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Algorithmic Methods in Combinatorial Algebra

Torstensson, Anna LU (2003)
Abstract (Swedish)
Popular Abstract in Swedish

Avhandlingen, som består av tre delar, handlar om olika metoder för att undersöka algebraiska begrepp med hjälp av datorberäkningar.



I första delen visar vi att en viss Liealgebra inte, till skillnad från många andra Liealgebror av liknande typ, kan brytas ned i vinkelräta komponenter av en speciellt enkel typ.



Andra delen handlar om vilka polynom som kan bildas genom att i godtyckliga polynom p(y,z) i två variabler y och z substituera y och z med två givna polynom f och g i en variabel x. Vi intresserar oss främst för vilka gradtal polynomen p(f(x),g(x)) kan ha och visar bl a att om gradtalen för f och g saknar gemensamma faktorer så kan endast gradtal på... (More)
Popular Abstract in Swedish

Avhandlingen, som består av tre delar, handlar om olika metoder för att undersöka algebraiska begrepp med hjälp av datorberäkningar.



I första delen visar vi att en viss Liealgebra inte, till skillnad från många andra Liealgebror av liknande typ, kan brytas ned i vinkelräta komponenter av en speciellt enkel typ.



Andra delen handlar om vilka polynom som kan bildas genom att i godtyckliga polynom p(y,z) i två variabler y och z substituera y och z med två givna polynom f och g i en variabel x. Vi intresserar oss främst för vilka gradtal polynomen p(f(x),g(x)) kan ha och visar bl a att om gradtalen för f och g saknar gemensamma faktorer så kan endast gradtal på formen a deg(f)+b deg(g), där a och b är naturliga tal, förekomma.



Tredje delen av avhandlingen handlar om isometrier, dvs längd och vinkelbevarande avbildningar från tredimensionella hyperboliska mångfalder (en generalisering av ytor) till sig själva. Vi söker grupper av isometrier som hör till mångfalder som har maximalt antal isometrier per volymsenhet. Dessa kan beskrivas som kvoter av en speciell ändligt presenterad grupp och med hjälp av datoralgebrasystemet Magma har vi med olika metoder funnit en mängd sådana grupper. Bland annat den av minst ordning och tre oändliga sekvenser av dem. (Less)
Abstract
This thesis consists of a collection of articles all using and/or developing algorithmic methods for the investigation of different algebraic structures.



Part A concerns orthogonal decompositions of simple Lie algebras. The main result of this part is that the symplectic Lie algebra C3 has no orthogonal decomposition of so called monomial type. This was achieved by developing an algorithm for finding all monomial orthogonal decompositions and implementing it in Maple.



In part B we study subalgebras on two generators of the univariate polynomial ring and the semigroups of degrees associated to such subalgebras. The generators f and g of the subalgebra constitute a so called SAGBI basis if and only if... (More)
This thesis consists of a collection of articles all using and/or developing algorithmic methods for the investigation of different algebraic structures.



Part A concerns orthogonal decompositions of simple Lie algebras. The main result of this part is that the symplectic Lie algebra C3 has no orthogonal decomposition of so called monomial type. This was achieved by developing an algorithm for finding all monomial orthogonal decompositions and implementing it in Maple.



In part B we study subalgebras on two generators of the univariate polynomial ring and the semigroups of degrees associated to such subalgebras. The generators f and g of the subalgebra constitute a so called SAGBI basis if and only if the semigroup of degrees is generated by deg(f) and deg(g). We show that this occurs exactly when both the generators are contained in a subalgebra k[h] for some polynomial h of degree equal to the greatest common divisor of the degrees of f and g. In particular this is the case whenever the degrees of f and g are relatively prime. There is an algorithmic test to check if a set polynomials constitute a SAGBI basis and our proof is by showing that the condition of this test is satisfied. We present two ways of proving this. The first one uses the fact that g is integral over k[f] and therefore satisfies a polynomial equation over k[f], while the second one gives this equation explicitly as a resultant related to f and g.



Part C of the thesis is about maximal symmetry groups of hyperbolic three-manifolds. Those are groups of orientation preserving isometries of three-dimensional hyperbolic manifolds that are of maximal order in relation to the volume of the manifold. One can show that maximal symmetry groups are the quotients by normal torsion free subgroups of a certain finitely presented group. We use different computational methods to find such quotients. Our main results are the following: PGL(2,9) is the smallest maximal symmetry group, and for each prime p there is some prime power q=pk such that either PSL(2,q) or PGL(2,q) is a maximal symmetry group, and all but finitely many alternating and symmetric groups are maximal symmetry groups. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Fröberg, Ralf, Stockholm University
organization
publishing date
type
Thesis
publication status
published
subject
keywords
algebraic geometry, field theory, Number Theory, maximal symmetry group, resultant, SAGBI basis, Orthogonal decomposition, algebra, group theory, Talteori, fältteori, algebraisk geometri, gruppteori
pages
132 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Sal C, matematikhuset, Sölvegatan 18, Lund
defense date
2003-06-06 13:15
ISSN
1404-0034
ISBN
91-628-5710-X
language
English
LU publication?
yes
id
0b6c14d2-6cb8-4322-ab18-4190e57bde51 (old id 465933)
date added to LUP
2007-09-27 16:18:25
date last changed
2018-05-29 10:27:59
@phdthesis{0b6c14d2-6cb8-4322-ab18-4190e57bde51,
  abstract     = {This thesis consists of a collection of articles all using and/or developing algorithmic methods for the investigation of different algebraic structures.<br/><br>
<br/><br>
Part A concerns orthogonal decompositions of simple Lie algebras. The main result of this part is that the symplectic Lie algebra C3 has no orthogonal decomposition of so called monomial type. This was achieved by developing an algorithm for finding all monomial orthogonal decompositions and implementing it in Maple.<br/><br>
<br/><br>
In part B we study subalgebras on two generators of the univariate polynomial ring and the semigroups of degrees associated to such subalgebras. The generators f and g of the subalgebra constitute a so called SAGBI basis if and only if the semigroup of degrees is generated by deg(f) and deg(g). We show that this occurs exactly when both the generators are contained in a subalgebra k[h] for some polynomial h of degree equal to the greatest common divisor of the degrees of f and g. In particular this is the case whenever the degrees of f and g are relatively prime. There is an algorithmic test to check if a set polynomials constitute a SAGBI basis and our proof is by showing that the condition of this test is satisfied. We present two ways of proving this. The first one uses the fact that g is integral over k[f] and therefore satisfies a polynomial equation over k[f], while the second one gives this equation explicitly as a resultant related to f and g.<br/><br>
<br/><br>
Part C of the thesis is about maximal symmetry groups of hyperbolic three-manifolds. Those are groups of orientation preserving isometries of three-dimensional hyperbolic manifolds that are of maximal order in relation to the volume of the manifold. One can show that maximal symmetry groups are the quotients by normal torsion free subgroups of a certain finitely presented group. We use different computational methods to find such quotients. Our main results are the following: PGL(2,9) is the smallest maximal symmetry group, and for each prime p there is some prime power q=pk such that either PSL(2,q) or PGL(2,q) is a maximal symmetry group, and all but finitely many alternating and symmetric groups are maximal symmetry groups.},
  author       = {Torstensson, Anna},
  isbn         = {91-628-5710-X},
  issn         = {1404-0034},
  keyword      = {algebraic geometry,field theory,Number Theory,maximal symmetry group,resultant,SAGBI basis,Orthogonal decomposition,algebra,group theory,Talteori,fältteori,algebraisk geometri,gruppteori},
  language     = {eng},
  pages        = {132},
  publisher    = {Centre for Mathematical Sciences, Lund University},
  school       = {Lund University},
  title        = {Algorithmic Methods in Combinatorial Algebra},
  year         = {2003},
}