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Identifier
Values
['A',1] => 3
['A',2] => 8
['B',2] => 5
['G',2] => 7
['A',3] => 15
['B',3] => 7
['C',3] => 14
['A',4] => 24
['B',4] => 9
['C',4] => 27
['D',4] => 28
['F',4] => 26
['A',5] => 35
['B',5] => 11
['C',5] => 44
['D',5] => 45
['A',6] => 48
['B',6] => 13
['C',6] => 65
['D',6] => 66
['E',6] => 78
['A',7] => 63
['B',7] => 15
['C',7] => 90
['D',7] => 91
['E',7] => 133
['A',8] => 80
['B',8] => 17
['C',8] => 119
['D',8] => 120
['E',8] => 248
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Description
The dimension of the quasi-minuscule representation of the Lie group of given type.
For every simple type there is a unique quasi-minuscule representation, and the unique dominant short root is its highest weight, see [2].
References
[1] wikipedia:Minuscule representation
[2] van Leeuwen, M. quasi-minuscule representations MathOverflow:129985
Code
def statistic(C):
    n = C.rank()
    T = C.type()
    if T == "A":
        return n^2+2*n # adjoint
    if T == "B":
        return 2*n+1 # vector
    if T == "C":
        return 2*n^2-n-1
    if T == "D":
        return 2*n^2-n # adjoint
    if T == "E":
        if n == 6:
            return 78 # adjoint
        if n == 7:
            return 133 # adjoint
        if n == 8:
            return 248 # adjoint
    if T == "F":
        return 26
    if T == "G": 
        return 7

def statistic_alternative(C):
    n = C.rank()
    T = C.type()
    W = WeylCharacterRing(C)
    for r in W.positive_roots():
        if r.is_dominant() and r.is_short_root():
            return W(r).degree()
    return W.adjoint_representation().degree()
Created
Apr 19, 2018 at 13:25 by Martin Rubey
Updated
Apr 19, 2018 at 14:48 by Martin Rubey