- Let $u_i$ be an object in the fundamental of SU(2). Let $\bar u_i$ be an object in the anti-fundamental of SU(2). We can consider the objects combining several $u_i$ and $\bar u_i$, following the rule of representation (rep) of SU(2),
we know that
$$2 \otimes 2 =1 \oplus 3$$
$$2 \otimes \bar 2 =1 \oplus 3$$
so if we combine two of SU(2) fundamental or anti-fundamentals, say
$$u_a \otimes u_b = v_{ab} $$
we can get a trivial representation of SU(2) (that is 1 in the above equations). The object $v_{ab}$ in this trivial rep 1 does not transform under SU(2).
We can say the object $v_{ab}$ in this trivial rep 1 is invariant under SU(2).
We can also get a nontrivial (adjoint) rep of SU(2) which is also a vector rep of SO(3) (that is the 3 in the above equations). The object $v_{ab}$ in this vector rep 3 of SO(3) does not transform under the full SU(2), but only under the SO(3).
We can say the object $v_{ab}$ in this vector rep 3 of SO(3) is not invariant under the SO(3), nor in the full SU(2).
My question: Which subgroup of SU(2) is this object [$v_{ab}$ in this vector rep 3 of SO(3)] invariant under? [Naively, it looks that it is a U(1) subgroup of SU(2). The $v_{ab}$ fixes a direction in SO(3), but is still a U(1) rotational degree of freedom within SO(3).] In particular, are there are some additional finite subgroup $\mathbb{Z}/2\mathbb{Z}$?
- Let $w_i$ be an object in the fundamental of SU(3). Let $\bar w_i$ be an object in the anti-fundamental of SU(3). We know that $$3 \otimes 3 =\bar 3 \oplus 6$$ $$3 \otimes \bar 3 =1 \oplus 8$$ $$3^2 \otimes \bar 3^2 =27 \oplus 10 \oplus \bar{10} \oplus 2 (1) \oplus 4(8)$$ Say, in the 1st case, 2nd and 3rd case, we denote, the objects obtained from multiplications of SU(3) fundamentals (also anti-fundamentals) as: $$w_a \otimes w_b \equiv W_{ab} $$ $$w_a \otimes \bar w_b \equiv W_{a \bar b}$$ $$w_a \otimes w_b\otimes w_c \otimes w_d \equiv W_{ab \bar c \bar d} $$
My question: Which subgroup of SU(3) are these objects, (here $W_{ab}$, $W_{a \bar b}$, and $ W_{ab \bar c \bar d}$ respectively), invariant under within each sector of their multiplication decomposition of simple Lie groups irreducible Representations? (say they are in part of their own decompositions, respectively, in $3 \oplus 6$, in $1 \oplus 8$, and in $27 \oplus 10 \oplus \bar{10} \oplus 2 (1) \oplus 4(8)$?) In particular, are there are some finite subgroup factors?