- FindStat

Identifier

St000107: Finite Cartan types ⟶ ℤ

Values

=>
                            Cc0022;cc-rep
                            
                            
                            

                        ['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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Description

The dimension of the representation $V(\Lambda_1)$.
The sizes of $E_6$ and $E_7$ can be seen in [1].

References

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

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()

Created

Jun 14, 2013 at 16:37 by Travis Scrimshaw

Updated

Dec 29, 2016 at 09:20 by Christian Stump