Help finding matrix elements for $SU(3)$ tensors

Question

I'm trying to solve the following problem:

Find the matrix elements $\langle u|T_a|v \rangle$ where $T_a$ are the $SU(3)$ generators and $|u\rangle$ and $|v\rangle$ are tensors in the adjoint representation of SU(3) with components $u^i_j$ and $v^i_j$. Write the results in terms of the tensor components and the $\lambda_a$ matrices.

First order of business, I don't think I understand exactly what's meant by the adjoint representation here. The text is Georgi's on Lie algebras and Lie groups. Is this the adjoint representation of the Lie algebra or the Lie group? Can it only be one for some reason?

Also, what exactly are the objects $|u\rangle$ and $|v\rangle$? Are they vectors in the adjoint representation, so linear combinations of the gell-mann matrices? For the lie algebra, to get the hexagonal weight decomposition, it requires 6 ladder operators and 2 Cartan generators. Are these the actual basis of the adjoint representation of the Lie algebra? If so, why?

Depending on the answer to the first question will help me understand how to put matrix elements here in terms of the gell-mann matrices. Can both $|u\rangle$ and $|v\rangle$ be written in terms of the gell-mann matrices (via some basis I guess?).

Lastly, will $T_a$ act on tensors $|u\rangle$ and $|v\rangle$ via commutation since we're working in the adjoint representation? If not, how exactly do indices contract here? Does $T_a$ have mixed, solely upper, or solely lower indices?

Is $T_a |v\rangle$ supposed to live in 3 $\otimes$ 8?

Edit: Georgi's text defines the action of the SU(3) generators on an arbitrary tensor $|v\rangle$ in the representation (n,m) as

$$ T_a v^{j_1 ... j_n}_{i_1 ... i_m} = \sum_{l=1}^n [T_a]^{j_l}_k v^{j_1 ... k ... j_n}_{i_1 ... i_m} - \sum_{l=1}^m [T_a]^{k}_{i_l } v^{j_1 ... j_n}_{i_1 ... k ... i_m} $$

but I don't understand why or where this would come from.

Answer 1

I don't really understand the $v_j^i$ notation but I suppose $\vert v\rangle$ and $\vert u\rangle$ are basis for the carrier space of the adjoint irrep (1,1) or $\boldsymbol{8}$ as you write it.

Denote by $T_a $ the $8\times 8$ realization of $\lambda_a$. Then you need $T_a\vert v\rangle$, which is the action of $\lambda_a$ on the basis vector $\vert v\rangle$. Because we are dealing with the adjoint, where the action of the algebra on itself is the commutator, this is by definition equivalent to $$ [\lambda_a,\lambda_{v}] $$ where $\lambda_v$ is the Gell-Mann matrix corresponding to $\vert v\rangle$. Thus, your matrix elements are basically the structure constants.

If fact, $T_a\vert v\rangle$ lives in $\boldsymbol{8}\otimes \boldsymbol{8}$ but the action of the generators does not change the irrep so the result is also in $\boldsymbol{8}$.