Generating long root subgroup geometries of classical groups over finite prime fields
For a point-line geometry \(\Gamma=(P,L)\) with point set \(P\) and line set \(L\) the generating rank, \(gr(\Gamma)\), of \(\Gamma\) is the minimal cardinality of a generating set of \(\Gamma\). If \(\Gamma\) is embeddable, the embedding rank, \(er(\Gamma)\), of \(\Gamma\) is the maximal dimension of a vector space \(V\) for which there exists a projective embedding \(e:P\to PG(V)\) into the projective space \(PG(V)\). One always has \(er(\Gamma)\leq gr(\Gamma)\). In fact, both ranks are equal in many cases, but there are examples where \(er(\Gamma)\) is strictly less than \(gr(\Gamma)\). The author determines the generating ranks of the (long) root subgroup geometries of \(SL(n,{\mathbb F})\), \(n\geq 3\), and the orthogonal groups \(\Omega(k;{\mathbb F})\) of isometries for a non-singular orthogonal space of dimension \(k\) and Witt index \(n\geq 3\) (so that \(k=2n, 2n+1\) or \(2n+2\)), where \({\mathbb F}\) is a finite field of prime order. Points of these geometries are the conjugates of the subgroups associated with the highest root; lines correspond to those conjugates contained in subgroups generated by two root subgroups. Building on the classification of these spaces by \textit{W. M. Kantor} [Trans. Am. Math. Soc. 248, 347-379 (1979; Zbl 0406.20040)] it is shown that the embedding and generating ranks are equal in these cases and that the long root subgeometries have generating ranks \(n^2-1\) and \({k\choose 2}\), respectively. The author works with equivalent geometries which allows him to express points and lines of the long root subgroup geometries in terms of subspaces of a projective space \(PG(V)\) for a suitable vector space \(V\). In case \(SL(n,{\mathbb F})\) it is known that the root subgroup geometry has embedding rank \(n^2-1\). In case of the orthogonal groups \(\Omega(k;{\mathbb F})\) the 2-fold exterior product \(\Lambda^2(V)\) of \(V\) affords an embedding of the geometry so that the embedding and hence generating rank is at least \({k\choose 2}\). It is then shown that the generating rank is at most this lower bound by specifying suitable generating sets in each of the four cases separately. Since an embedding in a projective space \(PG(V)\) with dim \(V=gr(\Gamma)\) is relatively universal, the result provides a simple proof that the embeddings used are relatively universal and that the given dimensions are the universal embedding dimensions. The paper concludes with a brief review of what is known about the generating ranks of other Lie incidence geometries.
scientific article; zbMATH DE number 1266789