***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000854 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of orbits of reflections of a finite Cartan type. Let $W$ be the Weyl group of a Cartan type. The reflections in $W$ are closed under conjugation, and this statistic counts the number of conjugacy classes of $W$ that are reflections. It is well-known that there are either one or two such conjugacy classes. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(cartan_type): W = ReflectionGroup(cartan_type) return sum( 1 for w in W.conjugacy_classes_representatives() if w.is_reflection() ) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 1 ['B',2] => 2 ['G',2] => 2 ['A',3] => 1 ['B',3] => 2 ['C',3] => 2 ['A',4] => 1 ['B',4] => 2 ['C',4] => 2 ['D',4] => 1 ['F',4] => 2 ['A',5] => 1 ['B',5] => 2 ['C',5] => 2 ['D',5] => 1 ['A',6] => 1 ['B',6] => 2 ['C',6] => 2 ['D',6] => 1 ['E',6] => 1 ['A',7] => 1 ['B',7] => 2 ['C',7] => 2 ['D',7] => 1 ['E',7] => 1 ['A',8] => 1 ['B',8] => 2 ['C',8] => 2 ['D',8] => 1 ['E',8] => 1 ----------------------------------------------------------------------------- Created: Jun 25, 2017 at 10:15 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 25, 2017 at 10:15 by Christian Stump