Dear GAP Forum,
Dmitrii Pasechnik asked the following two questions.
Could someone provide me with a reference to the algorithm used
in the function Eigenvalues which computes the eigenvalues
of a representative of a conjugacy class in a given irreducible
(I cannot find this in standard texts on character theory.)
The only reference to the computation of eigenvalues from characters
in the literature I am aware of is on page 231
of the article by J. Neub\"user, H. Pahlings, and W. Plesken
that is cited in the GAP manual.
So the idea seems to be folklore, and in fact it is straightforward.
Namely, let us assume we have the character of an ordinary representation
$D$ (which need not be irreducible) of the group $G$,
and that you want to compute the eigenvalues for the element $g$
Consider the restriction $D'$ of $D$ to the cyclic group $C$
generated by $g$.
The matrix $D(g)$ is equivalent to a diagonal matrix with diagonal
entries certain complex $n$-th roots of unity,
where $n$ is the order of $g$.
The multiplicity of the eigenvalue $\epsilon$ as diagonal entry in
this matrix is exactly the multiplicity of the irreducible character
of $C$ that maps $g$ to $\epsilon$, in the decomposition of the
character afforded by the representation $D'$,
and this multiplicity can be computed by computing scalar products
Since we need only the character of the restriction $D'$ and not
the representation itself, and since the embedding of the cyclic group
$C$ into $G$ is described by the power maps of the character table of
$G$, it is clear that the character of $D$ and the power maps of $G$
suffice to compute the eigenvalues.
I would like to know if there is a quickier way to compute the
characteristic polynomial f=f(G) of a representative G
of a conjugacy class mentioned
above, than using Eigenvalues().
(The latter function involves using algebraic numbers,
whereas it might happen
that f has rational or integer coefficients ,
i.e. all the irrationalies cancel)
For example, if the character values in question are rational
one can use Galois sums of the irreducible characters of the cyclic
subgroup instead of the irreducible characters when computing
because the multiplicities of Galois conjugate eigenvalues are equal
in such a case.
This avoids computations with non-rational numbers.