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Statistic identifier: St000107

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Collection: Finite Cartan types

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Description: The dimension of the representation $V(\Lambda_1)$.

The sizes of $E_6$ and $E_7$ can be seen in [1].

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References: [1]   Jones, B., Schilling, A. Affine structures and a tableau model for $E_6$ crystals [[MathSciNet:2684152]]

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Code:
def statistic(ct):
    if ct.letter in ['A','B','C','D']:
        return crystals.Letters(ct).cardinality()
    elif ct.letter == 'E':
        if ct.rank() == 6:
            B = KirillovReshetikhinCrystal(['E',6,1], 1,1)
            return B.cardinality()
        elif ct.rank() == 7:
            La = C.root_system().weight_lattice().fundamental_weight(1)     
            T = HighestWeightCrystal(La)
            return T.cardinality()
        elif ct.rank() == 8:
            RC = RiggedConfigurations(['E',8,1], [[1,1]])
            return CT.cardinality()
    elif ct.letter == 'F' and ct.rank() == 4:
        RC = RiggedConfigurations(['F',4,1], [[1,1]])
        return RC.cardinality()

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Statistic values:

['A',1]  => 2
['A',2]  => 3
['B',2]  => 5
['G',2]  => 7
['A',3]  => 4
['B',3]  => 7
['C',3]  => 6
['A',4]  => 5
['B',4]  => 9
['C',4]  => 8
['D',4]  => 8
['F',4]  => 52
['A',5]  => 6
['B',5]  => 11
['C',5]  => 10
['D',5]  => 10
['A',6]  => 7
['B',6]  => 13
['C',6]  => 12
['D',6]  => 12
['E',6]  => 27
['A',7]  => 8
['B',7]  => 15
['C',7]  => 14
['D',7]  => 14
['E',7]  => 133
['A',8]  => 9
['B',8]  => 17
['C',8]  => 16
['D',8]  => 16
['C',2]  => 4
['A',9]  => 10
['B',9]  => 19
['C',9]  => 18
['D',9]  => 18
['A',10] => 11
['B',10] => 21
['C',10] => 20
['D',10] => 20

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Created: Jun 14, 2013 at 16:37 by Travis Scrimshaw

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Last Updated: Dec 29, 2016 at 09:20 by Christian Stump