Research Articles - Julianne Geering Rainbolt
The Gelfand-Graev Representation of
U(3,q), Journal of Algebra, 188 648-685 (1997).
Abstract: In this paper we explicitly calculate the irreducible
representations of the endomorphism algebra of the Gelfand-Graev representation
of the unitary group U(3,q). In addition, we compute the structure constants of
this endomorphism algebra.
The Generalized Gelfand-Graev
Representation of U(3,q), Journal of Algebra, 202 44-71
(1998).
Abstract: There are three distinct generalized Gelfand-Graev
representations of the unitary group U(3,q). One is the regular representation
and one is the usual Gelfand-Graev representation. The third generalized
Gelfand-Graev representation is the one we examine in this paper.
Images of Periodic Linear Groups, Archiv
der Mathematik, 71 97-106 (1998), joint work with Richard E.
Phillips.
Abstract: Let G be a periodic subgroup of GL(n,K) for some field K and
let N be a normal subgroup of G. It is not, in general, true that G/N has a
faithful K-linear representation. The following will show that if G is periodic
and does not contain any normal unipotent subgroups then G/N has a faithful
K-linear representation. In addition, we show that, in this case, the degree of
the representation is bounded by a function of n.
Using
GAP in an Abstract Algebra Class, in Innovations in Teaching Abstract
Algebra, Allen Hibbard and Ellen Maycock editors, Mathematical Association
of America, (2002), 77-83.
Abstract
Algebra with GAP, A manual to be
used with Contemporary Abstract Algebra by Joseph Gallian, joint with
Joseph Gallian. This manual is
published electronically by Houghton Mifflin, January 2002 at
http://college.hmco.com/mathematics/gallian/abstract_algebra/5e/students/gap.html
The
Irreducible Representations of the Heck Algebras Constructed from the
Gelfand-Graev Representations of GL(3,q) and U(3,q), Communications in
Algebra, 30 (9), 4085-4103
(2002).
Extensions of Periodic Linear Groups, to
appear in Communications in Algebra, joint work with Richard Phillips,
Jon Hall and Ulrich Meierfrankenfeld.
Abstract:
A group is called p-linear if it is isomorphic to a subgroup of GL(n,K) for some field K of characteristic p and some integer n. Let H be a normal
subgroup of G and assume that both H and G/H are periodic and
p-linear. In addition, assume that both
H and G/H have finite unipotent
radicals and that the Hirsch-Plotkin radical of H is
cernikov. The main result of this article
is a proof that under these assumptions G is p-linear. An example is provided
showing the result is false if the assumption regarding the Hirsch-Plotkin radical is removed.
The Multiplicity Free Permutation Representations of the Ree
Groups and the Suzuki Groups and their Automorphism Groups, to appear in Communications in Algebra, joint work with Jagat
Sheth.
Abstract:
Let G a simple group of type $\sp 2B_2(q)$ or $\sp 2G_2(q)$, where q is an odd power of 2 or
3, respectively. The main goal of this paper is to determine the multiplicity free permutation
representations of G and A \leq Aut(G)$ where A is a
subgroup containing a copy of G. Let B be a Borel subgroup of G. If
$G={}\sp2B_2(q)$ we show that there is only one non-trivial multiplicity free permutation representation, namely
the representation ofG associated to the action on G/B. If $G={}\sp2G_2(q)$ we show that there are exactly two such non-trivial
representations, namely the representations of G associated to
the action on G/B and the action on
G/M, where M=UC with U the maximal unipotent subgroup of B and C the unique subgroup of
index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B
if $G={}\sp 2B_2(q)$, and containing $M$ if $G={}\sp
2G_2(q)$. The problem of determining the multiplicity free representations of the finite simple groups is
important, for example, in the classification of distance-transitive graphs.
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