The Algebra Seminar
FALL 2003
The seminar meets Tuesdays in room LN 2205 at 2:50 p.m. There will be refreshments in the Anderson Reading Room at 4:00.
Organizer: Marcin Mazur
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September 9: Organizational meeting
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September 16 : Dikran Karagueuzian
Title: Kirillov Theory for Finite Groups
Abstract: A fundamental result in the representation theory of
nilpotent Lie groups is the Kirillov Correspondence, which is a
bijection between the irreducible unitary representations of the Lie
group and its coadjoint orbits. (Orbits of the group acting by
conjugation on the dual of the Lie algebra.) We will discuss an
analog of this correspondence for finite 2-groups arising from
nilpotent local algebras. Our main result shows that certain
representations of these groups are not realizable over the real
numbers.
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September 23 : Marcin Mazur
Title: Groups normal in the unit groups of their group rings
Abstract: We investigate groups G and rings R such that G is normal in
the group U of units of the group ring RG, or such that U is of finite
exponent (or torsion) over its center.
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September 30 : Omar Saldarriaga
Title: Orbits, Young Tableaux and Fusion Rules
Abstract: A method of computing fusion coefficients
for Lie algebras of type A_{n-1} on level k was recently developed by
A. Feingold and M. Weiner using orbits of Z_n^k under the permutation action
of S_k on k-tuples.
They got the fusion coefficients only for n = 2 and 3.
In this talk we will extend this method to all n >= 2 and all k >= 1.
First we show a connection between Young Tableaux and S_k-orbits of Z_n^k,
and using Pieri rules we prove that this method works for certain
specific weights
that generate the fusion algebra. Then we show that the orbit method
does not
work in general, but with the help of Schur functions, we give an
iterative method
to reproduce all type A fusion products.
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October 7 : John Cossey, Australian National University.
Title: Abelian groups as inner mapping groups of loops
Abstract: The question of which abelian groups can occur as inner
mapping groups of loops can be translated into the following group
theoretic question. Let G be a group, H a subgroup of G. Two transversals
A, B of H in G are called H-connected if [A, B] < H.
We can then ask for which abelian groups H can we have a group G with a corefree
subgroup isomorphic to H and H-connected transversals A, B with < A, B >=G.
Kepka and Niemenmaa have given a number of restrictions on the
structure of G and H. I will talk about extending their results.
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October 14 : Anni Neumann, University of Tuebingen
Title: Conjugacy of BN-Injectors in Finite Groups
Abstract:
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October 21 : Luise-Charlotte Kappe
Title: On commutators in p-groups
Abstract: For a given prime p, what is the smallest integer n
such that there
exists a group of order p^n in which the commutators are not a subgroup? We
show that n=6 for any odd prime and n=7 for p=2.
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October 28 : Luise-Charlotte Kappe
Title: On commutators in p-groups, part 2.
Abstract: For a given prime p, what is the smallest integer n
such that there
exists a group of order p^n in which the commutators are not a subgroup? We
show that n=6 for any odd prime and n=7 for p=2.
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November 4 : Thomas Zaslavsky
Title: The Associative Law for Groups
Abstract: A group without associativity is called a ``quasigroup''. (Technically,
a quasigroup also has no identity, but that part is not too significant.)
Groups can be used to construct a certain combinatorial structure called
a ``Dowling geometry'' of any dimension, but for quasigroups that
construction is possible only in dimension 2 or less. The reason is that
dimension 3 implies the associative law. This was shown by Kahn and Kung
around 1980 by using the combinatorial structure to construct a
multiplication operation which, in dimension 3, can be shown to be
associative. I will give a new and more natural proof using the axioms
for a group in terms of division -- an axiomatization that is not as well
known as it ought to be.
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Thursday, November 13, 4.30-5.30 : Gaywalee Yamskulna
Title: On certain vertex algebras and their modules associated
with vertex algebroids
Abstract: We study the family of vertex algebras constructed from vertex
algebroids. As the main result, we classify all the graded simple modules
for such vertex algebras and we show that graded simple modules one-to-one
correspond to simple modules for the Lie algebroids associated with the
vertex algebroids.
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November 18 : Alex Feingold
Title: Representations of Infinite Dimensional Lie Algebras,
Power Series Identities and Partitions
Abstract: I will show how the representation theory of the
infinite dimensional Heisenberg and Virasoro Lie algebras is connected to
combinatorics and number theory through power series identities and partitions.
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November 25 : Gabriela Mendoza
Title: On the Power Structure of Finite p-Groups
Abstract: Phillip Hall introduced regular p-groups as groups which have a
power structure similar to abelian ones. We consider various conditions
on the power structure of finite p-groups which are consequences of
regularity and investigate their interdependencies.
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December 2 : Joseph Smith
Title: Groups with two Normalizers
Abstract: The structure of groups with every subgroup normal (Dedekind
groups) are well known. In this talk I will discuss the structure of
groups with exactly two normalizer subgroups.
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December 9 : Gina Baird
Title: Connected Transversals and Multiplication Groups of Loops
Abstract: Two transversals A,B to a subgroup H in G are called
H-connected if [A,B]< H. Connected transversals play an important role in the study of
multiplication groups of loops and are also interesting from a group
theoretical standpoint. We discuss results of Niemenmaa and Kepka in
both areas.
(This is related to a previous seminar talk by John Cossey.)
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