// Description of sigma(W) and f_W for the Coxeter group: Coxeter group: Finitely presented group on 5 generators Relations s * t = t * s s * u * s = u * s * u s * v = v * s s * w = w * s t * u * t = u * t * u t * v = v * t t * w = w * t u * v * u = v * u * v u * w = w * u v * w * v = w * v * w s^2 = Id($) t^2 = Id($) u^2 = Id($) v^2 = Id($) w^2 = Id($) // W has order 1920. // f_W is not defined on 1 elements: u := [ u * v * u * w * v * u * s * t * u * v * w ]; // 176 elements of W do not lie in \sigma(W). // They are: c := [ s * u * s * t * v * u * s, t * u * s * t * v * u * t, s * t * u * v * u * s * t, u * s * t * u * v * u * s * t, s * t * u * v * u * s * t * u, u * s * t * u * v * u * s * t * u, u * v * u * s * t * u * v, s * u * s * t * v * u * s * w, t * u * s * t * v * u * t * w, s * t * u * v * u * s * t * w, u * s * t * u * v * u * s * t * w, s * t * u * v * u * s * t * u * w, u * s * t * u * v * u * s * t * u * w, u * v * u * s * t * u * v * w, s * u * s * t * v * u * s * w * v, t * u * s * t * v * u * t * w * v, s * t * u * v * u * s * t * w * v, u * s * t * u * v * u * s * t * w * v, s * t * u * v * u * s * t * u * w * v, u * s * t * u * v * u * s * t * u * w * v, u * v * u * s * t * u * v * w * v, s * u * s * t * v * u * s * w * v * u, t * u * s * t * v * u * t * w * v * u, s * t * u * v * u * s * t * w * v * u, u * s * t * u * v * u * s * t * w * v * u, s * t * u * v * u * s * t * u * w * v * u, u * s * t * u * v * u * s * t * u * w * v * u, u * v * u * s * t * u * v * w * v * u, s * u * s * t * w * v * u * s, s * u * s * t * v * u * s * w * v * u * s, t * u * s * t * v * u * t * w * v * u * s, s * v * u * s * t * w * v * u * s, s * t * u * v * u * s * t * w * v * u * s, u * s * v * u * s * t * w * v * u * s, s * u * s * v * u * s * t * w * v * u * s, t * u * s * v * u * s * t * w * v * u * s, s * t * u * s * v * u * s * t * w * v * u * s, s * u * s * t * v * u * s * t * w * v * u * s, t * u * s * t * v * u * s * t * w * v * u * s, s * t * u * s * t * v * u * s * t * w * v * u * s, u * s * t * u * v * u * s * t * w * v * u * s, t * u * s * t * u * v * u * s * t * w * v * u * s, s * t * u * v * u * s * t * u * w * v * u * s, s * u * s * v * u * s * t * u * w * v * u * s, s * t * u * s * v * u * s * t * u * w * v * u * s, s * u * t * v * u * s * t * u * w * v * u * s, s * u * s * t * v * u * s * t * u * w * v * u * s, t * u * s * t * v * u * s * t * u * w * v * u * s, s * t * u * s * t * v * u * s * t * u * w * v * u * s, u * s * t * u * v * u * s * t * u * w * v * u * s, t * u * s * t * u * v * u * s * t * u * w * v * u * s, u * v * u * s * t * u * v * w * v * u * s, t * u * s * t * v * u * s * t * u * v * w * v * u * s, t * u * s * t * w * v * u * t, s * u * s * t * v * u * s * w * v * u * t, t * u * s * t * v * u * t * w * v * u * t, t * v * u * s * t * w * v * u * t, s * t * u * v * u * s * t * w * v * u * t, u * t * v * u * s * t * w * v * u * t, s * u * t * v * u * s * t * w * v * u * t, t * u * t * v * u * s * t * w * v * u * t, s * t * u * t * v * u * s * t * w * v * u * t, s * u * s * t * v * u * s * t * w * v * u * t, t * u * s * t * v * u * s * t * w * v * u * t, s * t * u * s * t * v * u * s * t * w * v * u * t, u * s * t * u * v * u * s * t * w * v * u * t, s * u * s * t * u * v * u * s * t * w * v * u * t, s * t * u * v * u * s * t * u * w * v * u * t, t * u * s * v * u * s * t * u * w * v * u * t, t * u * t * v * u * s * t * u * w * v * u * t, s * t * u * t * v * u * s * t * u * w * v * u * t, s * u * s * t * v * u * s * t * u * w * v * u * t, t * u * s * t * v * u * s * t * u * w * v * u * t, s * t * u * s * t * v * u * s * t * u * w * v * u * t, u * s * t * u * v * u * s * t * u * w * v * u * t, s * u * s * t * u * v * u * s * t * u * w * v * u * t, u * v * u * s * t * u * v * w * v * u * t, s * u * s * t * v * u * s * t * u * v * w * v * u * t, s * t * u * w * v * u * s * t, u * s * t * u * w * v * u * s * t, s * t * u * v * w * v * u * s * t, s * t * u * s * t * v * w * v * u * s * t, u * s * t * u * v * w * v * u * s * t, s * t * v * u * w * v * u * s * t, u * s * t * v * u * w * v * u * s * t, s * u * s * t * v * u * w * v * u * s * t, t * u * s * t * v * u * w * v * u * s * t, s * t * u * s * t * v * u * w * v * u * s * t, s * t * u * s * t * u * v * u * w * v * u * s * t, s * u * s * t * v * u * s * w * v * u * s * t, t * u * s * t * v * u * t * w * v * u * s * t, s * t * u * v * u * s * t * w * v * u * s * t, s * u * s * t * v * u * s * t * w * v * u * s * t, t * u * s * t * v * u * s * t * w * v * u * s * t, u * s * t * u * v * u * s * t * w * v * u * s * t, v * u * s * t * u * w * v * u * s * t, u * v * u * s * t * u * w * v * u * s * t, s * u * v * u * s * t * u * w * v * u * s * t, t * u * v * u * s * t * u * w * v * u * s * t, s * t * u * v * u * s * t * u * w * v * u * s * t, t * u * s * v * u * s * t * u * w * v * u * s * t, s * u * t * v * u * s * t * u * w * v * u * s * t, s * u * s * t * v * u * s * t * u * w * v * u * s * t, t * u * s * t * v * u * s * t * u * w * v * u * s * t, u * s * t * u * v * u * s * t * u * w * v * u * s * t, v * u * s * t * u * v * w * v * u * s * t, u * v * u * s * t * u * v * w * v * u * s * t, s * t * u * v * u * s * t * u * v * w * v * u * s * t, u * s * t * u * v * u * s * t * u * v * w * v * u * s * t, s * t * u * w * v * u * s * t * u, u * s * t * u * w * v * u * s * t * u, s * t * u * v * w * v * u * s * t * u, u * s * t * u * v * w * v * u * s * t * u, s * t * u * s * t * u * v * w * v * u * s * t * u, s * t * v * u * w * v * u * s * t * u, u * s * t * v * u * w * v * u * s * t * u, s * u * s * t * v * u * w * v * u * s * t * u, t * u * s * t * v * u * w * v * u * s * t * u, s * t * u * s * t * v * u * w * v * u * s * t * u, s * t * u * s * t * u * v * u * w * v * u * s * t * u, s * u * s * t * v * u * s * w * v * u * s * t * u, t * u * s * t * v * u * t * w * v * u * s * t * u, s * t * u * v * u * s * t * w * v * u * s * t * u, u * s * t * u * v * u * s * t * w * v * u * s * t * u, v * u * s * t * u * w * v * u * s * t * u, u * v * u * s * t * u * w * v * u * s * t * u, s * u * v * u * s * t * u * w * v * u * s * t * u, t * u * v * u * s * t * u * w * v * u * s * t * u, s * t * u * v * u * s * t * u * w * v * u * s * t * u, t * u * s * v * u * s * t * u * w * v * u * s * t * u, s * u * t * v * u * s * t * u * w * v * u * s * t * u, u * s * t * u * v * u * s * t * u * w * v * u * s * t * u, v * u * s * t * u * v * w * v * u * s * t * u, s * t * v * u * s * t * u * v * w * v * u * s * t * u, u * v * u * s * t * u * v * w * v * u * s * t * u, s * t * u * v * u * s * t * u * v * w * v * u * s * t * u, u * s * t * v * u * s * t * u * v * w * v * u * s * t * u, u * s * t * u * v * u * s * t * u * v * w * v * u * s * t * u, u * w * v * u * s * t * u * v, s * t * u * v * w * v * u * s * t * u * v, u * s * t * u * v * w * v * u * s * t * u * v, s * t * u * s * t * u * v * w * v * u * s * t * u * v, v * u * w * v * u * s * t * u * v, u * v * u * w * v * u * s * t * u * v, s * u * v * u * w * v * u * s * t * u * v, t * u * v * u * w * v * u * s * t * u * v, s * t * u * v * u * w * v * u * s * t * u * v, u * s * t * u * v * u * w * v * u * s * t * u * v, s * t * u * s * t * u * v * u * w * v * u * s * t * u * v, s * u * s * t * v * u * s * w * v * u * s * t * u * v, t * u * s * t * v * u * t * w * v * u * s * t * u * v, s * t * u * v * u * s * t * w * v * u * s * t * u * v, u * s * t * u * v * u * s * t * w * v * u * s * t * u * v, s * t * u * v * u * s * t * u * w * v * u * s * t * u * v, u * s * t * u * v * u * s * t * u * w * v * u * s * t * u * v, v * u * s * t * u * v * w * v * u * s * t * u * v, s * t * v * u * s * t * u * v * w * v * u * s * t * u * v, u * v * u * s * t * u * v * w * v * u * s * t * u * v, s * t * u * v * u * s * t * u * v * w * v * u * s * t * u * v, u * s * t * v * u * s * t * u * v * w * v * u * s * t * u * v, u * s * t * u * v * u * s * t * u * v * w * v * u * s * t * u * v, u * w * v * u * s * t * u * v * w, v * w * v * u * s * t * u * v * w, u * v * w * v * u * s * t * u * v * w, v * u * w * v * u * s * t * u * v * w, s * u * s * t * v * u * s * w * v * u * s * t * u * v * w, t * u * s * t * v * u * t * w * v * u * s * t * u * v * w, s * t * u * v * u * s * t * w * v * u * s * t * u * v * w, u * s * t * u * v * u * s * t * w * v * u * s * t * u * v * w, s * t * u * v * u * s * t * u * w * v * u * s * t * u * v * w, u * s * t * u * v * u * s * t * u * w * v * u * s * t * u * v * w, u * v * u * s * t * u * v * w * v * u * s * t * u * v * w, s * u * v * u * s * t * u * v * w * v * u * s * t * u * v * w, t * u * v * u * s * t * u * v * w * v * u * s * t * u * v * w, u * s * v * u * s * t * u * v * w * v * u * s * t * u * v * w, u * t * v * u * s * t * u * v * w * v * u * s * t * u * v * w ]; // Values of f_W on the above elements (critical sets): p := [ [ s * u * s * t * v * u * s , s * u * s ], [ t * u * s * t * v * u * t , t * u * t ], [ s * t * u * v * u * s * t , s * t * v ], [ u * s * t * u * v * u * s * t , u * s * t * v ], [ s * t * u * v * u * s * t * u , s * t * v * u ], [ u * s * t * u * v * u * s * t * u , u * s * t * v * u ], [ u * v * u * s * t * u * v , u * v * u ], [ s * u * s * t * v * u * s * w , s * u * s * w ], [ t * u * s * t * v * u * t * w , t * u * t * w ], [ s * t * u * v * u * s * t * w , s * t * v * w ], [ u * s * t * u * v * u * s * t * w , u * s * t * v * w ], [ s * t * u * v * u * s * t * u * w , s * t * v * u * w ], [ u * s * t * u * v * u * s * t * u * w , u * s * t * v * u * w ], [ u * v * u * s * t * u * v * w , u * v * u * w ], [ s * u * s * t * v * u * s * w * v , s * u * s * w * v ], [ t * u * s * t * v * u * t * w * v , t * u * t * w * v ], [ s * t * u * v * u * s * t * w * v , s * t * v * w * v ], [ u * s * t * u * v * u * s * t * w * v , u * s * t * v * w * v ], [ s * t * u * v * u * s * t * u * w * v , s * t * v * u * w * v ], [ u * s * t * u * v * u * s * t * u * w * v , u * s * t * v * u * w * v ], [ u * v * u * s * t * u * v * w * v , u * v * u * w * v ], [ s * u * s * t * v * u * s * w * v * u , s * u * s * w * v * u , s * u * s * v * u * s ], [ t * u * s * t * v * u * t * w * v * u , t * u * t * v * u * t , t * u * t * w * v * u ], [ s * t * u * v * u * s * t * w * v * u , s * t * v * w * v * u ], [ u * s * t * u * v * u * s * t * w * v * u , u * s * t * v * w * v * u ], [ s * t * u * v * u * s * t * u * w * v * u , s * t * v * u * w * v * u ], [ u * s * t * u * v * u * s * t * u * w * v * u , u * s * t * v * u * w * v * u ], [ u * v * u * s * t * u * v * w * v * u , u * v * u * w * v * u ], [ s * u * s * t * w * v * u * s , s * u * s * w ], [ s * u * s * t * v * u * s * w * v * u * s , s * u * s * w * v * u * s ], [ t * u * s * t * v * u * t * w * v * u * s , t * u * t * w * v * u * s , t * u * t * v * u * s * t ], [ s * v * u * s * t * w * v * u * s , s * v * u * s * w ], [ s * t * u * v * u * s * t * w * v * u * s , s * t * v * w * v * u * s ], [ u * s * v * u * s * t * w * v * u * s , s * u * s * v * u * s , u * s * v * u * s * w ], [ s * u * s * v * u * s * t * w * v * u * s , s * u * s * v * u * s * w ], [ t * u * s * v * u * s * t * w * v * u * s , t * u * s * v * u * s * w , s * t * u * s * v * u * s ], [ s * t * u * s * v * u * s * t * w * v * u * s , s * t * u * s * v * u * s * w ], [ s * u * s * t * v * u * s * t * w * v * u * s , s * u * s * v * u * s ], [ t * u * s * t * v * u * s * t * w * v * u * s , t * u * t * v * u * s * t * u , t * u * s * t * u * v * u * s ], [ s * t * u * s * t * v * u * s * t * w * v * u * s , s * t * u * s * v * u * s ], [ u * s * t * u * v * u * s * t * w * v * u * s , u * s * t * v * w * v * u * s ], [ t * u * s * t * u * v * u * s * t * w * v * u * s , t * u * s * t * u * v * u * s * w ], [ s * t * u * v * u * s * t * u * w * v * u * s , s * t * v * u * w * v * u * s ], [ s * u * s * v * u * s * t * u * w * v * u * s , s * u * s * v * u * s ], [ s * t * u * s * v * u * s * t * u * w * v * u * s , s * t * u * s * v * u * s ], [ s * u * t * v * u * s * t * u * w * v * u * s , s * u * s * v * u * s ], [ s * u * s * t * v * u * s * t * u * w * v * u * s , s * u * s * t * u * v * u * s * t * u * v ], [ t * u * s * t * v * u * s * t * u * w * v * u * s , t * u * t * v * u * s * t * u * v ], [ s * t * u * s * t * v * u * s * t * u * w * v * u * s , s * t * u * s * t * u * v * u * s * t * u * v , s * t * u * t * v * u * s * t * u * v ], [ u * s * t * u * v * u * s * t * u * w * v * u * s , u * s * t * v * u * w * v * u * s ], [ t * u * s * t * u * v * u * s * t * u * w * v * u * s , t * u * s * t * u * v * u * s ], [ u * v * u * s * t * u * v * w * v * u * s , u * v * u * w * v * u * s ], [ t * u * s * t * v * u * s * t * u * v * w * v * u * s , t * u * t * v * u * s * t * u * v * w ], [ t * u * s * t * w * v * u * t , t * u * t * w ], [ s * u * s * t * v * u * s * w * v * u * t , s * u * s * w * v * u * t , s * u * s * v * u * s * t ], [ t * u * s * t * v * u * t * w * v * u * t , t * u * t * w * v * u * t ], [ t * v * u * s * t * w * v * u * t , t * v * u * t * w ], [ s * t * u * v * u * s * t * w * v * u * t , s * t * v * w * v * u * t ], [ u * t * v * u * s * t * w * v * u * t , t * u * t * v * u * t , u * t * v * u * t * w ], [ s * u * t * v * u * s * t * w * v * u * t , s * t * u * t * v * u * t , s * u * t * v * u * t * w ], [ t * u * t * v * u * s * t * w * v * u * t , t * u * t * v * u * t * w ], [ s * t * u * t * v * u * s * t * w * v * u * t , s * t * u * t * v * u * t * w ], [ s * u * s * t * v * u * s * t * w * v * u * t , s * u * s * v * u * s * t * u , s * u * s * t * u * v * u * t ], [ t * u * s * t * v * u * s * t * w * v * u * t , t * u * t * v * u * t ], [ s * t * u * s * t * v * u * s * t * w * v * u * t , s * t * u * t * v * u * t ], [ u * s * t * u * v * u * s * t * w * v * u * t , u * s * t * v * w * v * u * t ], [ s * u * s * t * u * v * u * s * t * w * v * u * t , s * u * s * t * u * v * u * t * w ], [ s * t * u * v * u * s * t * u * w * v * u * t , s * t * v * u * w * v * u * t ], [ t * u * s * v * u * s * t * u * w * v * u * t , t * u * t * v * u * t ], [ t * u * t * v * u * s * t * u * w * v * u * t , t * u * t * v * u * t ], [ s * t * u * t * v * u * s * t * u * w * v * u * t , s * t * u * t * v * u * t ], [ s * u * s * t * v * u * s * t * u * w * v * u * t , s * u * s * v * u * s * t * u * v ], [ t * u * s * t * v * u * s * t * u * w * v * u * t , t * u * s * t * u * v * u * s * t * u * v ], [ s * t * u * s * t * v * u * s * t * u * w * v * u * t , s * t * u * s * v * u * s * t * u * v , s * t * u * s * t * u * v * u * s * t * u * v ], [ u * s * t * u * v * u * s * t * u * w * v * u * t , u * s * t * v * u * w * v * u * t ], [ s * u * s * t * u * v * u * s * t * u * w * v * u * t , s * u * s * t * u * v * u * t ], [ u * v * u * s * t * u * v * w * v * u * t , u * v * u * w * v * u * t ], [ s * u * s * t * v * u * s * t * u * v * w * v * u * t , s * u * s * v * u * s * t * u * v * w ], [ s * t * u * w * v * u * s * t , s * t * w * v ], [ u * s * t * u * w * v * u * s * t , u * s * t * w * v ], [ s * t * u * v * w * v * u * s * t , s * t * v * w * v , s * t * w ], [ s * t * u * s * t * v * w * v * u * s * t , s * t * u * s * t * u * w ], [ u * s * t * u * v * w * v * u * s * t , u * s * t * w , u * s * t * v * w * v ], [ s * t * v * u * w * v * u * s * t , s * t * v * w * v ], [ u * s * t * v * u * w * v * u * s * t , u * s * t * v * w * v ], [ s * u * s * t * v * u * w * v * u * s * t , s * u * s * t * v * w * v ], [ t * u * s * t * v * u * w * v * u * s * t , t * u * s * t * v * w * v ], [ s * t * u * s * t * v * u * w * v * u * s * t , s * t * u * s * t * u * w * v , s * t * u * s * t * v * w * v ], [ s * t * u * s * t * u * v * u * w * v * u * s * t , s * t * u * s * t * u * v * w * v ], [ s * u * s * t * v * u * s * w * v * u * s * t , s * u * s * w * v * u * s * t ], [ t * u * s * t * v * u * t * w * v * u * s * t , t * u * t * w * v * u * s * t ], [ s * t * u * v * u * s * t * w * v * u * s * t , s * t * v * w * v * u * s * t , s * t * v * u * s * t * u * w ], [ s * u * s * t * v * u * s * t * w * v * u * s * t , s * u * s * v * u * s * t ], [ t * u * s * t * v * u * s * t * w * v * u * s * t , t * u * t * v * u * s * t ], [ u * s * t * u * v * u * s * t * w * v * u * s * t , u * s * t * v * u * s * t * u * w , u * s * t * v * w * v * u * s * t , s * t * u * s * t * u * w ], [ v * u * s * t * u * w * v * u * s * t , v * u * s * t * w * v ], [ u * v * u * s * t * u * w * v * u * s * t , u * v * u * s * t * w * v ], [ s * u * v * u * s * t * u * w * v * u * s * t , s * u * v * u * s * t * w * v ], [ t * u * v * u * s * t * u * w * v * u * s * t , t * u * v * u * s * t * w * v ], [ s * t * u * v * u * s * t * u * w * v * u * s * t , s * t * v * u * s * t * u * w * v , s * t * v * w * v , s * t * v * u * w * v * u * s * t , s * t * u * v * u * s * t * w * v ], [ t * u * s * v * u * s * t * u * w * v * u * s * t , t * u * t * v * u * s * t ], [ s * u * t * v * u * s * t * u * w * v * u * s * t , s * u * s * v * u * s * t ], [ s * u * s * t * v * u * s * t * u * w * v * u * s * t , s * t * u * s * t * u * v * u * s * t * u * v , s * u * s * t * v * u * s * t * u * v ], [ t * u * s * t * v * u * s * t * u * w * v * u * s * t , t * u * s * t * v * u * s * t * u * v , s * t * u * s * t * u * v * u * s * t * u * v ], [ u * s * t * u * v * u * s * t * u * w * v * u * s * t , s * t * u * s * t * u * w * v , u * s * t * v * u * s * t * u * w * v , u * s * t * v * w * v , u * s * t * v * u * w * v * u * s * t , u * s * t * u * v * u * s * t * w * v ], [ v * u * s * t * u * v * w * v * u * s * t , u * v * u * s * t * w * v , v * u * s * t * w , s * t * v * w * v ], [ u * v * u * s * t * u * v * w * v * u * s * t , u * v * u * w * v * u * s * t ], [ s * t * u * v * u * s * t * u * v * w * v * u * s * t , s * t * v * u * s * t * u * v * w * v ], [ u * s * t * u * v * u * s * t * u * v * w * v * u * s * t , u * s * t * v * u * s * t * u * v * w * v ], [ s * t * u * w * v * u * s * t * u , s * t * w * v * u ], [ u * s * t * u * w * v * u * s * t * u , u * s * t * w * v * u ], [ s * t * u * v * w * v * u * s * t * u , s * t * u * w , s * t * v * w * v * u ], [ u * s * t * u * v * w * v * u * s * t * u , u * s * t * v * w * v * u , u * s * t * u * w , s * t * u * s * t * u * w ], [ s * t * u * s * t * u * v * w * v * u * s * t * u , s * t * u * s * t * u * w ], [ s * t * v * u * w * v * u * s * t * u , s * t * v * w * v * u ], [ u * s * t * v * u * w * v * u * s * t * u , u * s * t * v * w * v * u ], [ s * u * s * t * v * u * w * v * u * s * t * u , s * u * s * t * v * w * v * u ], [ t * u * s * t * v * u * w * v * u * s * t * u , t * u * s * t * v * w * v * u ], [ s * t * u * s * t * v * u * w * v * u * s * t * u , s * t * u * s * t * v * w * v * u , s * t * u * s * t * u * w * v * u , s * t * u * s * t * u * w ], [ s * t * u * s * t * u * v * u * w * v * u * s * t * u , s * t * u * s * t * u * v * w * v * u ], [ s * u * s * t * v * u * s * w * v * u * s * t * u , s * u * s * w * v * u * s * t * u ], [ t * u * s * t * v * u * t * w * v * u * s * t * u 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