- FindStat

Identifier

St001678: Finite Cartan types ⟶ ℤ

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
                    

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

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