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Statistic identifier: St001944

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Collection: Finite Cartan types

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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 $c(a_1,a_2) = \{ (ga_1g^{-1},ga_2g^{-1}) \ | \ g \in G \}.$

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References: [1] (reference broken)  [[mathoverflow 468354]]

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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


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Statistic values:

['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

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Created: Apr 05, 2024 at 11:43 by Martin Rubey

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Last Updated: Apr 05, 2024 at 11:43 by Martin Rubey