# What is the physical meaning of Lie congruence classes?

Any weight $\lambda$ characterising a representation of $\mathfrak{su}(N)$ is an element of one of the $N$ congruence classes defined by (ref.1) $$\lambda_1+2\lambda_2+\cdots+(N-1)\lambda_{N-1}\quad\text{mod}\ N$$

These classes are the elements of the coset defined by taking the weight lattice and quotienting out the root lattice. (This admits a straightforward generalisation to arbitrary simple algebras).

In the case of $\mathfrak{su}(2)$, these two classes have a clear and very important physical meaning: integer and half-integer spins, which classify particles into bosons and fermions (and classify Lorentz fields into spinorial and regular ones, i.e., commuting and anti-commuting).

By extension, I would expect that the congruence classes above classify some physical objects into classes with some physical meaning. I don't even know what the three classes represent in the case of $\mathfrak{su}(3)$ corresponding to the colour symmetry.

Question: Is there a physical interpretation of these classes? What do they classify, in physical terms?

For definiteness, one may consider that $\mathfrak{su}(N)$ is the gauge group (or flavour group) of some QFT. Other systems are welcome though.

References.

1. Di Francesco, Mathieu and Sénéchal - Conformal Field Theory, eq. 13.77.
• See this paper: Congruence number, a generalization of SU(3) triality, by F. Lemire and J. Patera, J. Math. Phys. 21 (8), August 1980 aip.scitation.org/doi/pdf/10.1063/1.524711 – ZeroTheHero Mar 26 '18 at 16:14
• In particular, triality (for su(3)) is preserved in the decomposition of the tensor product, i.e. the product of two irreps with triality $a$ and $b$ resp. will contain only irreps of triality $a+2b$ (mod 3)$, generalizing the rule that the decomposition of two spinor irreps gives true irreps, the decomposition of one spinor and one true irrep gives a sum of spinor irreps etc. – ZeroTheHero Mar 26 '18 at 16:16 • It depends on the physics model. – Qmechanic Mar 26 '18 at 16:50 • @Qmechanic It sure does. I would settle for the case where$\mathfrak{su}(N)\$ is the gauge group (or flavour group) of some QFT. – AccidentalFourierTransform Mar 26 '18 at 16:56