# What exactly means “is a singlet under $SU(N)$” [duplicate]

I don't get a grip of what that exactly means. What IS an abstract singlet, doublet,... under $SU(N)$ or other groups?

• Essentially a duplicate of physics.stackexchange.com/q/41424/2451 – Qmechanic Nov 12 '13 at 13:38
• In addition to the linked thread: A SU(2) doublet is a pair of states (particles) which can be rotated into each other by SU(2) (gauge) transformations. You have weak isospin ($T_3$) up and down, but you are free to e.g. replace up <-> down everywhere without changing the physics. A singlet has no SU(2) "charge" and it not affected by the transformation. In the SM, left-handed leptons are SU(2) doublets $(e, \nu)_L$, $T_3 = \pm 1/2$, and right-handed ones SU(2) singlets $e_R$, $T_3$ = 0. In the same way, you have color triplets (quarks), singlets (leptons), and octets (gluons) under SU(3). – jdm Nov 26 '13 at 12:45

"Singlet under $SU(N)$" means that the related representation is invariant under $SU(N)$.
"Doublet" is related in general to $SU(2)$, and corresponds to the fundamental ($2$- dimensional) representation of $SU(2)$
For instance, looking at the "weak interaction", left-handed particles transform as a doublet under $SU(2)_L$ , while right-handed particles transform as a singlet under $SU(2)_L$