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Identifier
Values
=>
Cc0022;cc-rep
['A',1]=>1 ['A',2]=>1 ['B',2]=>2 ['G',2]=>2 ['A',3]=>2 ['B',3]=>2 ['C',3]=>2 ['A',4]=>2 ['B',4]=>3 ['C',4]=>3 ['D',4]=>2 ['F',4]=>3 ['A',5]=>3 ['B',5]=>4 ['C',5]=>4 ['D',5]=>3 ['A',6]=>4 ['B',6]=>4 ['C',6]=>4 ['D',6]=>4 ['E',6]=>4 ['A',7]=>5 ['B',7]=>5 ['C',7]=>5 ['D',7]=>4 ['E',7]=>5 ['A',8]=>6 ['B',8]=>6 ['C',8]=>6 ['D',8]=>5 ['E',8]=>7
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Description
The minimal size of a base for the Weyl group of the Cartan type.
A base of a permutation group is a set $B$ such that the pointwise stabilizer of $B$ is trivial. For example, a base of the symmetric group on $n$ letters must contain all but one letter.
Any base has at least $\log |G|/n$ elements, where $n$ is the degree of the group, i.e., the size of its domain.
References
[1] wikipedia:Base_(group_theory)
Code
def minimal_base(G):
    """
    EXAMPLES::

        sage: [G for G in SymmetricGroup(7).conjugacy_classes_subgroups() if G.base() != minimal_base(G)]
        [Subgroup generated by [(2,4,5,7,3), (2,3)(4,7), (1,6)(2,7,3,4)] of (Symmetric group of order 7! as a permutation group)]

    """
    b = G.base()
    for i in range(ceil(log(G.cardinality())/log(G.degree())), len(b)):
        for s in Subsets(G.domain(), i):
            nb = G.base(seed=s)
            if len(nb) < len(b):
                return nb
    return b

def statistic(ct):
    return len(minimal_base(WeylGroup(ct, implementation="permutation")))
Created
Jul 05, 2024 at 11:04 by Martin Rubey
Updated
Jul 05, 2024 at 11:04 by Martin Rubey