The limits of Schur multipliers in P\'olya conversion problems for the $q$-permanent function

This paper studies generalized P\'olya conversion problems for the $q$-permanent $$\operatorname{P}_q(A)=\sum_{\sigma\in S_n} q^{\ell(\sigma)} a_{1,\sigma(1)} \cdots a_{n,\sigma(n)},$$ where $q\in\mathbb{C}^*$ and $\ell(\sigma)$ is the permutation length. We show that for $n\geq 3$ and $q\neq \pm1$, the $q$-permanent is not linearly convertible to the determinant or the permanent, and we completely classify and give a geometric interpretation of the special case $n=2$. Focusing on Schur multiplier transformations, we characterize the space of Schur multiplier preservers. For $|q|\neq1$, the preserver exponents is a $(2n-2)$-dimensional vector space consisting of additive matrices satisfying a discrete Monge relation. In contrast, for $q$ on the unit circle, the solution space becomes a countable union of affine lattices. For lower Hessenberg matrices, we prove that the rigidity phenomenon disappears, yielding an explicit determinantal reduction of the $q$-permanent and an $O(n^3)$ evaluation algorithm. The central results of this paper establish sharp rigidity thresholds governing permutational symmetries and mixed conversion identities. First, we classify permutational converter exponents and show that, for $n\geq4$, the admissible symmetries are precisely the elements of the dihedral group. Second, we solve a mixed conversion problem that expresses the $q$-permanent as a linear combination of the determinant and the permanent, and prove that the corresponding solution space is nonempty if and only if $n\leq4$, in which case it decomposes into finitely many affine components modeled on the preserver exponent space. This mixed formulation yields a direct algebraic characterization of the $q$-permanent's zero locus for $n \le 4$ via a generalized P\'olya identity.