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26. CJM 2010 (vol 63 pp. 413)

Konvalinka, Matjaž; Skandera, Mark
 Generating Functions for Hecke Algebra Characters Certain polynomials in $n^2$ variables that serve as generating functions for symmetric group characters are sometimes called ($S_n$) character immanants. We point out a close connection between the identities of Littlewood--Merris--Watkins and Goulden--Jackson, which relate $S_n$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra $H_n(q)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood--Merris--Watkins identities and selected Goulden--Jackson identities that relate $H_n(q)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis--L\^e--Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity. Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ringCategories:15A15, 20C08, 81R50

27. CJM 2010 (vol 62 pp. 1310)

Lee, Kyu-Hwan
 Iwahori--Hecke Algebras of $SL_2$ over $2$-Dimensional Local Fields In this paper we construct an analogue of Iwahori--Hecke algebras of $\operatorname{SL}_2$ over $2$-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori--Matsumoto type relations. Categories:22E50, 20G25

28. CJM 2010 (vol 62 pp. 481)

Casals-Ruiz, Montserrat; Kazachkov, Ilya V.
 Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups The first main result of the paper is a criterion for a partially commutative group $\mathbb G$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb G$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb G$ (of a coordinate group over $\mathbb G$) to the elementary theories of the direct factors of $\mathbb G$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\mathbb H$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb H$ has quantifier elimination and that arbitrary first-order formulas lift from $\mathbb H$ to $\mathbb H\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable. Categories:20F10, 03C10, 20F06

29. CJM 2009 (vol 62 pp. 34)

Campbell, Peter S.; Nevins, Monica
 Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples. Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groupsCategories:20G25, 20G05

30. CJM 2009 (vol 61 pp. 950)

Tange, Rudolf
 Infinitesimal Invariants in a Function Algebra Let $G$ be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let $\g$ be its Lie algebra. First we extend a well-known result about the Picard group of a semi-simple group to reductive groups. Then we prove that if the derived group is simply connected and $\g$ satisfies a mild condition, the algebra $K[G]^\g$ of regular functions on $G$ that are invariant under the action of $\g$ derived from the conjugation action is a unique factorisation domain. Categories:20G15, 13F15

31. CJM 2009 (vol 61 pp. 740)

Caprace, Pierre-Emmanuel; Haglund, Frédéric
 On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if $\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0) realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings. Keywords:Coxeter group, flat rank, $\cat0$ space, buildingCategories:20F55, 51F15, 53C23, 20E42, 51E24

32. CJM 2009 (vol 61 pp. 691)

Yu, Xiaoxiang
 Prehomogeneity on Quasi-Split Classical Groups and Poles of Intertwining Operators Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit, connected, reductive classical group $G$ defined over a non-Archimedean field and $A$ is the standard intertwining operator attached to a tempered representation of $G$ induced from $M$. In this paper we determine all the cases in which $\Lie(N)$ is prehomogeneous under $\Ad(m)$ when $N$ is non-abelian, and give necessary and sufficient conditions for $A$ to have a pole at $0$. Categories:22E50, 20G05

33. CJM 2009 (vol 61 pp. 708)

Zelenyuk, Yevhen
 Regular Homeomorphisms of Finite Order on Countable Spaces We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following. \begin{compactenum}[\rm(a)] \item Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets. \item If $G$ is a countably infinite Abelian group with finitely many elements of order $2$ and $\beta G$ is the Stone--\v Cech compactification of $G$ as a discrete semigroup, then for every idempotent $p\in\beta G\setminus\{0\}$, the subset $\{p,-p\}\subset\beta G$ generates algebraically the free product of one-element semigroups $\{p\}$ and~$\{-p\}$. \end{compactenum} Keywords:Homeomorphism, homogeneous space, topological group, resolvability, Stone-\v Cech compactificationCategories:22A30, 54H11, 20M15, 54A05

34. CJM 2008 (vol 60 pp. 1001)

Cornulier, Yves de; Tessera, Romain; Valette, Alain
 Isometric Group Actions on Hilbert Spaces: Structure of Orbits Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group. Keywords:affine actions, Hilbert spaces, minimal actions, nilpotent groupsCategories:22D10, 43A35, 20F69

35. CJM 2007 (vol 59 pp. 828)

Ortner, Ronald; Woess, Wolfgang
 Non-Backtracking Random Walks and Cogrowth of Graphs Let $X$ be a locally finite, connected graph without vertices of degree $1$. Non-backtracking random walk moves at each step with equal probability to one of the forward'' neighbours of the actual state, \emph{i.e.,} it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of old results concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but \emph{small cycles are dense in} $X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen. Keywords:graph, oriented line grap, covering tree, random walk, cogrowth, amenabilityCategories:05C75, 60G50, 20F69

36. CJM 2007 (vol 59 pp. 449)

 $\SL_n$, Orthogonality Relations and Transfer Let $\pi$ be a square integrable representation of $G'=\SL_n(D)$, with $D$ a central division algebra of finite dimension over a local field $F$ \emph{of non-zero characteristic}. We prove that, on the elliptic set, the character of $\pi$ equals the complex conjugate of the orbital integral of one of the pseudocoefficients of~$\pi$. We prove also the orthogonality relations for characters of square integrable representations of $G'$. We prove the stable transfer of orbital integrals between $\SL_n(F)$ and its inner forms. Category:20G05

37. CJM 2007 (vol 59 pp. 418)

Stoimenow, A.
 On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials It is known that the Brandt--Lickorish--Millett--Ho polynomial $Q$ contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from $Q$ is an open problem. We show that this is not so up to degree $9$. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree $d \le 10$ are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems. Categories:57M25, 57M27, 20F36, 57M50

38. CJM 2007 (vol 59 pp. 296)

Chein, Orin; Goodaire, Edgar G.
 Bol Loops of Nilpotence Class Two Call a non-Moufang Bol loop \emph{minimally non-Moufang} if every proper subloop is Moufang and \emph{minimally nonassociative} if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to $1$. In the process, we derive many commutator and associator identities which hold in such loops. Keywords:Bol loop, Moufang loop, nilpotent, commutator, associator, minimally nonassociativeCategory:20N05

39. CJM 2006 (vol 58 pp. 1144)

Hamana, Masamichi
 Partial $*$-Automorphisms, Normalizers, and Submodules in Monotone Complete $C^*$-Algebras For monotone complete $C^*$-algebras $A\subset B$ with $A$ contained in $B$ as a monotone closed $C^*$-subalgebra, the relation $X = AsA$ gives a bijection between the set of all monotone closed linear subspaces $X$ of $B$ such that $AX + XA \subset X$ and $XX^* + X^*X \subset A$ and a set of certain partial isometries $s$ in the normalizer" of $A$ in $B$, and similarly for the map $s \mapsto \Ad s$ between the latter set and a set of certain partial $*$-automorphisms" of $A$. We introduce natural inverse semigroup structures in the set of such $X$'s and the set of partial $*$-automorphisms of $A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough $B$ the homomorphism becomes surjective and all the partial $*$-automorphisms of $A$ are realized via partial isometries in $B$. In particular, the inverse semigroup associated with a type ${\rm II}_1$ von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the $C^*$-algebra version of these results. Categories:46L05, 46L08, 46L40, 20M18

40. CJM 2006 (vol 58 pp. 897)

Courtès, François
 Distributions invariantes sur les groupes rÃ©ductifs quasi-dÃ©ployÃ©s Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \a l'espace de leurs restrictions \a l'alg\ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis. Keywords:reductive $p$-adic groups, orbital integrals, invariant distributionsCategories:22E35, 20G40

41. CJM 2006 (vol 58 pp. 23)

Dabbaghian-Abdoly, Vahid
 Constructing Representations of Finite Simple Groups and Covers Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. An efficient and simple method to construct representations of finite groups is applicable whenever $G$ has a subgroup $H$ such that $\chi_H$ has a linear constituent with multiplicity $1$. In this paper we show (with a few exceptions) that if $G$ is a simple group or a covering group of a simple group and $\chi$ is an irreducible character of $G$ of degree less than 32, then there exists a subgroup $H$ (often a Sylow subgroup) of $G$ such that $\chi_H$ has a linear constituent with multiplicity $1$. Keywords:group representations, simple groups, central covers, irreducible representationsCategories:20C40, 20C15

42. CJM 2005 (vol 57 pp. 1056)

Ozawa, Narutaka; Rieffel, Marc A.
 Hyperbolic Group $C^*$-Algebras and Free-Product $C^*$-Algebras as Compact Quantum Metric Spaces Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a Dirac'' operator for $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is a word-length function on $G$, then the topology from this metric coincides with the weak-$*$ topology (our definition of a compact quantum metric space''). We show that a convenient framework is that of filtered $C^*$-algebras which satisfy a suitable Haagerup-type'' condition. We also use this framework to prove an analogous fact for certain reduced free products of $C^*$-algebras. Categories:46L87, 20F67, 46L09

43. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
 Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup. Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-typesCategories:20G25, 22E35, 20H25

44. CJM 2005 (vol 57 pp. 416)

Wise, Daniel T.
 Approximating Flats by Periodic Flats in \\CAT(0) Square Complexes We investigate the problem of whether every immersed flat plane in a nonpositively curved square complex is the limit of periodic flat planes. Using a branched cover, we reduce the problem to the case of $\V$-complexes. We solve the problem for malnormal and cyclonormal $\V$-complexes. We also solve the problem for complete square complexes using a different approach. We give an application towards deciding whether the elements of fundamental groups of the spaces we study have commuting powers. We note a connection between the flat approximation problem and subgroup separability. Keywords:CAT(0), periodic flat planesCategories:20F67, 20F06

45. CJM 2004 (vol 56 pp. 945)

Helminck, Aloysius G.; Schwarz, Gerald W.
 Smoothness of Quotients Associated \\With a Pair of Commuting Involutions Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$ and $K:=G^\theta$ be the fixed point groups. We have an action $(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson \cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$. Categories:20G15, 20G20, 22E15, 22E46

46. CJM 2004 (vol 56 pp. 871)

Schocker, Manfred
 Lie Elements and Knuth Relations A coplactic class in the symmetric group $\Sym_n$ consists of all permutations in $\Sym_n$ with a given Schensted $Q$-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of $\Sym_n$ which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras. Keywords:symmetric group, descent set, coplactic relation, Hopf algebra,, convolution productCategories:17B01, 05E10, 20C30, 16W30

47. CJM 2004 (vol 56 pp. 246)

Bonnafé, Cédric
 ÃlÃ©ments unipotents rÃ©guliers des sous-groupes de Levi We investigate the structure of the centralizer of a regular unipotent element of a Levi subgroup of a reductive group. We also investigate the structure of the group of components of this centralizer in relation with the notion of cuspidal local system defined by Lusztig. We determine its unipotent radical, we prove that it admits a Levi complement, and we get some properties on its Weyl group. As an application, we prove some results which were announced in previous paper on regular unipotent elements. Nous \'etudions la structure du centralisateur d'un \'el\'ement unipotent r\'egulier d'un sous-groupe de Levi d'un groupe r\'eductif, ainsi que la structure du groupe des composantes de ce centralisateur en relation avec la notion de syst\`eme local cuspidal d\'efinie par Lusztig. Nous d\'eterminons son radical unipotent, montrons l'existence d'un compl\'ement de Levi et \'etudions la structure de son groupe de Weyl. Comme application, nous d\'emontrons des r\'esultats qui \'etaient annonc\'es dans un pr\'ec\'edent article de l'auteur sur les \'el\'ements unipotents r\'eguliers. Category:20G

48. CJM 2003 (vol 55 pp. 1080)

Kellerhals, Ruth
 Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$. Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40

49. CJM 2003 (vol 55 pp. 750)

Göbel, Rüdiger; Shelah, Saharon; Strüngmann, Lutz
 Almost-Free $E$-Rings of Cardinality $\aleph_1$ An $E$-ring is a unital ring $R$ such that every endomorphism of the underlying abelian group $R^+$ is multiplication by some ring element. The existence of almost-free $E$-rings of cardinality greater than $2^{\aleph_0}$ is undecidable in $\ZFC$. While they exist in G\"odel's universe, they do not exist in other models of set theory. For a regular cardinal $\aleph_1 \leq \lambda \leq 2^{\aleph_0}$ we construct $E$-rings of cardinality $\lambda$ in $\ZFC$ which have $\aleph_1$-free additive structure. For $\lambda=\aleph_1$ we therefore obtain the existence of almost-free $E$-rings of cardinality $\aleph_1$ in $\ZFC$. Keywords:$E$-rings, almost-free modulesCategories:20K20, 20K30, 13B10, 13B25

50. CJM 2003 (vol 55 pp. 822)

Kim, Djun Maximilian; Rolfsen, Dale
 An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements We define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic. By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups. Category:20F36
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