show that the
BumpDiaconisTracyWidom identity is a differentiated version of the
classical JacobiTrudi identity.
Keywords:Toeplitz matrices, JacobiTrudi identity, SzegÅ limit theorem, Heine identity, WienerHopf factorization Categories:47B35, 05E05, 20G05 

41. CMB 2011 (vol 54 pp. 297)
 Johnson, Marianne; Stöhr, Ralph

Lie Powers and PseudoIdempotents
We give a new factorisation of the classical Dynkin operator,
an element of the integral group ring of the symmetric group that
facilitates projections of tensor powers onto Lie powers.
As an application we show that the iterated Lie power $L_2(L_n)$ is
a module direct summand of the Lie power $L_{2n}$ whenever the
characteristic of the ground field does not divide $n$. An explicit
projection of the latter onto the former is exhibited in this case.
Categories:17B01, 20C30 

42. CMB 2010 (vol 54 pp. 237)
 Creedon, Leo; Gildea, Joe

The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$
Let $RG$ denote the group ring of the group $G$ over
the ring $R$. Using an isomorphism between $RG$ and a
certain ring of $n \times n$ matrices in conjunction with other
techniques, the structure of the unit group of the group algebra
of the dihedral group of order $8$ over any
finite field of chracteristic $2$ is determined in
terms of split extensions of cyclic groups.
Categories:16U60, 16S34, 20C05, 15A33 

43. CMB 2010 (vol 54 pp. 3)
 Bakonyi, M.; Timotin, D.

Extensions of Positive Definite Functions on Amenable Groups
Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and
$S^{1}=S$. The main result of this paper states that if the Cayley
graph of $G$ with respect to $S$ has a certain combinatorial property,
then every positive definite operatorvalued function on $S$ can be
extended to a positive definite function on $G$. Several known
extension results are obtained as corollaries. New applications are
also presented.
Categories:43A35, 47A57, 20E05 

44. CMB 2010 (vol 54 pp. 39)
 Chapman, S. T.; GarcíaSánchez, P. A.; Llena, D.; Marshall, J.

Elements in a Numerical Semigroup with Factorizations of the Same Length
Questions concerning the lengths of factorizations into irreducible
elements in numerical monoids
have gained much attention in the recent literature. In this note,
we show that a numerical monoid has an element with two different
irreducible factorizations of the same length if and only if its
embedding dimension is greater than
two. We find formulas in embedding dimension three for the smallest
element with two different irreducible factorizations of the same
length and the largest element whose different irreducible
factorizations all have distinct lengths. We show that these
formulas do not naturally extend to higher embedding
dimensions.
Keywords:numerical monoid, numerical semigroup, nonunique factorization Categories:20M14, 20D60, 11B75 

45. CMB 2010 (vol 53 pp. 706)
46. CMB 2010 (vol 53 pp. 629)
47. CMB 2010 (vol 53 pp. 602)
48. 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
2squashed 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 

49. 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 

50. 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 3for2
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 onesided condition.
Keywords:category, pushout, amalgamation Categories:18A30, 18B40, 20L17 
