This worksheet contains commands to check the various 
computational claims made in the paper "Generalized 
Cluster Complexes and Coxeter Combinatorics" by
Sergey Fomin and Nathan Reading.

The procedures called here are defined in three files called 
compat_maple, f_maple and fake_maple.  All of them use 
the coxeter package written by John Stembridge and some 
use Stembridge's weyl package as well.

First, to check that compatibility restricts to parabolic subgroups:
(Proposition {prop:restrict} (Currently numbered 1.6).
This should take about 2 hours.

read(compat_maple):
for R in [H3,H4,F4,E6,E7,E8] do check_res_all(R); od;

Next, to check the correction factors given in Theorems 
{th:product-f} and{th:product-f} (Currently 3.5 and 4.3)

read(f_maple):
for R in [E6,E7,E8,F4,H3,H4] do
	printf("\n");
	print(R,cox_number(R));
	for i from 0 to rank(R) do 
			print(i,correction_f(R,i)); 
	od;
od;

for R in [D8,E6,E7,E8,F4,H3,H4] do
	printf("\n");
	print(R,cox_number(R));
	for i from 0 to rank(R) do 
			print(i,correction_h(R,i)); 
	od;
od;


Finally, to check the computational results claimed for "fake invariants" of 
infinite Coxeter groups.

To get the fake Coxeter number by the three methods, do:

cox_euler(R);

cox_sym(R);

or

cox_recip(R);

where R is the name of a finite Coxeter group legal
for the coxeter package, or one of the names of infinite groups
defined (usually ad hoc) in the file fake_maple.

To get the fake exponents, do 

fake_exp(R,h_proc)

where h_proc is one of cox_euler, cox_sym or cox_recip.

First, checking the complete graph with all labels 3.

cox_euler(k4_333333);  # fails silently because it is asked to solve a tautology.
cox_sym(k4_333333);    # fails with division by zero
cox_recip(k4_333333);  # fails silently because it is asked to solve a tautology.

Next, the 4-cycle with labels 3434.

cox_euler(k4_324423);  # fails silently because it is asked to solve a tautology.
cox_sym(k4_324423);    # fails with division by zero
cox_recip(k4_324423);  # fails silently because it is asked to solve a tautology.

We check a few finite Coxeter groups:

for R in [A3,A4,A5,A6,A7,A8,B3,B4,B5,B6,B7,B8,D4,D5,D6,D7,D8,E6,E7,E8,F4,H3,H4] do 
	print(R,cox_euler(R),cox_sym(R),cox_recip(R),fake_exp(R,cox_euler)); 
od;

The affine groups.  If you want to actually check these to rank12, you can alter the commands in the obvious ways.
It doesn't take terribly long to do these checks, but produces an ugly mass of output when the methods fail.

for R in [tildeA2,tildeA3,tildeA4,tildeA5,tildeA6,tildeA7,tildeA8,tildeA9] do 
	print(R,cox_euler(R),cox_sym(R),cox_recip(R),fake_exp(R,cox_euler)); 
od;
for R in [tildeB3,tildeB4] do print(R,cox_euler(R),fake_exp(R,cox_euler)); od;
for R in [tildeB3,tildeB4] do print(R,cox_sym(R),fake_exp(R,cox_sym)); od;
for R in [tildeB3,tildeB4] do print(R,cox_recip(R),fake_exp(R,cox_recip)); od;
for R in [tildeC2,tildeC3,tildeC4] do print(R,cox_euler(R),fake_exp(R,cox_euler)); od;
for R in [tildeC2,tildeC3,tildeC4] do print(R,cox_sym(R),fake_exp(R,cox_sym)); od;
for R in [tildeC2,tildeC3,tildeC4] do print(R,cox_recip(R),fake_exp(R,cox_recip)); od;
for R in [tildeD4,tildeD5] do print(R,cox_euler(R),fake_exp(R,cox_euler)); od;
for R in [tildeD4,tildeD5] do print(R,cox_sym(R),fake_exp(R,cox_sym)); od;
for R in [tildeD4,tildeD5] do print(R,cox_recip(R),fake_exp(R,cox_recip)); od;
for R in [tildeE6,tildeE7,tildeE8,tildeF4,tildeG2] do print(R,cox_recip(R),fake_exp(R,cox_recip)); od;

Some other groups:

for R in [ring4e4,ring5e4,ring4e5] do print(R,cox_euler(R),cox_sym(R),cox_recip(R),fake_exp(R,cox_euler)); od;
for R in [k4_322424,k4_422325] do print(R,cox_euler(R),cox_sym(R),cox_recip(R),fake_exp(R,cox_euler)); od;