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51. CJM 1999 (vol 51 pp. 1307)

Johnson, Norman W.; Weiss, Asia Ivić
 Quadratic Integers and Coxeter Groups Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)$-space $\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of $\mbox{H}^{n+1}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\PSL_2 (\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$. Categories:11F06, 20F55, 20G20, 20H10, 22E40

52. CJM 1999 (vol 51 pp. 1240)

Monson, B.; Weiss, A. Ivić
 Realizations of Regular Toroidal Maps We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types $\{3,6\}$ and $\{6,3\}$. Keywords:regular maps, realizations of polytopesCategories:51M20, 20F55

53. CJM 1999 (vol 51 pp. 1226)

McKay, John
 Semi-Affine Coxeter-Dynkin Graphs and $G \subseteq \SU_2(C)$ The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types. Categories:20C99, 05C25, 14B05

54. CJM 1999 (vol 51 pp. 1194)

Lusztig, G.
 Subregular Nilpotent Elements and Bases in $K$-Theory In this paper we describe a canonical basis for the equivariant $K$-theory (with respect to a $\bc^*$-action) of the variety of Borel subalgebras containing a subregular nilpotent element of a simple complex Lie algebra of type $D$ or~$E$. Category:20G99

55. CJM 1999 (vol 51 pp. 1149)

Cohen, A. M.; Cuypers, H.; Sterk, H.
 Linear Groups Generated by Reflection Tori A reflection is an invertible linear transformation of a vector space fixing a given hyperplane, its axis, vectorwise and a given complement to this hyperplane, its center, setwise. A reflection torus is a one-dimensional group generated by all reflections with fixed axis and center. In this paper we classify subgroups of general linear groups (in arbitrary dimension and defined over arbitrary fields) generated by reflection tori. Categories:20Hxx, 20Gxx, 51A50

56. CJM 1999 (vol 51 pp. 881)

Witherspoon, Sarah J.
 The Representation Ring and the Centre of a Hopf Algebra When $H$ is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of $H$ which generalizes the conjugacy class sums of a finite group, and express the class equation of $H$ in terms of this basis. We show that the representation ring and the centre of $H$ are dual character algebras (or signed hypergroups). Categories:16W30, 20N20

57. CJM 1999 (vol 51 pp. 658)

Shumyatsky, Pavel
 Nilpotency of Some Lie Algebras Associated with $p$-Groups Let $L=L_0+L_1$ be a $\mathbb{Z}_2$-graded Lie algebra over a commutative ring with unity in which $2$ is invertible. Suppose that $L_0$ is abelian and $L$ is generated by finitely many homogeneous elements $a_1,\dots,a_k$ such that every commutator in $a_1,\dots,a_k$ is ad-nilpotent. We prove that $L$ is nilpotent. This implies that any periodic residually finite $2'$-group $G$ admitting an involutory automorphism $\phi$ with $C_G(\phi)$ abelian is locally finite. Categories:17B70, 20F50

58. CJM 1998 (vol 50 pp. 1176)

Dobson, Edward
 Isomorphism problem for metacirculant graphs of order a product of distinct primes In this paper, we solve the isomorphism problem for metacirculant graphs of order $pq$ that are not circulant. To solve this problem, we first extend Babai's characterization of the CI-property to non-Cayley vertex-transitive hypergraphs. Additionally, we find a simple characterization of metacirculant Cayley graphs of order $pq$, and exactly determine the full isomorphism classes of circulant graphs of order $pq$. Categories:05, 20

59. CJM 1998 (vol 50 pp. 1007)

Elder, G. Griffith
 Galois module structure of ambiguous ideals in biquadratic extensions Let $N/K$ be a biquadratic extension of algebraic number fields, and $G=\Gal (N/K)$. Under a weak restriction on the ramification filtration associated with each prime of $K$ above $2$, we explicitly describe the $\bZ[G]$-module structure of each ambiguous ideal of $N$. We find under this restriction that in the representation of each ambiguous ideal as a $\bZ[G]$-module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone. For a given group, $G$, define ${\cal S}_G$ to be the set of indecomposable $\bZ[G]$-modules, ${\cal M}$, such that there is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$ is a $\bZ[G]$-module summand of an ambiguous ideal of $N$. Can ${\cal S}_G$ ever be infinite? In this paper we answer this question of Chinburg in the affirmative. Keywords:Galois module structure, wild ramificationCategories:11R33, 11S15, 20C32

60. CJM 1998 (vol 50 pp. 719)

Göbel, Rüdiger; Shelah, Saharon
 Indecomposable almost free modules---the local case Let $R$ be a countable, principal ideal domain which is not a field and $A$ be a countable $R$-algebra which is free as an $R$-module. Then we will construct an $\aleph_1$-free $R$-module $G$ of rank $\aleph_1$ with endomorphism algebra End$_RG = A$. Clearly the result does not hold for fields. Recall that an $R$-module is $\aleph_1$-free if all its countable submodules are free, a condition closely related to Pontryagin's theorem. This result has many consequences, depending on the algebra $A$ in use. For instance, if we choose $A = R$, then clearly $G$ is an indecomposable almost free' module. The existence of such modules was unknown for rings with only finitely many primes like $R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime $p$. The result complements a classical realization theorem of Corner's showing that any such algebra is an endomorphism algebra of some torsion-free, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorial-algebraic techniques related with what we call {\it rigid tree-elements\/} coming from a module generated over a forest of trees. Keywords:indecomposable modules of local rings, $\aleph_1$-free modules of rank $\aleph_1$, realizing rings as endomorphism ringsCategories:20K20, 20K26, 20K30, 13C10

61. CJM 1998 (vol 50 pp. 829)

Putcha, Mohan S.
 Conjugacy classes and nilpotent variety of a reductive monoid We continue in this paper our study of conjugacy classes of a reductive monoid $M$. The main theorems establish a strong connection with the Bruhat-Renner decomposition of $M$. We use our results to decompose the variety $M_{\nil}$ of nilpotent elements of $M$ into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite. Categories:20G99, 20M10, 14M99, 20F55

62. CJM 1998 (vol 50 pp. 525)

Brockman, William; Haiman, Mark
 Nilpotent orbit varieties and the atomic decomposition of the $q$-Kostka polynomials We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here $\mu'$ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer~\cite{Springer76,Springer78}. The famous $q$-Kostka polynomial~$\Klmt(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$. \LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition of~$\Klmt(q)$ as a sum of `atomic'' polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model. Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen~\cite{Mehta&vanderKallen} imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer~\cite{Broer}. This gives a direct-sum decomposition of the ideals yielding the $k[\Cmubar\cap \hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials. Keywords:$q$-Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varietiesCategories:05E10, 14M99, 20G05, 05E15

63. CJM 1998 (vol 50 pp. 401)

Li, Yuanlin
 The hypercentre and the $n$-centre of the unit group of an integral group ring In this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most $2$. We then give a complete characterization of the $n$-centre of that unit group. The $n$-centre of the unit group is either the centre or the second centre (for $n \geq 2$). Categories:16U60, 20C05

64. CJM 1998 (vol 50 pp. 312)

Dokuchaev, Michael A.; Singer, Maria Lucia Sobral
 Units in group rings of free products of prime cyclic groups Let $G$ be a free product of cyclic groups of prime order. The structure of the unit group ${\cal U}(\Q G)$ of the rational group ring $\Q G$ is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups of ${\cal U}(\Q G)$, up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in $\Z G$ is proved. A strong version of the Tits Alternative for ${\cal U}(\Q G)$ is obtained as a corollary of the structural result. Keywords:Free Products, Units in group rings, Zassenhaus ConjectureCategories:20C07, 16S34, 16U60, 20E06

65. CJM 1998 (vol 50 pp. 167)

Halverson, Tom; Ram, Arun
 Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of the complex reflection groups $G(r,p,n)$ Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,p,n)$ were constructed recently in the work of Ariki and Koike~\cite{AK}, Brou\'e and Malle \cite{BM}, and Ariki~\cite{Ari}. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper ~\cite{HR} in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene~\cite{Gre}, who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of $G(r,p,n)$ given by Ariki and Koike~\cite{AK} and Ariki ~\cite{Ari}. Categories:20C05, 05E05

66. CJM 1998 (vol 50 pp. 3)

Amberg, B.; Dickenschied, O.; Sysak, Ya. P.
 Subgroups of the adjoint group of a radical ring It is shown that the adjoint group $R^\circ$ of an arbitrary radical ring $R$ has a series with abelian factors and that its finite subgroups are nilpotent. Moreover, some criteria for subgroups of $R^\circ$ to be locally nilpotent are given. Categories:16N20, 20F19

67. CJM 1997 (vol 49 pp. 788)

Lichtman, A. I.
 Trace functions in the ring of fractions of polycyclic group rings, II We prove the existence of trace functions in the rings of fractions of polycyclic-by-finite group rings or their homomorphic images. In particular a trace function exists in the ring of fractions of $KH$, where $H$ is a polycyclic-by-finite group and $\char K > N$, where $N$ is a constant depending on $H$. Categories:20C07, 16A08, 16A39

68. CJM 1997 (vol 49 pp. 722)

Elder, G. Griffith; Madan, Manohar L.
 Galois module structure of the integers in wildly ramified $C_p\times C_p$ extensions Let $L/K$ be a finite Galois extension of local fields which are finite extensions of $\bQ_p$, the field of $p$-adic numbers. Let $\Gal (L/K)=G$, and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$, respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We determine, explicitly in terms of specific indecomposable $\bZ_p[G]$-modules, the $\bZ_p[G]$-module structure of $\euO_L$ and $\euP_L$, for $L$, a composite of two arithmetically disjoint, ramified cyclic extensions of $K$, one of which is only weakly ramified in the sense of Erez \cite{erez}. Keywords:Galois module structure---integral representation.Categories:11S15, 20C32

69. CJM 1997 (vol 49 pp. 263)

Hamel, A. M.
 Determinantal forms for symplectic and orthogonal Schur functions Symplectic and orthogonal Schur functions can be defined combinatorially in a manner similar to the classical Schur functions. This paper demonstrates that they can also be expressed as determinants. These determinants are generated using planar decompositions of tableaux into strips and the equivalence of these determinants to symplectic or orthogonal Schur functions is established by Gessel-Viennot lattice path techniques. Results for rational (also called {\it composite}) Schur functions are also obtained. Categories:05E05, 05E10, 20C33

70. CJM 1997 (vol 49 pp. 405)

Vulakh, L. Ya.
 On Hurwitz constants for Fuchsian groups Explicit bounds for the Hurwitz constants for general cofinite Fuchsian groups have been found. It is shown that the bounds obtained are exact for the Hecke groups and triangular groups with signature $(0:2,p,q)$. Categories:11J04, 20H10

71. CJM 1997 (vol 49 pp. 133)

Reeder, Mark
 Exterior powers of the adjoint representation Exterior powers of the adjoint representation of a complex semisimple Lie algebra are decomposed into irreducible representations, to varying degrees of satisfaction. Keywords:Lie algebras, adjoint representation, exterior algebraCategories:20G05, 20C30, 22E10, 22E60
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