- FindStat

Identifier

St001849: Finite Cartan types ⟶ ℤ

Values

=>
                            Cc0022;cc-rep
                            
                            
                            

                        ['A',1]=>1
['A',2]=>4
['B',2]=>7
['G',2]=>16
['A',3]=>10
['B',3]=>22
['C',3]=>22
['A',4]=>20
['B',4]=>50
['C',4]=>50
['D',4]=>28
['F',4]=>110
['A',5]=>35
['B',5]=>95
['C',5]=>95
['D',5]=>60
['A',6]=>56
['B',6]=>161
['C',6]=>161
['D',6]=>110
['E',6]=>156
['A',7]=>84
['B',7]=>252
['C',7]=>252
['D',7]=>182
['E',7]=>399
['A',8]=>120
['B',8]=>372
['C',8]=>372
['D',8]=>280
['E',8]=>1240
                    

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Description

The atomic length of the longest element.
The atomic length of an element $w$ of a Weyl group is the sum of the heights of the inversions of $w$.

References

[1] Chapelier-Laget, N., Gerber, T. Atomic length in Weyl groups arXiv:2211.12359

Code

def atomic_length(pi):
    """
    EXAMPLES::

        sage: l = [atomic_length(SignedPermutations(n).long_element()) for n in range(1,8)]
        sage: l
        sage: fricas.guess(l)[0].sage().factor()
        1/6*(4*n + 3)*(n + 2)*(n + 1)

    """
    W = WeylGroup(pi.parent().coxeter_type())
    w = W.from_reduced_word(pi.reduced_word())
    return sum(a.height() for a in w.inversions(inversion_type="roots"))

def statistic(ct):
    return atomic_length(WeylGroup(ct).long_element())

Created

Nov 23, 2022 at 16:34 by Martin Rubey

Updated

Nov 23, 2022 at 16:34 by Martin Rubey