Just in case some other new person sees this there is one point I should
correct/clarify from the original post.
In fact a1r1, a1r2, and a1r3 are *not *three distinct representations of
the same mathematical object. The
representation of the first simple root as an element of the ambient space
is r3[13], not r3[0].
sage: a1r3=r3[13]
sage: a1r3.to_weight_space()
2*Lambda[1] - Lambda[2]
sage: a1r3.to_simple_root()
1
sage: a1r3==a1r1
True
sage: a1r3==a1r2
True
On Wednesday, July 27, 2016 at 4:26:06 AM UTC-4, Joseph Hundley wrote:
>
> 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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