*****************************************************************************
*       www.FindStat.org - The Combinatorial Statistic Finder               *
*                                                                           *
*       Copyright (C) 2019 The FindStatCrew <info@findstat.org>             *
*                                                                           *
*    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: St001678

-----------------------------------------------------------------------------
Collection: Finite Cartan types

-----------------------------------------------------------------------------
Description: The symmetric bilinear form applied to the highest root and the Weyl vector of a finite Cartan type.

The Weyl vector is half the sum of the positive roots, or the sum of the fundamental weights.

-----------------------------------------------------------------------------
References: 

-----------------------------------------------------------------------------
Code:
def statistic(ct):
    # work around https://trac.sagemath.org/ticket/31410
    R = RootSystem(ct)
    P = R.root_space()
    rho = 1/2*sum(P.positive_roots())
    return (P.highest_root()).symmetric_form(rho)


-----------------------------------------------------------------------------
Statistic values:

['A',1] => 1
['A',2] => 2
['B',2] => 4
['G',2] => 9
['A',3] => 3
['B',3] => 8
['C',3] => 6
['A',4] => 4
['B',4] => 12
['C',4] => 8
['D',4] => 5
['F',4] => 16
['A',5] => 5
['B',5] => 16
['C',5] => 10
['D',5] => 7
['A',6] => 6
['B',6] => 20
['C',6] => 12
['D',6] => 9
['E',6] => 11
['A',7] => 7
['B',7] => 24
['C',7] => 14
['D',7] => 11
['E',7] => 17
['A',8] => 8
['B',8] => 28
['C',8] => 16
['D',8] => 13
['E',8] => 29

-----------------------------------------------------------------------------
Created: Feb 06, 2021 at 21:45 by Martin Rubey

-----------------------------------------------------------------------------
Last Updated: Feb 17, 2021 at 12:32 by Martin Rubey