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); cox_recip(R); or

cox_mg(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.

read(fake_maple):

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. cox_mg(k4_333333); # fails with division by zero

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. cox_mg(k4_324423); # fails with division by zero

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),cox_mg(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),cox_mg(R),fake_exp(R,cox_euler)); od;

for R in [tildeA2,tildeA3,tildeA4,tildeA5,tildeA6,tildeA7,tildeA8,tildeA9] do print(R,mg(R)); 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 [tildeB3,tildeB4] do print(R,cox_mg(R),fake_exp(R,cox_mg)); od;

for R in [tildeB3,tildeB4,tildeB5,tildeB6,tildeB7,tildeB8,tildeB9] do print(R,mg(R)); 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 [tildeC2,tildeC3,tildeC4] do print(R,cox_mg(R),fake_exp(R,cox_mg)); od;

for R in [tildeC2,tildeC3,tildeC4,tildeC5,tildeC6,tildeC7,tildeC8,tildeC9] do print(R,mg(R)); 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 [tildeD4,tildeD5,tildeD6] do print(R,cox_mg(R),fake_exp(R,cox_mg)); od;

for R in [tildeD4,tildeD5,tildeD6,tildeD7,tildeD8,tildeD9] do print(R,mg(R)); od;

for R in [tildeE6,tildeE7,tildeE8,tildeF4,tildeG2] do print(R,cox_euler(R),fake_exp(R,cox_euler)); od;

for R in [tildeE6,tildeE7,tildeE8,tildeF4,tildeG2] do print(R,cox_sym(R),fake_exp(R,cox_sym)); od;

for R in [tildeE6,tildeE7,tildeE8,tildeF4,tildeG2] do print(R,cox_recip(R),fake_exp(R,cox_recip)); od;

for R in [tildeE6,tildeE7,tildeE8,tildeF4,tildeG2] do print(R,cox_mg(R),fake_exp(R,cox_mg)); od;

for R in [tildeE6,tildeE7,tildeE8,tildeF4,tildeG2] do print(R,mg(R)); od;

Some others:

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

for R in [ring4e4,ring5e4,ring4e5] do print(R,mg(R)); od;

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

for R in [k4_322424,k4_422325] do print(R,mg(R)); od;