From: Fred Lunnon <fred.lunnon@gmail.com> The following questions arose as a result of an enquiry raised on the list Geometric_Algebra <geometric_algebra@googlegroups.com> ---
SO(n) denotes the proper isometries of an (n-1)-sphere, embedded in standard fashion in a Cartesian coordinate basis (x_1,...,x_n); R_i(t) denotes rotation through angle t about axis the coline meet of coordinate hyperplanes x_(i-1), x_i . [ Ambiguity in rotation sense need not concern us just now. ] Question (A) : Does the set G = { R_2, ..., R_n } generate SO(n) (irredundantly)? --you forgot to say which t. Question (B) : Given an arbitrary isometry, is its minimum length as a word over G necessarily at most n_C_2 = n(n-1)/2 (sharply)? --first of all, letting R_ij(t) denote rotation in plane generated by i and j coordinates, thru angle t: any member of SO(n) is generated by the R_ij(t)'s, in fact is a word with n_C_2 factors, each an R_ij(t). The t's will all differ in general. The (i,j) will also all differ. Standard Givens rotation matrix factoring/reduction algorithms show this. You seem interested not in general (i,j) but want only to allow j=i+1. So obviously, if we could express R_ab(t) in terms of the R_(a, a+1) we would be done. In fact it obviously suffices to express R_(1,3) in terms of R_(1,2) and R_(2,3). In quaternion terms it suffices to express k in terms of i and j. Well, uh, gee. k = j*i. Proof done? Well, not quite. More precisely, R_12(t) is i^(2*t/pi) in quaternion terms. So we need to express k^p in terms of i^q and j^r for arbitrary real p and some real q and r. But, well, look. Plainly, we can PERMUTE the coordinates arbitrarily using only the R_{a, a+1}(pi). Proof: Bubble sort. Therefore plainly we can permute to move any two coordinates adjacent, then use an R_{a,a+1} to rotate just those two coords, then unpermute. Therefore, the R_{a,a+1} generate the full set of R_{a,b} which in turn via standard Givens algorithms generate the full SO(n). Question (C) : Is there a practical algorithm expressing an isometry, given in the form (say) of a numerical matrix, as a word over G ? --yes.