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Identifier
Values
=>
Cc0022;cc-rep
['A',1]=>2 ['A',2]=>5 ['B',2]=>8 ['G',2]=>13 ['A',3]=>15 ['B',3]=>38 ['C',3]=>38 ['A',4]=>52 ['B',4]=>218 ['C',4]=>218 ['D',4]=>75 ['F',4]=>637 ['A',5]=>203 ['B',5]=>1430 ['C',5]=>1430 ['D',5]=>428 ['A',6]=>877 ['B',6]=>10514 ['C',6]=>10514 ['D',6]=>2781 ['E',6]=>5079 ['A',7]=>4140 ['B',7]=>85202 ['C',7]=>85202 ['D',7]=>20093 ['E',7]=>107911 ['A',8]=>21147 ['B',8]=>751982 ['C',8]=>751982 ['D',8]=>159340 ['E',8]=>7591975 ['C',2]=>8
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Description
The number 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. This is the number of all pairwise different subgroups of $W$ obtained this way (including the trivial subgroup).
If $\Phi^+$ is an associated set of positive roots, then this also is the number of subsets $Y \subseteq \Phi^+$ such that $Y$ is a simple system of some type (including the empty system for type $A_0$).
Such a subset $Y$ is identified as simple system if for all $x \neq y \in Y$ we have $\langle x,y \rangle \leq 0$.
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)
    
    V = list(range(n))
    E = [[i, j] for i in range(n) for j in range(i) if P[i].inner_product(P[j]) <= 0]
    G = Graph([V,E])
    return IndependentSets(G, maximal=False, complement=True).cardinality()
Created
May 03, 2022 at 13:32 by Dennis Jahn
Updated
May 04, 2022 at 11:00 by Dennis Jahn