// Description of sigma(W) and f_W for the Coxeter group:
Coxeter group: Finitely presented group on 4 generators
Relations
    s * t * s = t * s * t
    s * u = u * s
    s * v = v * s
    t * u * t = u * t * u
    t * v * t = v * t * v
    u * v = v * u
    s^2 = Id($)
    t^2 = Id($)
    u^2 = Id($)
    v^2 = Id($)


// W has order 192.


// 7 elements of W do not lie in \sigma(W).
// They are:

c := [ 
t * v * t * s * u * t * v,
t * u * t * s * v * t * u,
s * u * t * v * t * s * u,
t * s * u * t * v * t * s * u,
s * u * t * v * t * s * u * t,
t * s * u * t * v * t * s * u * t,
s * t * s * u * v * t * s 
 ];


 // The b-basis on the elements not in Sigma(W):
 // (H(w) denotes the KL-basis elt corresponding to w.) 

(1)C(t * v * t) + (1)C(t * v * t * s * u * t * v).
(1)C(t * u * t) + (1)C(t * u * t * s * v * t * u).
(1)C(s * u * v) + (1)C(s * u * t * v * t * s * u).
(1)C(t * s * u * v) + (1)C(t * s * u * t * v * t * s * u).
(1)C(s * u * v * t) + (1)C(s * u * t * v * t * s * u * t).
(1)C(t * s * u * v * t) + (1)C(t * s * u * t * v * t * s * u * t).
(1)C(s * t * s) + (1)C(s * t * s * u * v * t * s).