***************************************************************************** * 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: St001156 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The Dynkin index of the Lie algebra of given type. This is the greatest common divisor of the Dynkin indices of the representations of the Lie algebra. It is computed in [2, prop.2.6]. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Dynkin index]] [2] Laszlo, Y., Sorger, C. The line bundles on the moduli of parabolic $G$-bundles over curves and their sections [[MathSciNet:1456243]] ----------------------------------------------------------------------------- Code: def statistic(C): n = C.rank() T = C.type() if T in ["A", "C"]: return 1 if T == "B": if n == 2: return 1 if n >= 3: return 2 if T == "D": if n >= 4: return 2 if T == "E": if n == 6: return 6 if n == 7: return 12 if n == 8: return 60 if T == "F": return 6 if T == "G": return 2 ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 1 ['B',2] => 1 ['G',2] => 2 ['A',3] => 1 ['B',3] => 2 ['C',3] => 1 ['A',4] => 1 ['B',4] => 2 ['C',4] => 1 ['D',4] => 2 ['F',4] => 6 ['A',5] => 1 ['B',5] => 2 ['C',5] => 1 ['D',5] => 2 ['A',6] => 1 ['B',6] => 2 ['C',6] => 1 ['D',6] => 2 ['E',6] => 6 ['A',7] => 1 ['B',7] => 2 ['C',7] => 1 ['D',7] => 2 ['E',7] => 12 ['A',8] => 1 ['B',8] => 2 ['C',8] => 1 ['D',8] => 2 ['E',8] => 60 ['A',9] => 1 ['B',9] => 2 ['C',9] => 1 ['D',9] => 2 ['A',10] => 1 ['B',10] => 2 ['C',10] => 1 ['D',10] => 2 ----------------------------------------------------------------------------- Created: Apr 20, 2018 at 21:31 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Apr 20, 2018 at 21:31 by Martin Rubey