IHere is the answer to question (1). The first thing to know is that the premise of the question is wrong: Indecomposable projectives do not always have some thoughts abouta distributive submodule lattice. Rather, $P_i$ has a distributive submodule lattice if and only if the other partsweight $\omega_i$ is minuscule. (Because we are in simply-laced type, but"minuscule" and "co-minscule" are synonyms.) In this case, the submodule lattice of $P_i$ is isomorphic to the lattice of order ideals in the heap $H(w^i)$.
The minuscule weights have been classified: See Table 1 in Thomas-Yong and ignore the rows for types $B$ and $C$. The corresponding heaps are also well known; see the Figures in Section 2.2 of the Thomas-Yong paper.
I needwill now explain how to work on other thingspiece these results together, using facts from Baumann and Kamnitzer and Dranowski, Elek, Kamnitzer and Morton-Ferguson.
We use the usual Coxeter notations: $\alpha_i$ are the simple roots, $s_i$ are the simple reflections and $\omega_i$ are the fundamental weights.
To every weight $\gamma$, Baumann and Kamnitzer define a preprojective module $N(\gamma)$, such that $N(\omega_i) = P_i$ (see Prop 10). Here are the key facts about $N(\gamma)$:
- If $\gamma$ is antidominant, then $N(\gamma) = 0$.
- If $s_j \gamma = \gamma + c \alpha_j$ for some $c \geq 0$, then we have a short exact sequence $0 \to N(\gamma) \to N(s_j \gamma) \to S_j^c \to 0$.
- As a corollary of the proceeding two statements, let $\gamma'$ be the unique antidominant weight in the same orbit as $\gamma$ and let $\gamma - \gamma' = \sum d_i \alpha_i$. Then $N(\gamma)$ has dimension $(d_1, d_2, \ldots, d_n)$.
- Let $\gamma$ and $\gamma'$ be as in the previous bullet point, and let $\gamma' = - \sum c_i \omega_i$. Then $N(\gamma)$ is uniquely characterized as the only preprojective module with dimension $(d_1, d_2, \ldots, d_n)$ and socle of dimension $(c_1, c_2, \ldots, c_n)$.
Now, let $\omega'_i$ be the unique fundamental root so that $-\omega'_i$ and $\omega_i$ are in the same orbit. Let $w^i$ be the Coxeter group element of minimal length such that $w^i \omega_i = - \omega'_i$ and let $s_{j_1} s_{j_2} \cdots s_{j_L}$ be a reduced word for $w^i$. Put $\gamma(k) = s_{j_k} \cdots s_{j_2} s_{j_1} (- \omega'_i)$, so $\gamma(0) = - \omega'_i$, $\gamma(L) = \omega_i$ and $\gamma(k) = \gamma(k-1) + c_k \alpha_{i_k}$ for some integer $c_k$; it turns out (proof omitted) that $c_k>0$. So the modules $N(\gamma(k))$ form a filtration of $N(\omega_i) = P_i$, with subquotients $S_{j_k}^{c_k}$.
In particular, if any $c_k$ is $\geq 2$, then the submodule lattice of $P_i$ is not distributive.
If all the $c_k$ are $1$, then $w^i$ is what is called $\omega_i$-minuscule (see Definition 2.4 in Dranowski, Elek, Kamnitzer and Morton-Ferguson), and is therefore minuscule, and the results of that paper apply. I'll summarize those results now.
We define a poset $H(w^i)$, called the heap of $w^i$, as follows: The ground set of $H(w^i)$ is $[L]$. The order relation is the transitive closure of the following: $a \prec b$ if $a<b$ and $s_{j_a} s_{j_b} \neq s_{j_b} s_{j_a}$. We have the following:
$H(w^i)$ indexes a basis for $\mathbb{C} H(w^i)$. To understand the maps in $\mathbb{C} H(w^i)$, take the Hasse diagram of $H(w^i)$ (see the Thomas-Yong paper) and place $1$'s and $-1$'s on the edges in order to make the preprojective relations hold.
Order ideals of $H(w^i)$ correspond to submodules of $\mathbb{C} H(w^i)$. These also correspond to the weak order interval $[e, w^i]$, and to roots in the $W$ orbit of $\omega_i$.
The submodule lattice of $\mathbb{C}H(w^i)$ is the lattice of order ideals in $H(w^i)$.
Total orders of $H(w^i)$ correspond to reduced words for $w^i$, which correspond to composition series of $P_i$.
In type $A_n$, $H(w^i)$ is the product of an $i$-vertex chain and an $(n+1-i)$ vertex chain. Order ideals correspond to partitions fitting in an $i \times (n+1-i)$ box, and the submodule lattice is this part of Young's lattice. Total orders on $H(w^i)$ are standard Young tableaux of shape $i \times (n+1-i)$.
