Background
I see a lot of confusing group representation terminology in physics writing. Here is a typical example, taken from D. J. Griffiths' Introduction to Elementary Particles, talking about quark flavor combinations in baryons:
As for flavor, there are $3^3 = 27$ possibilities: $uuu, uud, udu, udd, \dotsc, sss$, which we reshuffle into symmetric, antisymmetric, and mixed combinations; they form irreducible representations of $SU(3)$, just as the analogous spin combinations form representations of $SU(2)$.
Vantage point
I am very new to representation theory, but as I understand it, a representation is a homomorphism $\rho : G \to GL(V)$ from a group $G$ (of transformations) to the general linear group of a vector space $V$. Hence $V$ is where the spin (or isospin, or flavor, etc.) vectors "live", while the representation provides the matrices transforming elements of $V$. Now, some representations have a (proper nontrivial) invariant subspace $V_1 \subset V$, for which $\rho(g)v \in V_1$ if $v \in V_1$. Such a representation is called reducible (or more accurately decomposable or completely reducible, i believe) and may be block diagonalized, $$\rho_D(g) = U \rho(g) U^{-1} = \begin{pmatrix} \rho_1(g) & 0 \\ 0 & \rho_2(g) \end{pmatrix},$$ by some invertible matrix $U$. In other terms, $\rho_1$ and $\rho_2$ are representations of $G$ on $V_1$ and $V_2$ (where $V_2 = V \setminus V_1$ is also an invariant subspace), respectively, and we have $V = V_1 \oplus V_2$ and $\rho_D = \rho_1 \oplus \rho_2$. A representation that is not reducible is called an irreducible representation.
Question
So then, if the above is correct, isn't the quoted terminology wrong? Griffiths says that the symmetric, antisymmetric, and mixed flavor combinations form irreducible representations of $SU(3)$, but as I understand it these combinations are not even part of the representation, but elements in a vector space on which the representation acts. Would not the correct statement be that these combinations form invariant subspaces under $SU(3)$, and that each such invariant subspace transforms according to some irreducible representation of $SU(3)$?
