// 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 = v * t (u * v)^2 = (v * u)^2 s^2 = Id($) t^2 = Id($) u^2 = Id($) v^2 = Id($) // W has order 384. // f_W is not defined on 8 elements: u := [ t * u * t * v * u * t, t * s * u * t * s * v * u * t * s, v * u * t * s * v * u * t * v * u * v, s * u * t * v * u * t * s * v * u * v, s * t * s * u * t * s * v * u * t, t * s * u * t * v * u * t * s * v * u * t, s * t * s * u * v * u * t * s, s * t * s * u * t * s * v * u * t * s * v * u * t ]; // 180 elements of W do not lie in \sigma(W). // They are: c := [ v * u * v, u * v * u, v * u * t * v, u * v * u * t, v * u * t * v * u, u * t * v * u * t, t * v * u * v, u * t * v * u * v, v * u * t * v * u * v, (u * t * v)^2, u * v * u * t * v * u * v, t * u * v * u, t * u * t * v * u, (t * v * u)^2, t * v * u * t * v * u * v, u * t * v * u * t * v * u, u * t * v * u * t * v * u * v, t * u * v * u * t, t * u * v * u * t * v * u * v, t * u * t * v * u * t * v * u, s * v * u * v, s * u * v * u, v * u * t * s * v, u * v * u * t * s, v * u * t * s * v * u, u * t * v * u * t * s, t * s * v * u * v, u * t * s * v * u * v, v * u * t * s * v * u * v, u * t * v * u * t * s * v, u * v * u * t * s * v * u * v, t * s * u * v * u, t * u * t * s * v * u, t * v * u * t * s * v * u, t * v * u * t * s * v * u * v, u * t * v * u * t * s * v * u, u * t * v * u * t * s * v * u * v, t * u * v * u * t * s, t * u * v * u * t * s * v * u * v, t * u * t * v * u * t * s * v * u, s * v * u * t * v, s * u * v * u * t, s * v * u * t * s * v, s * u * v * u * t * s, v * u * t * s * v * u * t, u * t * s * v * u * t * s, t * s * v * u * t * v, u * t * s * v * u * t * v, v * u * t * s * v * u * t * v, (u * t * s * v)^2, u * v * u * t * s * v * u * t * v, t * s * u * v * u * t, t * u * t * s * v * u * t, t * v * u * t * s * v * u * t, t * v * u * t * s * v * u * t * v, u * t * v * u * t * s * v * u * t, u * t * v * u * t * s * v * u * t * v, t * s * u * v * u * t * s, t * u * t * s * v * u * t * s, t * u * v * u * t * s * v * u * t * v, t * u * t * v * u * t * s * v * u * t, s * v * u * t * v * u, s * u * t * v * u * t, s * v * u * t * s * v * u, s * u * t * v * u * t * s, (s * v * u * t)^2, s * u * t * s * v * u * t * s, t * s * v * u * t * v * u, u * t * s * v * u * t * v * u, v * u * t * s * v * u * t * v * u, u * t * s * v * u * t * s * v * u, u * v * u * t * s * v * u * t * v * u, t * s * u * t * v * u * t, t * s * u * t * s * v * u * t, t * s * v * u * t * s * v * u * t, t * v * u * t * s * v * u * t * v * u, u * t * s * v * u * t * s * v * u * t, u * t * v * u * t * s * v * u * t * v * u, t * s * u * t * v * u * t * s, t * u * v * u * t * s * v * u * t * v * u, t * u * t * s * v * u * t * s * v * u * t, s * t * v * u * v, s * u * t * v * u * v, s * v * u * t * v * u * v, s * u * t * v * u * t * v, s * u * v * u * t * v * u * v, s * t * s * v * u * v, s * u * t * s * v * u * v, s * v * u * t * s * v * u * v, s * u * t * v * u * t * s * v, s * u * v * u * t * s * v * u * v, s * v * u * t * s * v * u * t * v, s * u * t * s * v * u * t * s * v, t * s * u * t * v * u * v, t * s * v * u * t * v * u * v, t * s * u * t * s * v * u * v, t * s * v * u * t * s * v * u * v, u * t * s * v * u * t * v * u * v, u * t * s * v * u * t * s * v * u * v, u * v * u * t * s * v * u * t * v * u * v, t * s * u * t * v * u * t * v, t * s * u * v * u * t * v * u * v, t * s * u * t * s * v * u * t * v, t * s * v * u * t * s * v * u * t * v, t * u * t * s * v * u * t * v * u * v, u * t * s * v * u * t * s * v * u * t * v, t * s * u * t * v * u * t * s * v, t * s * u * v * u * t * s * v * u * v, t * u * t * s * v * u * t * s * v * u * v, t * u * t * s * v * u * t * s * v * u * t * v, s * t * u * v * u, s * t * u * t * v * u, s * t * v * u * t * v * u, s * t * v * u * t * v * u * v, s * u * t * v * u * t * v * u, s * u * t * v * u * t * v * u * v, s * t * s * u * v * u, s * t * u * t * s * v * u, s * t * v * u * t * s * v * u, s * t * v * u * t * s * v * u * v, s * u * t * v * u * t * s * v * u, s * t * s * u * t * v * u, s * t * s * v * u * t * v * u, s * t * s * u * t * s * v * u, s * t * s * v * u * t * s * v * u, s * u * t * s * v * u * t * v * u, s * v * u * t * s * v * u * t * v * u, s * u * t * s * v * u * t * s * v * u, s * u * v * u * t * s * v * u * t * v * u, s * t * s * v * u * t * v * u * v, s * u * t * s * v * u * t * v * u * v, s * v * u * t * s * v * u * t * v * u * v, s * u * t * s * v * u * t * s * v * u * v, s * u * v * u * t * s * v * u * t * v * u * v, t * s * u * t * v * u * t * v * u, t * s * u * t * v * u * t * v * u * v, t * s * u * t * s * v * u * t * v * u, t * s * v * u * t * s * v * u * t * v * u, t * s * u * t * s * v * u * t * v * u * v, u * t * s * v * u * t * s * v * u * t * v * u, t * s * u * t * v * u * t * s * v * u, t * s * u * t * s * v * u * t * s * v * u, t * s * u * v * u * t * s * v * u * t * v * u, t * u * t * s * v * u * t * s * v * u * t * v * u, s * t * u * v * u * t, s * t * u * v * u * t * v * u * v, s * t * u * t * v * u * t * v * u, s * t * s * u * v * u * t, s * t * u * t * s * v * u * t, s * t * v * u * t * s * v * u * t, s * t * v * u * t * s * v * u * t * v, s * u * t * v * u * t * s * v * u * t, s * t * s * u * t * v * u * t, s * t * v * u * t * s * v * u * t * v * u, s * u * t * s * v * u * t * s * v * u * t, s * t * s * u * v * u * t * v * u * v, s * t * u * t * s * v * u * t * v * u * v, s * u * t * s * v * u * t * s * v * u * t * v, s * t * s * u * t * v * u * t * v * u, s * t * s * u * t * s * v * u * t * v * u, s * t * s * v * u * t * s * v * u * t * v * u, s * u * t * s * v * u * t * s * v * u * t * v * u, t * s * u * t * s * v * u * t * s * v * u * t, t * s * u * t * s * v * u * t * s * v * u * t * v * u, s * t * u * v * u * t * s, s * t * u * v * u * t * s * v * u * v, s * t * u * t * v * u * t * s * v * u, s * t * u * v * u * t * s * v * u * t * v, s * t * u * t * v * u * t * s * v * u * t, s * t * u * v * u * t * s * v * u * t * v * u, s * t * u * t * s * v * u * t * s * v * u * t, s * t * s * u * v * u * t * s * v * u * v, s * t * u * t * s * v * u * t * s * v * u * v, s * t * u * t * s * v * u * t * s * v * u * t * v, s * t * s * u * t * v * u * t * s * v * u, s * t * s * u * t * s * v * u * t * s * v * u, s * t * s * u * v * u * t * s * v * u * t * v * u, s * t * u * t * s * v * u * t * s * v * u * t * v * u, s * t * s * u * t * v * u * t * s * v * u * t, s * t * s * u * t * s * v * u * t * s * v * u * t * v * u ]; // Values of f_W on the above elements (critical sets): p := [ [ v * u * v , v ], [ u * v * u , u ], [ v * u * t * v , t * v ], [ u * v * u * t , u * t ], [ v * u * t * v * u , t * v * u ], [ u * t * v * u * t , t * u * t ], [ t * v * u * v , t * v ], [ u * t * v * u * v , u * t * v ], [ v * u * t * v * u * v , v * u * t * v , t * v * u * v , (u * v)^2 , t * v ], [ (u * t * v)^2 , t * u * t * v ], [ u * v * u * t * v * u * v , u * v * u * t * v ], [ t * u * v * u , t * u ], [ t * u * t * v * u , t * u * t ], [ (t * v * u)^2 , t * v * u * t ], [ t * v * u * t * v * u * v , t * u * v * u * v ], [ u * t * v * u * t * v * u , u * t * v * u * t , t * u * t * v * u , t * u * t ], [ u * t * v * u * t * v * u * v , (u * v)^2 ], [ t * u * v * u * t , t * u * t , t ], [ t * u * v * u * t * v * u * v , t * u * v * u * t * v ], [ t * u * t * v * u * t * v * u , t * u * t * v * u * t ], [ s * v * u * v , s * v ], [ s * u * v * u , s * u ], [ v * u * t * s * v , t * s * v ], [ u * v * u * t * s , u * t * s ], [ v * u * t * s * v * u , t * s * v * u ], [ u * t * v * u * t * s , t * u * t * s ], [ t * s * v * u * v , t * s * v ], [ u * t * s * v * u * v , u * t * s * v ], [ v * u * t * s * v * u * v , s * u * v * u * v , t * s * v , v * u * t * s * v , t * s * v * u * v ], [ u * t * v * u * t * s * v , t * u * t * s * v ], [ u * v * u * t * s * v * u * v , u * v * u * t * s * v ], [ t * s * u * v * u , t * s * u ], [ t * u * t * s * v * u , t * u * t * s ], [ t * v * u * t * s * v * u , t * v * u * t * s ], [ t * v * u * t * s * v * u * v , t * s * u * v * u * v ], [ u * t * v * u * t * s * v * u , t * u * t * s , u * t * v * u * t * s , t * u * t * s * v * u ], [ u * t * v * u * t * s * v * u * v , s * u * v * u * v ], [ t * u * v * u * t * s , t * u * t * s , t * s ], [ t * u * v * u * t * s * v * u * v , t * u * v * u * t * s * v ], [ t * u * t * v * u * t * s * v * u , t * u * t * v * u * t * s ], [ s * v * u * t * v , s * t * v ], [ s * u * v * u * t , s * u * t ], [ s * v * u * t * s * v , s * t * s * v ], [ s * u * v * u * t * s , s * u * t * s ], [ v * u * t * s * v * u * t , t * s * v * u * t ], [ u * t * s * v * u * t * s , t * s * u * t * s ], [ t * s * v * u * t * v , s * t * s * v , t * v ], [ u * t * s * v * u * t * v , u * t * v , s * u * t * s * v ], [ v * u * t * s * v * u * t * v , s * v * u * t * s * v , t * s * v * u * t * v , v * u * t * v , s * u * v * u * t * v , s * t * s * v , t * v ], [ (u * t * s * v)^2 , t * s * u * t * s * v ], [ u * v * u * t * s * v * u * t * v , s * u * v * u * t * s * v ], [ t * s * u * v * u * t , t * s * u * t ], [ t * u * t * s * v * u * t , t * s * u * t * s , t * u * t ], [ t * v * u * t * s * v * u * t , t * s * v * u * t * s , t * v * u * t ], [ t * v * u * t * s * v * u * t * v , t * s * u * v * u * t * v ], [ u * t * v * u * t * s * v * u * t , u * t * v * u * t , t * s * u * t * s , t * u * t , u * t * s * v * u * t * s , t * u * t * s * v * u * t ], [ u * t * v * u * t * s * v * u * t * v , s * u * v * u * t * v ], [ t * s * u * v * u * t * s , t * s * u * t * s , s * t * s ], [ t * u * t * s * v * u * t * s , t * u * t * v * u * t ], [ t * u * v * u * t * s * v * u * t * v , t * s * u * v * u * t * s * v ], [ t * u * t * v * u * t * s * v * u * t , t * u * t * s * v * u * t * s ], [ s * v * u * t * v * u , s * t * v * u ], [ s * u * t * v * u * t , s * t * u * t ], [ s * v * u * t * s * v * u , s * t * s * v * u ], [ s * u * t * v * u * t * s , s * t * u * t * s ], [ (s * v * u * t)^2 , s * t * s * v * u * t ], [ s * u * t * s * v * u * t * s , s * t * s * u * t * s ], [ t * s * v * u * t * v * u , s * t * s * v * u , t * v * u ], [ u * t * s * v * u * t * v * u , s * u * t * s * v * u , u * t * v * u ], [ v * u * t * s * v * u * t * v * u , s * t * s * v * u , v * u * t * v * u , s * u * v * u * v , s * v * u * t * s * v * u , t * s * v * u * t * v * u , t * v * u , s * u * v * u * t * v * u ], [ u * t * s * v * u * t * s * v * u , t * s * u * t * s * v * u ], [ u * v * u * t * s * v * u * t * v * u , s * u * v * u * t * s * v * u ], [ t * s * u * t * v * u * t , s * t * s * u * t , t * u * t ], [ t * s * u * t * s * v * u * t , s * t * s * u * t * s ], [ t * s * v * u * t * s * v * u * t , s * t * s * v * u * t * s ], [ t * v * u * t * s * v * u * t * v * u , t * s * u * v * u * t * v * u ], [ u * t * s * v * u * t * s * v * u * t , s * u * t * s * v * u * t * s , t * s * u * t * s * v * u * t , s * t * s * u * t * s ], [ u * t * v * u * t * s * v * u * t * v * u , s * u * v * u * t * v * u ], [ t * s * u * t * v * u * t * s , t * u * t * s , s * t * s * u * t * s , s * t * s * u ], [ t * u * v * u * t * s * v * u * t * v * u , t * s * u * v * u * t * s * v * u ], [ t * u * t * s * v * u * t * s * v * u * t , t * s * u * t * s * v * u * t * s ], [ s * t * v * u * v , s * t * v ], [ s * u * t * v * u * v , s * u * t * v ], [ s * v * u * t * v * u * v , s * u * v * u * v , s * v * u * t * v , s * t * v * u * v , s * t * v ], [ s * u * t * v * u * t * v , s * t * u * t * v ], [ s * u * v * u * t * v * u * v , s * u * v * u * t * v ], [ s * t * s * v * u * v , s * t * s * v ], [ s * u * t * s * v * u * v , s * u * t * s * v ], [ s * v * u * t * s * v * u * v , s * v * u * t * s * v , s * t * s * v * u * v , s * t * s * v ], [ s * u * t * v * u * t * s * v , s * t * u * t * s * v ], [ s * u * v * u * t * s * v * u * v , s * u * v * u * v , s * u * v * u * t * s * v ], [ s * v * u * t * s * v * u * t * v , s * t * s * v * u * t * v ], [ s * u * t * s * v * u * t * s * v , s * t * s * u * t * s * v ], [ t * s * u * t * v * u * v , t * s * u * t * v ], [ t * s * v * u * t * v * u * v , t * s * v * u * t * v , t * v * u * v , s * t * s * v * u * v , t * s * u * v * u * v , s * t * s * v , t * v ], [ t * s * u * t * s * v * u * v , t * s * u * t * s * v ], [ t * s * v * u * t * s * v * u * v , t * s * v * u * t * s * v ], [ u * t * s * v * u * t * v * u * v , s * u * v * u * v , u * t * v , u * t * s * v * u * t * v , u * t * v * u * v , s * u * t * s * v * u * v , t * u * t * s * v * u * v , s * u * t * s * v ], [ u * t * s * v * u * t * s * v * u * v , t * s * u * t * s * v , (u * t * s * v)^2 , t * s * u * t * s * v * u * v ], [ u * v * u * t * s * v * u * t * v * u * v , s * u * v * u * v ], [ t * s * u * t * v * u * t * v , s * t * s * u * t * v , t * u * t * v ], [ t * s * u * v * u * t * v * u * v , t * s * u * v * u * t * v ], [ t * s * u * t * s * v * u * t * v , s * t * s * u * t * s * v ], [ t * s * v * u * t * s * v * u * t * v , s * t * s * v * u * t * s * v ], [ t * u * t * s * v * u * t * v * u * v , t * u * t * s * v * u * t * v ], [ u * t * s * v * u * t * s * v * u * t * v , s * t * s * u * t * s * v , s * u * t * s * v * u * t * s * v , t * s * u * t * s * v * u * t * v ], [ t * s * u * t * v * u * t * s * v , s * t * s * u * t * s * v , s * t * s * u * v , t * u * t * s * v ], [ t * s * u * v * u * t * s * v * u * v , t * s * u * v * u * v , t * s * u * v * u * t * s * v ], [ t * u * t * s * v * u * t * s * v * u * v , t * u * t * s * v * u * v , t * u * t * s * v * u * t * s * v ], [ t * u * t * s * v * u * t * s * v * u * t * v , t * s * u * t * s * v * u * t * s * v ], [ s * t * u * v * u , s * t * u ], [ s * t * u * t * v * u , s * t * u * t ], [ s * t * v * u * t * v * u , s * t * v * u * t ], [ s * t * v * u * t * v * u * v , s * t * u * v * u * v ], [ s * u * t * v * u * t * v * u , s * t * u * t * v * u , s * t * u * t , s * u * t * v * u * t ], [ s * u * t * v * u * t * v * u * v , s * u * v * u * v ], [ s * t * s * u * v * u , s * t * s * u ], [ s * t * u * t * s * v * u , s * t * u * t * s ], [ s * t * v * u * t * s * v * u , s * t * v * u * t * s ], [ s * t * v * u * t * s * v * u * v , s * u * v * u * v , s * t * s * u * v * u * v ], [ s * u * t * v * u * t * s * v * u , s * t * u * t * s , s * u * v * u * v , s * u * t * v * u * t * s , s * t * u * t * s * v * u ], [ s * t * s * u * t * v * u , s * t * s * u * t ], [ s * t * s * v * u * t * v * u , s * t * s * v * u * t ], [ s * t * s * u * t * s * v * u , s * t * s * u * t * s ], [ s * t * s * v * u * t * s * v * u , s * t * s * v * u * t * s ], [ s * u * t * s * v * u * t * v * u , s * u * t * s * v * u * t ], [ s * v * u * t * s * v * u * t * v * u , (s * v * u * t)^2 , s * t * s * v * u * t * v * u , s * t * s * v * u * t ], [ s * u * t * s * v * u * t * s * v * u , s * u * t * s * v * u * t * s , s * t * s * u * t * s * v * u , s * t * s * u * t * s ], [ s * u * v * u * t * s * v * u * t * v * u , s * u * v * u * t * s * v * u * t ], [ s * t * s * v * u * t * v * u * v , s * t * s * u * v * u * v ], [ s * u * t * s * v * u * t * v * u * v , s * t * u * t * s * v * u * v ], [ s * v * u * t * s * v * u * t * v * u * v , s * u * v * u * v ], [ s * u * t * s * v * u * t * s * v * u * v , s * t * s * u * t * s * v * u * v ], [ s * u * v * u * t * s * v * u * t * v * u * v , s * u * t * v * u * t * s * v * u * v ], [ t * s * u * t * v * u * t * v * u , t * s * u * t * v * u * t , t * u * t * v * u , s * t * s * u * t * v * u , s * t * s * u * t , t * u * t ], [ t * s * u * t * v * u * t * v * u * v , t * s * u * v * u * v ], [ t * s * u * t * s * v * u * t * v * u , t * s * u * t * s * v * u * t , s * t * s * u * t * s * v * u , s * t * s * u * t * s ], [ t * s * v * u * t * s * v * u * t * v * u , t * s * v * u * t * s * v * u * t , s * t * s * v * u * t * s * v * u , s * t * s * v * u * t * s ], [ t * s * u * t * s * v * u 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u * t * s ], [ s * t * s * u * v * u * t * s * v * u * t * v * u , s * t * s * u * v * u * t * s * v * u * t ], [ s * t * u * t * s * v * u * t * s * v * u * t * v * u , s * t * s * u * t * s * v * u * t * s , s * t * u * t * s * v * u * t * s * v * u * t , s * t * s * u * t * s * v * u * t * s * v * u ], [ s * t * s * u * t * v * u * t * s * v * u * t , s * t * s * u * t * s ], [ s * t * s * u * t * s * v * u * t * s * v * u * t * v * u , s * t * s * u * t * s * v * u * t * s * v * u * t ], ];