  Obtaining eigenvalues of permutations acting on the product space of $N$
representations of SU($n$) usually involves either diagonalising their
representation matrices on total-weight subspaces or decomposing their
characters, which can be obtained from Frobenius' formula or via graphical
methods using Young tableaux. For products of fundamental representations of
SU($n$), Schuricht and one of us proposed the method of extended Young
Tableaux, which allows reading the eigenvalues of the cyclic permutation $C_N$
directly off the, slightly modified, standard Young tableaux labelling an
irreducible SU($n$) representation. Here we generalise the method to all
symmetric representations of SU($n$), and show that $C_N$ eigenvalue
computation based on extended Young tableaux is at least linearly faster than
the standard methods mentioned.
