- FindStat

Identifier

St001789: Finite Cartan types ⟶ ℤ

Values

['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

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

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

$F_{2} = q^{3} + q^{4} + q^{5}$

$F_{3} = q^{5} + 2\ q^{9}$

$F_{4} = q^{7} + q^{8} + 2\ q^{17} + q^{19}$

$F_{5} = q^{11} + q^{14} + 2\ q^{31}$

$F_{6} = q^{15} + q^{21} + q^{23} + 2\ q^{57}$

$F_{7} = q^{22} + q^{35} + q^{41} + 2\ q^{98}$

$F_{8} = q^{30} + q^{56} + q^{72} + 2\ q^{166}$

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