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Identifier
Values
=>
Cc0022;cc-rep
['A',1]=>2 ['A',2]=>3 ['B',2]=>4 ['G',2]=>5 ['A',3]=>5 ['B',3]=>9 ['C',3]=>9 ['A',4]=>7 ['B',4]=>17 ['C',4]=>17 ['D',4]=>8 ['F',4]=>19 ['A',5]=>11 ['B',5]=>31 ['C',5]=>31 ['D',5]=>14 ['A',6]=>15 ['B',6]=>57 ['C',6]=>57 ['D',6]=>23 ['E',6]=>21 ['A',7]=>22 ['B',7]=>98 ['C',7]=>98 ['D',7]=>35 ['E',7]=>41 ['A',8]=>30 ['B',8]=>166 ['C',8]=>166 ['D',8]=>56 ['E',8]=>72 ['C',2]=>4
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Description
The number of types of reflection subgroups of the associated Weyl group.
Let $\mathcal{R} \subseteq W$ be the set of reflections in the Weyl group $W$.
A (possibly empty) subset $X \subseteq \mathcal{R}$ generates a subgroup of $W$ that is again a reflection group of some (not necessarily reduced) finite type. This is the number of all pairwise different types of subgroups of $W$ obtained this way (including type $A_0$).
Code
def statistic(cartanType):
    from sage.graphs.independent_sets import IndependentSets
    W = WeylGroup(cartanType)
    P = [item.reflection_to_root().to_ambient() for item in W.reflections()]
    n = len(P)
    
    # calculate simple generating sets of reflection subgroups and angles between them
    V = list(range(n))
    E = []
    for i in range(n):
        for j in range(i):
            if P[i].inner_product(P[j]) <= 0:
                x = (P[i].inner_product(P[j]))^2
                y = P[i].inner_product(P[i]) * P[j].inner_product(P[j])
                E.append([i, j, x/y])
    G = Graph([V, E], weighted=True)
    C = IndependentSets(G, maximal=False, complement=True)
    
    # count different Cartan types
    Types = []
    for c in C:
        g = G.subgraph(c).canonical_label(edge_labels=True)
        if g not in Types:
            Types.append(g)
    return len(Types)
Created
May 03, 2022 at 14:44 by Dennis Jahn
Updated
May 04, 2022 at 11:29 by Dennis Jahn