***************************************************************************** * 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: St001147 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of minuscule dominant weights in the weight lattice of a finite Cartan type. In short, this is the number of simple roots that appear with multiplicity one in the hightest root of the root system. By definition, a weight $\lambda \neq 0$ in the weight lattice is '''dominant''' if $\langle \lambda, \alpha\rangle \geq 0$ for all simple roots $\alpha$ and a dominant weight is '''minuscule''' if $\langle \lambda, \beta\rangle \in \{0,\pm 1\}$ for all roots $\beta$. Since $\langle \lambda, \alpha\rangle \in \{0,1\}$ for simple roots $\alpha$, we have that $\lambda$ is minuscule if and only if it is fundamental and $\langle \lambda, \rho\rangle = 1$ for the unique highest root $\rho$. The number of minuscule dominant weights is one less than the determinant of the Cartan matrix [[St000821]]. They index the nontrivial minuscule representations, see [1]. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Minuscule_representation]] ----------------------------------------------------------------------------- Code: def statistic(ct): rho = RootSystem(ct).root_lattice().highest_root() return tuple(vector(rho)).count(1) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 2 ['B',2] => 1 ['G',2] => 0 ['A',3] => 3 ['B',3] => 1 ['C',3] => 1 ['A',4] => 4 ['B',4] => 1 ['C',4] => 1 ['D',4] => 3 ['F',4] => 0 ['A',5] => 5 ['B',5] => 1 ['C',5] => 1 ['D',5] => 3 ['A',6] => 6 ['B',6] => 1 ['C',6] => 1 ['D',6] => 3 ['E',6] => 2 ['A',7] => 7 ['B',7] => 1 ['C',7] => 1 ['D',7] => 3 ['E',7] => 1 ['A',8] => 8 ['B',8] => 1 ['C',8] => 1 ['D',8] => 3 ['E',8] => 0 ['A',9] => 9 ['B',9] => 1 ['C',9] => 1 ['D',9] => 3 ['A',10] => 10 ['B',10] => 1 ['C',10] => 1 ['D',10] => 3 ----------------------------------------------------------------------------- Created: Apr 19, 2018 at 09:07 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Apr 19, 2018 at 09:48 by Martin Rubey