[GAP Forum] problem with OrthogonalEmbeddings

[Benjamin Sambale]

Dear GAP people,

according to the manual, the command OrthogonalEmbeddings does the 
following: Given an integral symmetric matrix M, compute all integral 
matrices X such that X^tr X = M where X^tr denotes the transpose of X. 
The solution matrices X are given up to permutations and signs of their 
rows.

If I do (with GAP 4.7.5) OrthogonalEmbeddings([[4]]), I only get one 
solution, namely X = [[2]]. However, there is another solution X = 
[[1],[1],[1],[1]] which is somehow missing! What is wrong here?
Apparently, the implementation is quite old and based on a paper by 
Plesken from 1995.

There is also an inaccuracy in the manual: It says: "the list L = [ x_1, 
x_2, ..., x_n ] of vectors that may be rows of a solution; these are 
exactly those vectors that fulfill the condition x_i ⋅ gram^{-1} ⋅ 
x_i^tr ≤ 1 (see ShortestVectors (25.6-2)), and we have gram = ∑_{i = 
1}^n x_i^tr ⋅ x_i".

The last equation is usually not true. The equation only holds for the 
set of vectors of a solution. Moreover, one should mention that the list 
of vectors is only up to signs.

Thanks and best wishes,
Benjamin

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