[GAP Forum] Root systems representation in GAP

[Willem Adriaan De Graaf]

Dear Nikolay Hodyunya,

> For A_3 positive roots subsystem GAP gives me these vectors: [ [ 2,
> -1, 0 ], [ -1, 2, -1 ], [ 0, -1, 2 ], [ 1, 1, -1 ], [ -1, 1, 1 ], [ 1,
> 0, 1 ] ]. What do they mean?


Of course, there are many ways to represent the roots of a root system.
In this case, the roots record the eigenvalues of the elements of the
Cartan subalgebra,
which are contained in ChevalleyBasis(L)[3] (assuming your Lie algebra is
denoted L).
If h1, h2, h3 are these elements, and x is an element of the second root
space
(for example), then h1*x = -x, h2*x = 2*x, h3*x = -x. In a GAP session:


gap> L:= SimpleLieAlgebra("A",3,Rationals);
<Lie algebra of dimension 15 over Rationals>
gap> ch:= ChevalleyBasis(L);
[ [ v.1, v.2, v.3, v.4, v.5, v.6 ], [ v.7, v.8, v.9, v.10, v.11, v.12 ],
  [ v.13, v.14, v.15 ] ]
gap> ch[3]*ch[1][2];
[ (-1)*v.2, (2)*v.2, (-1)*v.2 ]



Best wishes,

Willem de Graaf

On 10 November 2016 at 16:19, Nikolay Hodyunya <nkhodyunya at gmail.com> wrote:

> Hello all,
>
> Could someone explain me please how does the GAP represent root
> systems? I'd like to have representation as in Bourbaki IV-VI chapters
> (tables at the end of the book). For example, any root of A_n is of
> the form e_i - e_j.
>
> For A_3 positive roots subsystem GAP gives me these vectors: [ [ 2,
> -1, 0 ], [ -1, 2, -1 ], [ 0, -1, 2 ], [ 1, 1, -1 ], [ -1, 1, 1 ], [ 1,
> 0, 1 ] ]. What do they mean?
>
> Thanks.
>
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