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On the Loewy structure of the projective indecomposable representations of some stabilizer subgroups of A(8) in characteristic 2
Title
On the Loewy structure of the projective indecomposable representations of some stabilizer subgroups of A(8) in characteristic 2.
Creator
Hindman, Peter Blake, Florida Atlantic University, Klingler, Lee
Abstract/Description
Given a module over a ring for which the Jordan-Holder theorem is valid, the Loewy series is a filtration on the composition factors of the module yielding information on the structure in which they are arranged in the module. We derive subgroups of A8 by considering stabilizers of n-tuples derived from partitions of eight letters, and develop their representation theory over a field of characteristic 2, relying heavily on methods of passing information to groups from their subgroups, with...
Show moreGiven a module over a ring for which the Jordan-Holder theorem is valid, the Loewy series is a filtration on the composition factors of the module yielding information on the structure in which they are arranged in the module. We derive subgroups of A8 by considering stabilizers of n-tuples derived from partitions of eight letters, and develop their representation theory over a field of characteristic 2, relying heavily on methods of passing information to groups from their subgroups, with special attention toward obtaining the Loewy structure of their projective indecomposable representations.
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Date Issued
1993
PURL
http://purl.flvc.org/fcla/dt/14971
Subject Headings
Representations of groups, Projective modules (Algebra), Indecomposable modules
Format
Document (PDF)
Subgroups of bounded Abelian groups
Title
Subgroups of bounded Abelian groups.
Creator
Petroro, Carla., Florida Atlantic University, Schmidmeier, Markus
Abstract/Description
Birkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pn-bounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) --> (A', B') are morphisms f : B --> B' such that f (A) --> A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the Krull-Remak-Schmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities...
Show moreBirkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pn-bounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) --> (A', B') are morphisms f : B --> B' such that f (A) --> A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the Krull-Remak-Schmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities of the indecomposables are determined uniquely. The purpose of this thesis is to describe the decomposition of such pairs, (A, B), explicitly for n = 2 and n = 3. We describe explicitly how an indecomposable pair can possibly embed into a given pair (A, B). This construction gives rise to formulas for the multiplicity of an indecomposable in the direct sum decomposition of the pair (A, B). These decomposition numbers form a full set of relative invariant, as requested by Birkhoff.
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Date Issued
2004
PURL
http://purl.flvc.org/fcla/dt/13118
Subject Headings
Abelian groups, Modules (Algebra), Indecomposable modules, Representations of groups, Algebras, Linear
Format
Document (PDF)
Unique decomposition of direct sums of ideals
Title
Unique decomposition of direct sums of ideals.
Creator
Ay, Basak., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one...
Show moreWe say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one-dimensional reduced commutative Noetherian rings with the UDI property have the UDI property, also. In Chapter 5 we show that the UDI property implies the Krull-Schmidt property for direct sums of torsion-free rank one modules for a reduced local commutative Noetherian one-dimensional ring R.
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Date Issued
2010
PURL
http://purl.flvc.org/FAU/2683133
Subject Headings
Algebraic number theory, Modules (Algebra), Noetherian rings, Commutative rings, Algebra, Abstract
Format
Document (PDF)
Minimal zero-dimensional extensions
Title
Minimal zero-dimensional extensions.
Creator
Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zero-dimensional extensions of general ZPI-rings.
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Date Issued
2009
PURL
http://purl.flvc.org/FAU/210447
Subject Headings
Algebra, Abstract, Noetherian rings, Commutative rings, Modules (Algebra), Algebraic number theory
Format
Document (PDF)
class of rational surfaces with a non-rational singularity explicitly given by a single equation
Title
A class of rational surfaces with a non-rational singularity explicitly given by a single equation.
Creator
Harmon, Drake., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The family of algebraic surfaces X dened by the single equation zn = (y a1x) (y anx)(x 1) over an algebraically closed eld k of characteristic zero, where a1; : : : ; an 2 k are distinct, is studied. It is shown that this is a rational surface with a non-rational singularity at the origin. The ideal class group of the surface is computed. The terms of the Chase-Harrison-Rosenberg seven term exact sequence on the open complement of the ramication locus of X ! A2 are computed; the Brauer group...
Show moreThe family of algebraic surfaces X dened by the single equation zn = (y a1x) (y anx)(x 1) over an algebraically closed eld k of characteristic zero, where a1; : : : ; an 2 k are distinct, is studied. It is shown that this is a rational surface with a non-rational singularity at the origin. The ideal class group of the surface is computed. The terms of the Chase-Harrison-Rosenberg seven term exact sequence on the open complement of the ramication locus of X ! A2 are computed; the Brauer group is also studied in this unramied setting. The analysis is extended to the surface eX obtained by blowing up X at the origin. The interplay between properties of eX , determined in part by the exceptional curve E lying over the origin, and the properties of X is explored. In particular, the implications that these properties have on the Picard group of the surface X are studied.
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Date Issued
2013
PURL
http://purl.flvc.org/fcla/dt/3360782
Subject Headings
Mathematics, Galois modules (Algebra), Class field theory, Algebraic varieties, Integral equations
Format
Document (PDF)