- FindStat

Identifier

St000000: Finite Cartan types ⟶ ℤ

Values

25 entries:

['A',1]=>4
                
['A',2]=>8
                
['B',2]=>22
                
['G',2]=>32
                
['A',3]=>21
                
['B',3]=>84
                
['C',3]=>84
                
['A',4]=>39
                
['B',4]=>325
                
['C',4]=>325
                
['D',4]=>146
                
['F',4]=>441
                
['A',5]=>92
                
['B',5]=>1096
                
['C',5]=>1096
                
['D',5]=>274
                
['A',6]=>170
                
['B',6]=>3632
                
['C',6]=>3632
                
['D',6]=>1216
                
['E',6]=>448
                
['A',7]=>360
                
['B',7]=>11184
                
['C',7]=>11184
                
['D',7]=>2796

Description

The number conjugacy classes of pairs of commuting elements in the Weyl group of given Cartan type.
For any finite group $G$, this statistic is the cardinality of the set
$$ \{ c(a_1,a_2) \ | \ a_1,a_2 \in G \text{ with } a_1a_2 = a_2a_1\}, $$
where $c(a_1,a_2) = \{ (ga_1g^{-1},ga_2g^{-1}) \ | \ g \in G \}.$

References

[1] (reference broken) mathoverflow 468354

Code

def statistic(ct):
    G = WeylGroup(ct)
    r = 0
    for c in G.conjugacy_classes_representatives():
        C = G.centralizer(c)
        r += len(C.conjugacy_classes())
    return r

Created

Apr 05, 2024 at 11:43 by Martin Rubey

Updated

Apr 05, 2024 at 11:43 by Martin Rubey

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