by wendy » Fri Dec 20, 2013 7:49 am
I've had several bashes at this. It's quite messy.
In essence, you have to evaluate each flag separately, where that section gives an orthoscheme. The process amounts to something like this.
a_1 = p_0 - p_1, a_2 = p_1-p_2 etc right down to p_n. Then you multiply up all of the a_n, to get v_1. You proceed through each of the flags and add the volumes together. Then you multiply by G/n! . 2^n, to get the final volume. The stott-matrix can be used for the dot-matrix product, ie A dot B = S_ij A_i B_j. It's a sheer system, so addition and subtraction works as normal.
The next trick is to evaluate the flags. You do this from the dynkin symbol, supposing that 0 is the vertex, and the remainder of the nodes are 1-n. A path is any connected set, including connections to the vertex-node.
The flags of x3o5x are (1x)3(2o)5(3x), it has four in number. I have not figured out yet (ie it is not bleedingly obvious), on how one might do this from the dynkin symbol, but it will come.