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Identifier
Values
=>
Cc0022;cc-rep
['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
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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.
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)

Created
Feb 06, 2021 at 21:45 by Martin Rubey
Updated
Feb 17, 2021 at 12:32 by Martin Rubey