- FindStat

Identifier

St001755: Finite Cartan types ⟶ ℤ

Values

['A',1] => 1
['A',2] => 1
['B',2] => 3
['G',2] => 6
['A',3] => 1
['B',3] => 9
['C',3] => 9
['A',4] => 1
['B',4] => 35
['C',4] => 35
['D',4] => 4
['F',4] => 142
['A',5] => 1
['B',5] => 128
['C',5] => 128
['D',5] => 11
['A',6] => 1
['B',6] => 755
['C',6] => 755
['D',6] => 41
['E',6] => 77
['A',7] => 1
['B',7] => 4105
['C',7] => 4105
['D',7] => 162
['E',7] => 1516
['A',8] => 1
['E',8] => 132462
['C',2] => 3

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$F_{1} = q$

$F_{2} = q + q^{3} + q^{6}$

$F_{3} = q + 2\ q^{9}$

$F_{4} = q + q^{4} + 2\ q^{35} + q^{142}$

$F_{5} = q + q^{11} + 2\ q^{128}$

$F_{6} = q + q^{41} + q^{77} + 2\ q^{755}$

$F_{7} = q + q^{162} + q^{1516} + 2\ q^{4105}$

Description

The number of pairwise different full-rank 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 full-rank subgroups of

$W$ obtained this way.
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 and

$|Y| = n$ , where

$n$ is the rank of

$W$ .
For example the group of type

$B_2$ has two different subgroups of type

$A_1 \times A_1$ and itself as full-rank reflection subgroups.

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])
    C = IndependentSets(G, maximal=True, complement=True)
    return len([item for item in C if len(item) == W.rank()])

Created

Dec 13, 2021 at 13:51 by Dennis Jahn

Updated

May 04, 2022 at 11:23 by Dennis Jahn