Products of involutions in orthogonal groups. (English) Zbl 0643.20029
Combinatorics, Proc. Int. Conf. Incidence Geom. Comb. Struct., Passo della Mendola/Italy 1986, Ann. Discrete Math. 37, 231-247 (1988).
[For the entire collection see Zbl 0635.00001.]
Let F be a field of characteristic distinct from 2 and V a finite- dimensional vector space over F endowed with a symmetric bilinear form. This form may be degenerate. The group of all form preserving transformations is the orthogonal group O(V). Several subgroups of O(V) are generated by involutions, e.g. O(V) if V is regular, O \(*(V)=\{\pi \in O(V)|\pi |_{rad V}=1\}\), \(O^{\pm}(V)=\{\pi \in O(V)|\) (det \(\pi)\) \(2=1\}\), O \(+(V)=\{\pi \in O(V)|\) det \(\pi\) \(=1\}\), and \(GL^{\pm}(V)=\{\pi \in GL(V)|\) (det \(\pi)\) \(2=1\}\) if rad V\(=V\). Let \(\pi\) be an element in one of these groups. If \(\pi\) can be written as a product of t but not as a product of t-1 involutions, then t is called the length of \(\pi\). The author gives a survey on length results.
Let F be a field of characteristic distinct from 2 and V a finite- dimensional vector space over F endowed with a symmetric bilinear form. This form may be degenerate. The group of all form preserving transformations is the orthogonal group O(V). Several subgroups of O(V) are generated by involutions, e.g. O(V) if V is regular, O \(*(V)=\{\pi \in O(V)|\pi |_{rad V}=1\}\), \(O^{\pm}(V)=\{\pi \in O(V)|\) (det \(\pi)\) \(2=1\}\), O \(+(V)=\{\pi \in O(V)|\) det \(\pi\) \(=1\}\), and \(GL^{\pm}(V)=\{\pi \in GL(V)|\) (det \(\pi)\) \(2=1\}\) if rad V\(=V\). Let \(\pi\) be an element in one of these groups. If \(\pi\) can be written as a product of t but not as a product of t-1 involutions, then t is called the length of \(\pi\). The author gives a survey on length results.
Reviewer: E.W.Ellers
MSC:
| 20H20 | Other matrix groups over fields |
| 15A23 | Factorization of matrices |
| 51F25 | Orthogonal and unitary groups in metric geometry |
| 20F05 | Generators, relations, and presentations of groups |
