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1. CMB Online first

Leandro, Cagliero; Szechtman, Fernando
Jordan-Chevalley decomposition in Lie algebras
We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.

Keywords:solvable Lie algebra, Jordan-Chevalley decomposition, representation
Categories:17-08, 17B05, 20C40, 15A21

2. CMB 2018 (vol 61 pp. 553)

Karimianpour, Camelia
Branching Rules for $n$-fold Covering Groups of $\mathrm{SL}_2$ over a Non-Archimedean Local Field
Let $\mathtt{G}$ be the $n$-fold covering group of the special linear group of degree two, over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $\mathtt{G}$ to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

Keywords:local field, covering group, representation, Hilbert symbol, $\mathsf{K}$-type
Category:20G05

3. CMB Online first

Steinberg, Benjamin; van Gool, Samuel J.
Merge decompositions, two-sided Krohn-Rhodes, and aperiodic pointlikes
This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointlike theorem, using a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_1,T_2$, which together generate $T$, and the subsemigroup generated by their setwise product $T_1T_2$. In this sense we decompose $T$ by merging the subsemigroups $T_1$ and $T_2$. More generally, our technique merges semigroup homomorphisms from free semigroups.

Keywords:Krohn-Rhodes theorem, aperiodic pointlikes
Categories:20M07, 20M35, 68Q70

4. CMB 2018 (vol 61 pp. 848)

Schmidt, Simon; Weber, Moritz
Quantum Symmetries of Graph $C^*$-algebras
The study of graph $C^*$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph $C^*$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph $C^*$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Keywords:finite graph, graph automorphism, automorphism group, quantum automorphism, graph C*-algebra, quantum group, quantum symmetry
Categories:46LXX, 05CXX, 20B25

5. CMB 2017 (vol 61 pp. 405)

Yukita, Tomoshige
Growth Rates of 3-dimensional Hyperbolic Coxeter Groups are Perron Numbers
In this paper we consider the growth rates of 3-dimensional hyperbolic Coxeter polyhedra with at least one dihedral angle of the form $\frac{\pi}{k}$ for an integer $k\geq{7}$. Combining a classical result by Parry with a previous result of ours, we prove that the growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers.

Keywords:Coxeter group, growth function, growth rate, Perron number
Categories:20F55, 20F65

6. CMB 2017 (vol 61 pp. 174)

Roche, Alan; Vinroot, C. Ryan
A factorization result for classical and similitude groups
For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well known result of Mœglin, Vignéras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representation to its dual.

Keywords:classical group, similitude group, involution, $p$-adic group, dual of representation
Categories:20G15, 22E50

7. CMB 2017 (vol 61 pp. 572)

Koskivirta, Jean-Stefan
Normalization of closed Ekedahl-Oort strata
We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti-Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl-Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

Keywords:Ekedahl-Oort stratification, Shimura variety
Categories:14K10, 20G40, 11G18

8. CMB 2017 (vol 61 pp. 225)

Bichon, Julien; Kyed, David; Raum, Sven
Higher $\ell^2$-Betti Numbers of Universal Quantum Groups
We calculate all $\ell^2$-Betti numbers of the universal discrete Kac quantum groups $\hat{\mathrm U}^+_n$ as well as their half-liberated counterparts $\hat{\mathrm U}^*_n$.

Keywords:$\ell^2$-Betti number, free unitary quantum group, half-liberated unitary quantum group, free product formula, extension
Categories:16T05, 46L65, 20G42

9. CMB 2017 (vol 60 pp. 402)

Shravan Kumar, N.
Invariant Means on a Class of von Neumann Algebras Related to Ultraspherical Hypergroups II
Let $K$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi$ and let $VN(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K.$ In this note, we show that the set of invariant means on $VN(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$ in the $cb$-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

Keywords:ultraspherical hypergroup, Fourier algebra, Fourier-Stieltjes algebra, invariant mean, generalized translation, generalized invariant mean
Categories:43A62, 46J10, 43A30, 20N20

10. CMB 2017 (vol 61 pp. 301)

Józiak, Paweł
Remarks on Hopf Images and Quantum Permutation Groups $S_n^+$
Motivated by a question of A. Skalski and P.M. Sołtan (2016) about inner faithfulness of the S. Curran's map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, but also to classify the automorphisms of $S_4^+$, describe all the embeddings $O_{-1}(2)\subset S_4^+$ and show that all the copies of $O_{-1}(2)$ inside $S_4^+$ are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid.

Keywords:Hopf image, quantum permutation group, compact quantum group
Categories:20G42, 81R50, 46L89, 16W35

11. CMB 2017 (vol 60 pp. 774)

Jensen, Gerd; Pommerenke, Christian
On the Structure of the Schild Group in Relativity Theory
Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations. The present paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature we associate Lorentz transformations with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups.

Keywords:Lorentz transformation, integer lattice, Gaussian integers, Schild group, subgroup
Categories:22E43, 20H99, 83A05

12. CMB 2017 (vol 60 pp. 604)

Louder, Larsen; Wilton, Henry
Stackings and the $W$-cycles Conjecture
We prove Wise's $W$-cycles conjecture: Consider a compact graph $\Gamma'$ immersing into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma'$. Unless $\Lambda'$ is reducible, the degree of the covering map $\mathbb{S}\to S^1$ is bounded above by minus the Euler characteristic of $\Gamma'$. As a corollary, any finitely generated subgroup of a one-relator group has finitely generated Schur multiplier.

Keywords:free groups, one-relator groups, right-orderability
Category:20F65

13. CMB 2017 (vol 60 pp. 830)

Motegi, Kimihiko; Teragaito, Masakazu
Generalized Torsion Elements and Bi-orderability of 3-manifold Groups
It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of $3$-manifolds, and verify the conjecture for non-hyperbolic, geometric $3$-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic $3$-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group $F(2, m)$ ($m \gt 2$) is a generalized torsion element.

Keywords:generalized torsion element, bi-ordering, 3-manifold group
Categories:57M25, 57M05, 06F15, 20F05

14. CMB 2016 (vol 60 pp. 54)

Button, Jack
Tubular Free by Cyclic Groups Act Freely on CAT(0) Cube Complexes
We identify when a tubular group (the fundamental group of a finite graph of groups with $\mathbb{Z}^2$ vertex and $\mathbb{Z}$ edge groups) is free by cyclic and show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex.

Keywords:CAT(0), tubular group
Categories:20F65, 20F67, 20E08

15. CMB 2016 (vol 60 pp. 762)

Jantzen, Jens Carsten
Maximal Weight Composition Factors for Weyl Modules
Fix an irreducible (finite) root system $R$ and a choice of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group $G_k$ over $k$ with root system $k$. One associates to any dominant weight $\lambda$ for $R$ two $G_k$--modules with highest weight $\lambda$, the Weyl module $V (\lambda)_k$ and its simple quotient $L (\lambda)_k$. Let $\lambda$ and $\mu$ be dominant weights with $\mu \lt \lambda$ such that $\mu$ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists $k$ such that $L (\mu)_k$ is a composition factor of $V (\lambda)_k$, and they exhibit an example in type $E_8$ where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact, one that uses a classification of the possible pairs $(\lambda, \mu)$, and another one that relies only on the classification of root systems.

Keywords:algebraic groups, represention theory
Categories:20G05, 20C20

16. CMB 2016 (vol 60 pp. 12)

Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
Co-maximal Graphs of Subgroups of Groups
Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma(H)$ is connected and moreover the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

Keywords:co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group
Categories:05C25, 05E15, 20D10, 20D15

17. CMB 2016 (vol 60 pp. 154)

Liu, Ye
On Chromatic Functors and Stable Partitions of Graphs
The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated to simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is in general a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated to a finite graph restricted to the category $\mathrm{FI}$ of finite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups which is representation stable in the sense of Church-Farb.

Keywords:chromatic functor, stable partition, representation stability
Categories:05C15, 20C30

18. CMB 2016 (vol 60 pp. 77)

Christ, Michael; Rieffel, Marc A.
Nilpotent Group C*-algebras as Compact Quantum Metric Spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth
Categories:46L87, 20F65, 22D15, 53C23, 58B34

19. CMB 2016 (vol 59 pp. 682)

Carlson, Jon F.; Chebolu, Sunil K.; Mináč, Ján
Ghosts and Strong Ghosts in the Stable Category
Suppose that $G$ is a finite group and $k$ is a field of characteristic $p\gt 0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$-subgroup of $G$ is cyclic of order 2 or 3. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd's generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps which mimic ghosts in high degrees.

Keywords:Tate cohomology, ghost maps, stable module category, almost split sequence, periodic cohomology
Categories:20C20, 20J06, 55P42

20. CMB 2016 (vol 60 pp. 111)

Ghaani Farashahi, Arash
Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups
This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space $L^2(G/H,\mu)$.

Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formula
Categories:20G05, 43A85, 43A32, 43A40

21. CMB 2016 (vol 59 pp. 617)

Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei
Canonical Systems of Basic Invariants for Unitary Reflection Groups
It has been known that there exists a canonical system for every finite real reflection group. The first and the third authors obtained an explicit formula for a canonical system in the previous paper. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

Keywords:basic invariant, invariant theory, finite unitary reflection group
Categories:13A50, 20F55

22. CMB 2016 (vol 59 pp. 824)

Karpenko, Nikita A.
Incompressibility of Products of Pseudo-homogeneous Varieties
We show that the conjectural criterion of $p$-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties which includes the norm varieties associated to symbols in Galois cohomology of arbitrary degree.

Keywords:algebraic groups, projective homogeneous varieties, Chow groups and motives, canonical dimension and incompressibility
Categories:20G15, 14C25

23. CMB 2016 (vol 59 pp. 234)

Beardon, Alan F.
Non-discrete Frieze Groups
The classification of Euclidean frieze groups into seven conjugacy classes is well known, and many articles on recreational mathematics contain frieze patterns that illustrate these classes. However, it is only possible to draw these patterns because the subgroup of translations that leave the pattern invariant is (by definition) cyclic, and hence discrete. In this paper we classify the conjugacy classes of frieze groups that contain a non-discrete subgroup of translations, and clearly these groups cannot be represented pictorially in any practical way. In addition, this discussion sheds light on why there are only seven conjugacy classes in the classical case.

Keywords:frieze groups, isometry groups
Categories:51M04, 51N30, 20E45

24. CMB 2016 (vol 59 pp. 244)

Cao, Wensheng; Huang, Xiaolin
A Note on Quaternionic Hyperbolic Ideal Triangle Groups
In this paper, the quaternionic hyperbolic ideal triangle groups are parameterized by a real one-parameter family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is the tangent of the quaternionic angular invariant of a triple of points in $\partial \mathbf{H}_{\mathbb{h}}^2 $ forming this ideal triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is not a discrete embedding, and if $s \leq \sqrt{35}$ then $\phi_s$ is a discrete embedding.

Keywords:quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant
Categories:20F67, 22E40, 30F40

25. CMB 2016 (vol 59 pp. 392)

Prajapati, S. K.; Sarma, R.
Total Character of a Group $G$ with $(G,Z(G))$ as a Generalized Camina Pair
We investigate whether the total character of a finite group $G$ is a polynomial in a suitable irreducible character of $G$. When $(G,Z(G))$ is a generalized Camina pair, we show that the total character is a polynomial in a faithful irreducible character of $G$ if and only if $Z(G)$ is cyclic.

Keywords:finite groups, group characters, total characters
Category:20C15
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