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Identifier
Values
=>
Cc0022;cc-rep
['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