According to Yu Qiu 's Ext-quivers of hearts of A-type and the orientation of associahedron,
We know hearts in $\mathcal{D}^b(A_n)$ one by one corresponds to the Ext quivers and are precisely the associated quivers of graded gentle trees with $n$ vertices.
And it's not hard to noticed
if two hearts if the shapes are equal then the inverse proposition of the proposition in the title holds.
Absolutely,the proposition and inverse proposition is trivia and I have proven the inverse proposition. But I can't find a simple proof to describe the title.
Example in $\mathcal{D}^b(A_3)$,$A_3$ has straight orientation, $\mathcal{H}_1=<S_1,S_2,S_3>$,$\mathcal{Q}(\mathcal{H}_1)=S_1\xrightarrow{1}S_2\xrightarrow{1}S_3$
$\mathcal{H}_2=<S_1[-1],P_1,S_3[1]>$,$\mathcal{Q}(\mathcal{H}_1)=S_3[1]\xrightarrow{1}P_1\xrightarrow{1}S_1[-1]$
the under graph of ext quiver are $\bullet\xrightarrow{1}\bullet\xrightarrow{1}\bullet$ , The two heart has the same shape (A positive triangle, an inverted triangle)
Question: My main problem is that when $n$ is large enough, Ext quiver itself will start to get complicated, and simply using Hom discussion is extremely tedious.Do we have a good idea or lemma to solve it
Thanks for everyone!
Remark:
Let $\mathcal{H}$ be a finite heart in a derived category $\mathcal{D}^b(A_n)$. The Ext-quiver $\mathcal{Q}(\mathcal{H})$ is the (positively) graded quiver whose vertices are the simples of $\mathcal{H}$ and whose degree $k$ arrows $S_i\rightarrow S_j$ correspond to a basis of $\mathrm{Hom}_\mathcal{D}(S_i,S_j[k])$.