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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

representation?

(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 $g$

in $G$.

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

of characters.

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

scalar products,

because the multiplicities of Galois conjugate eigenvalues are equal

in such a case.

This avoids computations with non-rational numbers.

Kind regards

Thomas

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