= Solution
For a finitely generated <associative algebra> $A$, the <representation variety of an associative algebra> consists of generator matrices satisfying all defining polynomial relations. With fixed orthogonal idempotents $1=\sum_{i=1}^me_i$, require a representation on $\bigoplus_i k^{n_i}$ to send $e_i$ to the standard vertex projector. This is the meaning of $\operatorname{Rep}_A(\mathbf n)$; without specified idempotents, use the usual single dimension $r$ and $\operatorname{Rep}_A(r)=\operatorname{Hom}_{k\text{-alg}}(A,M_r(k))$. Polynomial relations cut out a closed <affine variety> in the space of generator matrices.
The group $G=\prod_i\operatorname{GL}_{n_i}(k)$ acts by conjugation, preserving the prescribed projectors. Its stabilizer is $\operatorname{Aut}_A(X)$, a nonempty open subset of $\operatorname{End}_A(X)$. The <orbit dimension formula> consequently gives
$$
\boxed{\dim G-\dim\mathcal O_X=\dim\operatorname{Stab}_G(x)=\dim\operatorname{Aut}_A(X)=\dim\operatorname{End}_A(X)}.
$$
A <degeneration of a module> $X$ to $Y$ means that $\mathcal O_Y\subseteq\overline{\mathcal O_X}$, equivalently that one representative of $Y$ lies in this closure.
For $0\to X'\to X\to X''\to0$, choose a vector-space splitting, so every generator has block matrix $x_g=\left(\begin{smallmatrix}x'_g&b_g\\0&x''_g\end{smallmatrix}\right)$. Conjugation by $\operatorname{diag}(tI,I)$, $t\ne0$, gives
$$
x_g(t)=\begin{pmatrix}x'_g&t b_g\\0&x''_g\end{pmatrix}.
$$
This polynomial family extends to $t=0$, retains all algebra relations, and at zero represents $X'\oplus X''$. Thus <splitting an extension gives a module degeneration>. Iterating along a <composition series> gives \b[$X\rightsquigarrow\operatorname{gr}X=\bigoplus_jX_j/X_{j-1}$]. Degenerations are transitive because an orbit closure is closed and invariant under base change.
For the <one-arrow quiver> with dimension vector $(n_1,n_2)$,
$$
\boxed{\operatorname{Rep}_Q(n_1,n_2)=\operatorname{Hom}_k(k^{n_1},k^{n_2})\cong M_{n_2\times n_1}(k)}.
$$
The action is $A\mapsto g_2Ag_1^{-1}$. Its <rank orbits of a matrix under left-right multiplication> are indexed by $r=0,\ldots,\min(n_1,n_2)$, with a representative containing an $r\times r$ identity block and zeros elsewhere. Their dimensions are $r(n_1+n_2-r)$; their closures contain exactly matrices of rank at most $r$.
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