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Momentum classification of SU($n$) spin chains using extended Young Tableaux

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.