Rank reduction of string C-group representations

Abstract

We show that a rank reduction technique for string C-group representations first used in [3] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on d-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks 3 n d. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group Alt(11)-the only known group having 'rank gaps'-is perhaps more unusual than previously thought.

String C-group representations of almost simple groups: a survey

“…Rank reduction. The proof of Theorem 4.9 has inspired Peter Brooksbank and the author to prove a rank reduction theorem [4].…”

String C-group representations of alternating groups

“…The techniques we developed in this paper inspired Brooksbank and the second author to develop a general rank reduction technique, now available in [4].…”

“…cycles(1,2,3) and(4,5,6) (the vertices of the above graph on the right). Let l be the least integer such that (ρ 0 ρ 1 ) l fixes all the vertices of the component of the graph on the bottom.…”

On the ranks of string C-group representations for symplectic and orthogonal groups

“…That construction may be adapted in odd characteristic to generate O(5, F q ) as a string C-group of rank 5, but we must work harder to obtain such a representation for its simple subgroup Ω(5, F q ) of index 4. Having done so, we apply a recently discovered technique [BL2] to reduce the rank of this representation and obtain the desired constructions of ranks 3 and 4.…”

“…This 'Petrie like' construction was developed in [HL] and first used as a rank reduction technique for the symmetric groups in [FL1]. The general technique, along with the criteria in Theorem 4.2, was established in the recent paper [BL2].…”