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

TL;DR

This paper extends the method of extended Young Tableaux to efficiently compute eigenvalues of cyclic permutations in SU(n) spin chains, improving speed over traditional techniques.

Contribution

It generalizes the extended Young tableaux method to all symmetric SU(n) representations, enabling faster eigenvalue calculations for permutation operators.

Findings

Eigenvalue computation is at least linearly faster with the extended Young tableaux method.

The method applies to all symmetric representations of SU(n).

Provides a practical tool for analyzing SU(n) spin chains.

Abstract

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.