Numerical Invariants and Parameter Recovery for Quasi-Homogeneous Submodules $[z^k-w^{\mathrm{\ell }}]$

We study Yang’s higher numerical invariants for the quasi-homogeneous submodules $$M_{k,\mathrm{\ell }}=[z^k-w^{\mathrm{\ell }}]\subset H^2({\mathbb{D}}^2),k,\mathrm{\ell }\in \mathbb{N},k\mathrm{\ne }\mathrm{\ell }\mathrm{.}$$ The Hilbert–Schmidt property and the low-order invariants ${\mathrm{\Sigma }}_0$ and ${\mathrm{\Sigma }}_1$ for this family follow from earlier results on $M_{\theta ,\phi }$-type submodules. Our aim is to determine the complete higher-order sequence. Using the weighted homogeneous decomposition associated with $z^k-w^{\mathrm{\ell }}$, we construct explicit orthonormal bases for the two defect spaces and determine the exact support of the shifted defect inner products. For $j\ge 1$, set $$A_j=\lceil \frac{j}{\mathrm{\ell }}\rceil ,B_j=\lceil \frac{j}{k}\rceil \mathrm{.}$$ We obtain $${\mathrm{\Sigma }}_0(M_{k,\mathrm{\ell }})=\frac{{\pi }^2}{6},{\mathrm{\Sigma }}_j(M_{k,\mathrm{\ell }})=F(A_j,B_j),$$ where $F$ is an explicitly evaluated symmetric function on ${\mathbb{N}}^2$. The sequence is nonincreasing, tends to zero, and satisfies ${\mathrm{\Sigma }}_j>{\mathrm{\Sigma }}_{j+1}⟺k\mid j\text{ or }\mathrm{\ell }\mid j$. Hence its strict descent set is $k\mathbb{N}\cup \mathrm{\ell }\mathbb{N}$. This descent set determines the additive semigroup $⟨k,\mathrm{\ell }⟩$, while the complete numerical invariant sequence determines the unordered pair $\{k,\mathrm{\ell }\}$. The order of the two exponents is not recoverable because of coordinate-swap symmetry.