Expand all Collapse all | Results 51 - 71 of 71 |
51. CJM 1999 (vol 51 pp. 1307)
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)
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 polytopes Categories:51M20, 20F55 |
53. CJM 1999 (vol 51 pp. 1226)
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)
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)
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)
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)
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)
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)
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 ramification Categories:11R33, 11S15, 20C32 |
60. CJM 1998 (vol 50 pp. 719)
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 rings Categories:20K20, 20K26, 20K30, 13C10 |
61. CJM 1998 (vol 50 pp. 829)
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)
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 varieties Categories:05E10, 14M99, 20G05, 05E15 |
63. CJM 1998 (vol 50 pp. 401)
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)
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 Conjecture Categories:20C07, 16S34, 16U60, 20E06 |
65. CJM 1998 (vol 50 pp. 167)
Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of the complex reflection groups $G(r,p,n)$ |
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)
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)
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)
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)
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)
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)
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 algebra Categories:20G05, 20C30, 22E10, 22E60 |