On invariant Rings of Sylow subgroups of finite classical groups

• Main Author: Ferreira, Jorge Nélio Marques
• Publication Date: 2011
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10400.13/177
• topic: Modular invariant theory; Finite classical groups; p-Groups; Invariant fields; Invariant rings; SAGBI Bases; Complete Intersections; .

In this thesis we study the invariant rings for the Sylow p-subgroups of the nite classical groups. We have successfully constructed presentations for the invariant rings for the Sylow p-subgroups of the unitary groups GU(3; Fq2) and GU(4; Fq2 ), the symplectic group Sp(4; Fq) and the orthogonal group O+(4; Fq) with q odd. In all cases, we obtained a minimal generating set which is also a SAGBI basis. Moreover, we computed the relations among the generators and showed that the invariant ring for these groups are a complete intersection. This shows that, even though the invariant rings of the Sylow p-subgroups of the general linear group are polynomial, the same is not true for Sylow p-subgroups of general classical groups. We also constructed the generators for the invariant elds for the Sylow p-subgroups of GU(n; Fq2 ), Sp(2n; Fq), O+(2n; Fq), O-(2n + 2; Fq) and O(2n + 1; Fq), for every n and q. This is an important step in order to obtain the generators and relations for the invariant rings of all these groups.