# Algorithmic Methods in Combinatorial Algebra

Research output: ThesisDoctoral Thesis (compilation)

## 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 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.

## Details

Authors
Organisations
Research areas and keywords

• Mathematics

### Keywords

• algebraic geometry, field theory, Number Theory, maximal symmetry group, resultant, SAGBI basis, Orthogonal decomposition, algebra, group theory, Talteori, fältteori, algebraisk geometri, gruppteori
Original language English Doctor Mathematics (Faculty of Sciences) [unknown], [unknown], Supervisor, External person 2003 Jun 6 Centre for Mathematical Sciences, Lund University 91-628-5710-X Published - 2003 Research

## Bibliographic note

Defence details Date: 2003-06-06 Time: 13:15 Place: Sal C, matematikhuset, Sölvegatan 18, Lund External reviewer(s) Name: Fröberg, Ralf Title: [unknown] Affiliation: Stockholm University --- Article: Torstensson A: On the existence of orthogonaldecompositions of the simple Lie algebra of type C_3.Computer Science Journal of Moldova, Vol.8, No.1, pp. 16-41(2000). Article: Torstensson A: Canonical Bases for Subalgebrason two Generators in the Univariate Polynomial Ring. Beiträge zurAlgebra und Geometrie / Contributions to Algebra and Geometry,Vol. 43, No. 2, pp. 565-577 (2002). Article: Torstensson A, Ufnarovski V, Öfverbeck H: On SAGBI Basesand Resultants Proceedings of the Advanced Research Workshop onCommutative Algebra, Singularities and Computer Algebra, (2002). Article: Conder M, Martin G, Torstensson A: MaximalSymmetry Groups of Hyperbolic three-manifolds.