51. 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 

52. 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 

53. 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 

54. 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 

55. 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 

56. 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 

57. 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 

58. CMB 2011 (vol 55 pp. 188)
59. 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 

60. 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 

61. 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 

62. 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 

63. 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 

64. CMB 2010 (vol 53 pp. 629)
65. CMB 2010 (vol 53 pp. 602)
66. 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 

67. CMB 2010 (vol 53 pp. 706)
68. 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 

69. 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 

70. 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 

71. 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{skewsymmetric}
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 skewsymmetric elements of $RL$
commute if and only if
the characteristic of $R$ is $2$ or $4$.
Categories:20N05, 17D05 

72. CMB 2009 (vol 52 pp. 9)
 Chassé, Dominique; SaintAubin, 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 

73. 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 LittlewoodRichardson
coefficients that are nonzero, and a condition on Schur nonnegativity.
Keywords:polynomial representation, symmetric function, LittlewoodRichardson coefficient, Schur nonnegative Categories:05E05, 05E10, 20C30 

74. 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 

75. CMB 2008 (vol 51 pp. 134)