I've been looking around in the documentation for root systems. It's not
clear to me whether the individual roots in the system are accessible
objects. For example, I can take the root system F4.
sage: rs=RootSystem("F4")
sage: r1=rs.root_lattice().positive_roots()
sage: r2=rs.weight_lattice().positive_roots()
sage: r3=rs.ambient_space().positive_roots()
sage: r1
A recursively enumerated set with a graded structure (breadth first search)
sage: r2
A recursively enumerated set with a graded structure (breadth first search)
sage: r3
[(1, 0, 0, 0),
(0, 1, 0, 0),
...
(1/2, -1/2, -1/2, -1/2)]
sage: it1=r1.breadth_first_search_iterator()
sage: it2=r2.breadth_first_search_iterator()
sage: a1r1=next(it1)
sage: a1r2=next(it2)
sage: a1r3=r3[0]
sage: a1r1
alpha[1]
sage: a1r2
2*Lambda[1] - Lambda[2]
sage: a1r3
(1, 0, 0, 0)
sage: a1r1==a1r2
True
sage: a1r1 is a1r2
False
sage: a1r1 == a1r3
False
sage: a1r2==a1r3
False
If I understand what's going on, a1r1, a1r2 and a1r3 above are three
distinct representations of the same mathematical object (the first simple
root of F4). My question is whether there is a way to access the root
directly? (Say you have some function which takes a root or pair of roots
as input which you want to implement...)
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