// Description of sigma(W) and f_W for the Coxeter group: Coxeter group: Finitely presented group on 7 generators Relations s * t * s = t * s * t s * u = u * s s * v = v * s s * w = w * s s * x = x * s s * y = y * s t * u * t = u * t * u t * v = v * t t * w = w * t t * x = x * t t * y = y * t u * v * u = v * u * v u * w = w * u u * x = x * u u * y = y * u v * w * v = w * v * w v * x = x * v v * y = y * v w * x * w = x * w * x w * y = y * w x * y * x = y * x * y s^2 = Id($) t^2 = Id($) u^2 = Id($) v^2 = Id($) w^2 = Id($) x^2 = Id($) y^2 = Id($) // W has order 40320. // 38 elements of W do not lie in \sigma(W). // They are: c := [ u * t * s * w * v * u * t * x * w * v * u * y * x * w, v * u * t * s * w * v * u * t * x * w * v * u * y * x * w, t * s * v * u * t * w * v * u * x * w * v * u * t * s * y * x * w, u * t * s * v * u * t * w * v * u * x * w * v * u * t * s * y * x * w, u * t * s * w * v * u * t * x * w * v * u * y * x * w * v, v * u * t * s * w * v * u * t * x * w * v * u * y * x * w * v, s * t * u * w * v * u * t * s * x * w * v * u * t * y * x * w * v, s * t * u * v * w * v * u * t * s * x * w * v * u * t * y * x * w * v, s * u * t * v * u * w * v * u * t * s * x * y * x * w * v * u, t * s * u * t * v * u * w * v * u * t * s * x * y * x * w * v * u, s * u * t * v * u * w * v * u * t * s * x * w * y * x * w * v * u, t * s * u * t * v * u * w * v * u * t * s * x * w * y * x * w * v * u, t * u * v * u * t * s * w * x * w * v * u * t * y * x * w * v * u, s * t * u * v * u * t * s * w * x * w * v * u * t * y * x * w * v * u, s * t * u * w * v * u * t * s * x * w * v * u * t * y * x * w * v * u, t * u * v * w * v * u * t * s * x * w * v * u * t * y * x * w * v * u, s * t * u * v * w * v * u * t * s * x * w * v * u * t * y * x * w * v * u, s * u * t * v * u * w * x * w * v * u * t * s * y * x * w * v * u, t * s * u * t * v * u * w * x * w * v * u * t * s * y * x * w * v * u, s * u * t * v * u * w * v * x * w * v * u * t * s * y * x * w * v * u, t * s * u * t * v * u * w * v * x * w * v * u * t * s * y * x * w * v * u, s * u * t * v * u * w * v * u * t * s * x * y * x * w * v * u * t, t * s * u * t * v * u * w * v * u * t * s * x * y * x * w * v * u * t, u * t * s * v * u * t * w * v * u * x * w * y * x * w * v * u * t, s * u * t * v * u * w * v * u * t * s * x * w * y * x * w * v * u * t, t * s * u * t * v * u * w * v * u * t * s * x * w * y * x * w * v * u * t, u * t * s * v * u * t * w * v * u * x * w * v * y * x * w * v * u * t, s * u * t * v * u * w * x * w * v * u * t * s * y * x * w * v * u * t, t * s * u * t * v * u * w * x * w * v * u * t * s * y * x * w * v * u * t, s * u * t * v * u * w * v * x * w * v * u * t * s * y * x * w * v * u * t, t * s * u * t * v * u * w * v * x * w * v * u * t * s * y * x * w * v * u * t, u * t * s * v * u * t * w * v * u * x * w * y * x * w * v * u * t * s, t * s * v * u * t * w * v * u * x * w * v * y * x * w * v * u * t * s, u * t * s * v * u * t * w * v * u * x * w * v * y * x * w * v * u * t * s, s * t * u * t * s * v * w * v * u * t * x * w * v * y * x * w * v * u * t * s, s * t * u * v * u * t * s * w * v * u * t * x * w * v * y * x * w * v * u * t * s, s * t * u * t * s * v * w * v * u * t * x * w * v * u * y * x * w * v * u * t * s, s * t * u * v * u * t * s * w * v * u * t * x * w * v * u * y * x * w * v * u * t * s ]; // Values of f_W on the above elements (critical sets): p := [ [ u * t * s * w * v * u * t * x * w * v * u * y * x * w , t * u * t * w * x * w ], [ v * u * t * s * w * v * u * t * x * w * v * u * y * x * w , t * v * u * t * w * x * w ], [ t * s * v * u * t * w * v * u * x * w * v * u * t * s * y * x * w , s * t * s * v * w * v * x * w * v ], [ u * t * s * v * u * t * w * v * u * x * w * v * u * t * s * y * x * w , s * u * t * s * v * w * v * x * w * v ], [ u * t * s * w * v * u * t * x * w * v * u * y * x * w * v , t * u * t * w * x * w * v ], [ v * u * t * s * w * v * u * t * x * w * v * u * y * x * w * v , s * u * v * u * w * v * u * y , t * v * u * t * w * x * w * v ], [ s * t * u * w * v * u * t * s * x * w * v * u * t * y * x * w * v , s * t * s * v * w * v * x * w * v ], [ s * t * u * v * w * v * u * t * s * x * w * v * u * t * y * x * w * v , s * t * s * v * w * v * x * w * v * y ], [ s * u * t * v * u * w * v * u * t * s * x * y * x * w * v * u , s * u * v * u * w * v * u * y ], [ t * s * u * t * v * u * w * v * u * t * s * x * y * x * w * v * u , t * s * u * v * u * w * v * u * y ], [ s * u * t * v * u * w * v * u * t * s * x * w * y * x * w * v * u , s * u * v * u * w * v * u * y * x ], [ t * s * u * t * v * u * w * v * u * t * s * x * w * y * x * w * v * u , t * s * u * v * u * w * v * u * y * x ], [ t * u * v * u * t * s * w * x * w * v * u * t * y * x * w * v * u , t * u * t * v * u * t * x * y * x ], [ s * t * u * v * u * t * s * w * x * w * v * u * t * y * x * w * v * u , s * t * u * t * v * u * t * x * y * x ], [ s * t * u * w * v * u * t * s * x * w * v * u * t * y * x * w * v * u , s * t * s * v * w * v * x * w * v * u ], [ t * u * v * w * v * u * t * s * x * w * v * u * t * y * x * w * v * u , t * u * t * w * v * u * t * x * y * x ], [ s * t * u * v * w * v * u * t * s * x * w * v * u * t * y * x * w * v * u , s * t * s * v * w * v * x * w * v * u * y , s * t * u * t * w * v * u * t * x * y * x ], [ s * u * t * v * u * w * x * w * v * u * t * s * y * x * w * v * u , s * u * v * u * x * w * v * u * y ], [ t * s * u * t * v * u * w * x * w * v * u * t * s * y * x * w * v * u , t * s * u * v * u * x * w * v * u * y ], [ s * u * t * v * u * w * v * x * w * v * u * t * s * y * x * w * v * u , s * u * v * u * x * w * v * u * y * x ], [ t * s * u * t * v * u * w * v * x * w * v * u * t * s * y * x * w * v * u , t * s * u * v * u * x * w * v * u * y * x ], [ s * u * t * v * u * w * v * u * t * s * x * y * x * w * v * u * t , s * u * v * u * w * v * u * t * y ], [ t * s * u * t * v * u * w * v * u * t * s * x * y * x * w * v * u * t , t * s * u * v * u * w * v * u * t * y ], [ u * t * s * v * u * t * w * v * u * x * w * y * x * w * v * u * t , t * u * t * v * u * t * x * y * x ], [ s * u * t * v * u * w * v * u * t * s * x * w * y * x * w * v * u * t , s * u * v * u * w * v * u * t * y * x ], [ t * s * u * t * v * u * w * v * u * t * s * x * w * y * x * w * v * u * t , t * s * u * v * u * w * v * u * t * y * x ], [ u * t * s * v * u * t * w * v * u * x * w * v * y * x * w * v * u * t , t * u * t * v * u * t * x * y * x * w ], [ s * u * t * v * u * w * x * w * v * u * t * s * y * x * w * v * u * t , s * u * v * u * x * w * v * u * t * y ], [ t * s * u * t * v * u * w * x * w * v * u * t * s * y * x * w * v * u * t , t * s * u * v * u * x * w * v * u * t * y ], [ s * u * t * v * u * w * v * x * w * v * u * t * s * y * x * w * v * u * t , s * u * v * u * x * w * v * u * t * y * x ], [ t * s * u * t * v * u * w * v * x * w * v * u * t * s * y * x * w * v * u * t , t * s * u * v * u * x * w * v * u * t * y * x , s * t * s * u * t * s * w * x * w * y * x * w ], [ u * t * s * v * u * t * w * v * u * x * w * y * x * w * v * u * t * s , t * u * t * v * u * t * s * x * y * x ], [ t * s * v * u * t * w * v * u * x * w * v * y * x * w * v * u * t * s , s * t * s * v * w * v * y * x * w * v ], [ u * t * s * v * u * t * w * v * u * x * w * v * y * x * w * v * u * t * s , t * u * t * v * u * t * s * x * y * x * w , s * u * t * s * v * w * v * y * x * w * v ], [ s * t * u * t * s * v * w * v * u * t * x * w * v * y * x * w * v * u * t * s , s * t * s * u * t * s * w * x * w * y * x * w ], [ s * t * u * v * u * t * s * w * v * u * t * x * w * v * y * x * w * v * u * t * s , s * t * s * v * u * t * s * w * x * w * y * x * w ], [ s * t * u * t * s * v * w * v * u * t * x * w * v * u * y * x * w * v * u * t * s , s * t * s * u * t * s * w * x * w * y * x * w * v ], [ s * t * u * v * u * t * s * w * v * u * t * x * w * v * u * y * x * w * v * u * t * s , s * t * s * v * u * t * s * w * x * w * y * x * w * v ] ];