Rank reduction of string C-group representations

Abstract:

We show that a rank reduction technique for string C-group representations first used in [Adv. Math. 228 (2018), pp. 3207–3222] 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 $3\leqslant n\leqslant d$. 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 $\operatorname {Alt}(11)$—the only known group having “rank gaps”—is perhaps more unusual than previously thought.