isomorphism.at

root_permutations

find all Dynkin diagram isomorphisms: maps $f: #rank(rd1) -> #rank(rd2)$
transforming the Cartan matrices: $C_1[i,j]=C_2[f(i),f(j)]$ for all $i,j$,
where $C_k$ is the Cartan matrix of $rd_k$. This $f$ is represented by its
permutation matrix, the one the sends $e_j$ to $e_{f(j)}$, so $PC_1=C_2P$.
If the Dynkin diagrams are not isomorphic, this returns an empty list.

root_permutation

return (bool, one permutation matrix), reductive is allowed

local_isomorphism_long

find g\in GL(n,Q) and permutation matrix P so that
P is a root_permutation (as above), i.e.
P*Cartan_matrix_1=Cartan_matrix_2*P
gR_1=R_2P  (R_i = simple roots in rd_i)
these imply
^gS_2=S_1P^{-1}
this implies <\alpha,\beta^v>_1=<g\alpha,^g\beta^v>_2
this gives an isomorphism of root *systems*, i.e.
a local isomorphism since g,g^{-1} are
 in GL(n,Q) not necessarily GL(n,Z)
g gives an isomorphism of root systems (local isomorphism)
it also gives an isomorphism of root data if and only if g and g^{-1}
are both integral matrices
semisimple case only

local_isomorphism

return matrix of local isomorphism if it exists
abort with an error otherwise
note that the permutation is not returned
semisimple case only

isomorphism_long

to find an actual isomorphism, need to try all possible permutations
for example SL(2)xPGL(2) -> PGL(2)xSL(2)
also SO(8) has 6 local isomorphisms, only 1 non-trivial isomorphism
return (bool, matrix g, permutation P) satisfying gR_1=R_2P
semisimple case only

isomorphism

return matrix of isomorphism if it exists
abort with an error otherwise
note that the permutation is not returned
return value satisfies: gR_1=R_2P for some permutation P
semisimple case only

is_isomorphic

boolean
 gives the correct answer if both root data are semimple
 also: in reductive case, correctly returns false
  if the groups are not locally isomorphic
abort with an error if: the root data are locally
 isomorphic but not semisimple

also defined in W_reps.at, lietypes.at