// Description of sigma(W) and f_W for the Coxeter group:
Coxeter group: Finitely presented group on 3 generators
Relations
    s * t * s = t * s * t
    s * u = u * s
    (t * u)^2 = (u * t)^2
    s^2 = Id($)
    t^2 = Id($)
    u^2 = Id($)


// W has order 48.


// f_W is not defined on 1 elements: u := [ 
s * t * s * u * t * s 
 ];

// 20 elements of W do not lie in \sigma(W).
// They are:

c := [ 
u * t * u,
t * u * t,
u * t * s * u,
t * u * t * s,
u * t * s * u * t,
t * s * u * t * s,
s * u * t * u,
t * s * u * t * u,
u * t * s * u * t * u,
(t * s * u)^2,
t * u * t * s * u * t * u,
s * t * u * t,
s * t * s * u * t,
(s * u * t)^2,
s * u * t * s * u * t * u,
t * s * u * t * s * u * t,
t * s * u * t * s * u * t * u,
s * t * u * t * s,
s * t * u * t * s * u * t * u,
s * t * s * u * t * s * u * t 
 ];


 // Values of f_W on the above elements (critical sets):

 p := [ 
[ u * t * u , u  ],
[ t * u * t , t  ],
[ u * t * s * u , s * u  ],
[ t * u * t * s , t * s  ],
[ u * t * s * u * t , s * u * t  ],
[ t * s * u * t * s , s * t * s  ],
[ s * u * t * u , s * u  ],
[ t * s * u * t * u , t * s * u  ],
[ u * t * s * u * t * u , u * t * s * u , s * u * t * u , (t * u)^2 , s * u  ],
[ (t * s * u)^2 , s * t * s * u  ],
[ t * u * t * s * u * t * u , t * u * t * s * u  ],
[ s * t * u * t , s * t  ],
[ s * t * s * u * t , s * t * s  ],
[ (s * u * t)^2 , s * u * t * s  ],
[ s * u * t * s * u * t * u , s * t * u * t * u  ],
[ t * s * u * t * s * u * t , t * s * u * t * s , s * t * s * u * t , s * t * s  ],
[ t * s * u * t * s * u * t * u , (t * u)^2  ],
[ s * t * u * t * s , s * t * s , s  ],
[ s * t * u * t * s * u * t * u , s * t * u * t * s * u  ],
[ s * t * s * u * t * s * u * t , s * t * s * u * t * s  ],
 ];