51. CJM 2000 (vol 52 pp. 265)
 Brion, Michel; Helminck, Aloysius G.

On Orbit Closures of Symmetric Subgroups in Flag Varieties
We study $K$orbits in $G/P$ where $G$ is a complex connected
reductive group, $P \subseteq G$ is a parabolic subgroup, and $K
\subseteq G$ is the fixed point subgroup of an involutive
automorphism $\theta$. Generalizing work of Springer, we
parametrize the (finite) orbit set $K \setminus G \slash P$ and we
determine the isotropy groups. As a consequence, we describe the
closed (resp. affine) orbits in terms of $\theta$stable
(resp. $\theta$split) parabolic subgroups. We also describe the
decomposition of any $(K,P)$double coset in $G$ into
$(K,B)$double cosets, where $B \subseteq P$ is a Borel subgroup.
Finally, for certain $K$orbit closures $X \subseteq G/B$, and for
any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero
global sections, we show that the restriction map $\res_X \colon
H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and
that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we
describe the $K$module $H^0 (X, \mathcal{L})$. This gives
information on the restriction to $K$ of the simple $G$module $H^0
(G/B, \mathcal{L})$. Our construction is a geometric analogue of
Vogan and Sepanski's approach to extremal $K$types.
Keywords:flag variety, symmetric subgroup Categories:14M15, 20G05 

52. CJM 2000 (vol 52 pp. 438)
 Wallach, N. R.; Willenbring, J.

On Some $q$Analogs of a Theorem of KostantRallis
In the first part of this paper generalizations of Hesselink's
$q$analog of Kostant's multiplicity formula for the action of a
semisimple Lie group on the polynomials on its Lie algebra are given
in the context of the KostantRallis theorem. They correspond to the
cases of real semisimple Lie groups with one conjugacy class of Cartan
subgroup. In the second part of the paper a $q$analog of the
KostantRallis theorem is given for the real group $\SL(4,\mathbb{R})$
(that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This
example plays two roles. First it contrasts with the examples of the
first part. Second it has implications to the study of entanglement
of mixed 2 qubit states in quantum computation.
Categories:22E47, 20G05 

53. CJM 2000 (vol 52 pp. 197)
 Radjavi, Heydar

Sublinearity and Other Spectral Conditions on a Semigroup
Subadditivity, sublinearity, submultiplicativity, and other
conditions are considered for spectra of pairs of operators on a
Hilbert space. Sublinearity, for example, is a weakening of the
wellknown property~$L$ and means $\sigma(A+\lambda B) \subseteq
\sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The
effect of these conditions is examined on commutativity,
reducibility, and triangularizability of multiplicative semigroups
of operators. A sample result is that sublinearity of spectra
implies simultaneous triangularizability for a semigroup of compact
operators.
Categories:47A15, 47D03, 15A30, 20A20, 47A10, 47B10 

54. CJM 1999 (vol 51 pp. 1175)
 Lehrer, G. I.; Springer, T. A.

Reflection Subquotients of Unitary Reflection Groups
Let $G$ be a finite group generated by (pseudo) reflections in a
complex vector space and let $g$ be any linear transformation which
normalises $G$. In an earlier paper, the authors showed how to
associate with any maximal eigenspace of an element of the coset
$gG$, a subquotient of $G$ which acts as a reflection group on the
eigenspace. In this work, we address the questions of
irreducibility and the coexponents of this subquotient, as well as
centralisers in $G$ of certain elements of the coset. A criterion
is also given in terms of the invariant degrees of $G$ for an
integer to be regular for $G$. A key tool is the investigation of
extensions of invariant vector fields on the eigenspace, which
leads to some results and questions concerning the geometry of
intersections of invariant hypersurfaces.
Categories:51F15, 20H15, 20G40, 20F55, 14C17 

55. 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 3space. 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 

56. CJM 1999 (vol 51 pp. 1240)
57. CJM 1999 (vol 51 pp. 1226)
58. 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 

59. 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 onedimensional 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 

60. 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 

61. 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 adnilpotent. 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 

62. 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 CIproperty to
nonCayley vertextransitive 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 

63. 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 ramification Categories:11R33, 11S15, 20C32 

64. CJM 1998 (vol 50 pp. 719)
 Göbel, Rüdiger; Shelah, Saharon

Indecomposable almost free modulesthe 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 torsionfree,
reduced $R$module $G$ of countable rank. Its proof is based on new
combinatorialalgebraic techniques related with what we call {\it rigid
treeelements\/} 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 

65. 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 BruhatRenner 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 

66. 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
schemetheoretic
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
nonnegative integer coefficients, and Lascoux proposed a
corresponding decomposition in the cohomology model.
Our work provides a geometric interpretation of the atomic
decomposition. The Frobeniussplitting results of Mehta and van der
Kallen~\cite{Mehta&vanderKallen} imply a directsum 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 directsum
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 

67. CJM 1998 (vol 50 pp. 401)
68. 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 Conjecture Categories:20C07, 16S34, 16U60, 20E06 

69. CJM 1998 (vol 50 pp. 167)
 Halverson, Tom; Ram, Arun

MurnaghanNakayama rules for characters of IwahoriHecke algebras of the complex reflection groups $G(r,p,n)$
IwahoriHecke 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
MurnaghanNakayama 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
MurnaghanNakayama rules for the characters of the
IwahoriHecke 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 MurnaghanNakayama formula for irreducible symmetric
group characters by summing diagonal matrix entries in Young's
seminormal representations. We use the analogous representations
of the IwahoriHecke algebra of $G(r,p,n)$ given by Ariki and
Koike~\cite{AK} and Ariki ~\cite{Ari}.
Categories:20C05, 05E05 

70. CJM 1998 (vol 50 pp. 3)
71. 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
polycyclicbyfinite group rings or their homomorphic images. In
particular a trace function exists in the ring of fractions of $KH$,
where $H$ is a polycyclicbyfinite group and $\char K > N$, where
$N$ is a constant depending on $H$.
Categories:20C07, 16A08, 16A39 

72. 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 structureintegral representation. Categories:11S15, 20C32 

73. 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 GesselViennot lattice path
techniques. Results for rational (also called {\it composite}) Schur functions
are also obtained.
Categories:05E05, 05E10, 20C33 

74. 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 

75. 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 algebra Categories:20G05, 20C30, 22E10, 22E60 
