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26. CMB 2012 (vol 57 pp. 424)

Sołtan, Piotr M.; Viselter, Ami
A Note on Amenability of Locally Compact Quantum Groups
In this short note we introduce a notion called ``quantum injectivity'' of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups.

Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivity
Categories:20G42, 22D25, 46L89

27. CMB 2012 (vol 57 pp. 132)

Mubeena, T.; Sankaran, P.
Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-twisted conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that $\operatorname{SL}(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $\Gamma$ has the $R_\infty$-property where $\Gamma$ is a torsion free non-elementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}), \operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Keywords:twisted conjugacy classes, hyperbolic groups, lattices in Lie groups
Category:20E45

28. 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

29. 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

30. 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 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews--Curtis conjecture.

Keywords:coessential epimorphism, Nielsen transformations, Andrew-Curtis transformations
Categories:20F05, 20F99, 20J15

31. 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 Fréchet 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 Fréchet differentiability, representation
Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

32. 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

33. CMB 2011 (vol 55 pp. 673)

Aizenbud, Avraham; Gourevitch, Dmitry
Multiplicity Free Jacquet Modules
Let $F$ be a non-Archimedean 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

34. 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

35. 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 finite-dimensional $kG$-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

36. 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

37. 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 Richardson-Springer Poset. In particular, for $\Phi$ irreducible, the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$, for nontrivial $\theta$ is identical to the descending Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$.

Categories:20G15, 20G20, 22E15, 22E46, 43A85

38. 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

39. 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

40. CMB 2011 (vol 55 pp. 188)

Steinberg, Benjamin
Yet Another Solution to the Burnside Problem for Matrix Semigroups
We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

Keywords:Burnside problem, kernel category
Categories:20M30, 20M32

41. CMB 2011 (vol 54 pp. 487)

Kong, Xiangjun
Some Properties Associated with Adequate Transversals
In this paper, another relationship between the quasi-ideal 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 quasi-ideal 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, quasi-ideal
Category:20M10

42. CMB 2011 (vol 54 pp. 255)

Dehaye, Paul-Olivier
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 Wiener-Hopf 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 Bump-Diaconis-Tracy-Widom identity is a differentiated version of the classical Jacobi-Trudi identity.

Keywords:Toeplitz matrices, Jacobi-Trudi identity, Szegő limit theorem, Heine identity, Wiener-Hopf factorization
Categories:47B35, 05E05, 20G05

43. CMB 2011 (vol 54 pp. 297)

Johnson, Marianne; Stöhr, Ralph
Lie Powers and Pseudo-Idempotents
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

44. 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

45. 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 operator-valued 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

46. CMB 2010 (vol 54 pp. 39)

Chapman, S. T.; García-Sánchez, 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, non-unique factorization
Categories:20M14, 20D60, 11B75

47. CMB 2010 (vol 53 pp. 706)

Roberts, R.; Shareshian, J.
Non-Right-Orderable 3-Manifold Groups
We exhibit infinitely many hyperbolic $3$-manifold groups that are not right-orderable.

Categories:20F60, 57M05, 57M50

48. 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

49. 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

50. 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
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