- FindStat

Identifier

St001790: Finite Cartan types ⟶ ℤ

Values

['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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$F_{1} = q^{2}$

$F_{2} = q^{5} + q^{8} + q^{13}$

$F_{3} = q^{15} + 2\ q^{38}$

$F_{4} = q^{52} + q^{75} + 2\ q^{218} + q^{637}$

$F_{5} = q^{203} + q^{428} + 2\ q^{1430}$

$F_{6} = q^{877} + q^{2781} + q^{5079} + 2\ q^{10514}$

$F_{7} = q^{4140} + q^{20093} + 2\ q^{85202} + q^{107911}$

$F_{8} = q^{21147} + q^{159340} + 2\ q^{751982} + q^{7591975}$

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