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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.24349 |
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| _version_ | 1866914620224045056 |
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| author | Fahssi, Nour-Eddine |
| author_facet | Fahssi, Nour-Eddine |
| contents | This paper studies generalized Pólya conversion problems for the $q$-permanent. We establish a sharp threshold governing the transition from low-dimensional algebraic flexibility to higher-dimensional combinatorial rigidity. For $n \ge 3$ and $q \ne \pm 1$, we prove that the $q$-permanent is not linearly convertible to the determinant or permanent. Conversely, we completely classify the space of Schur multiplier preservers for $n=2$. Focusing on Schur multipliers, we characterize the preserver exponents as a $(2n-2)$-dimensional space of additive matrices. We show that for lower Hessenberg matrices, the general geometric obstruction disappears, yielding an explicit determinantal reduction and an $\mathcal{O}(n^3)$ evaluation algorithm. Furthermore, we classify permutational converter exponents, proving that for $n \ge 4$, admissible symmetries are strictly constrained to the dihedral group. Finally, we resolve a mixed conversion problem, showing the solution space is nonempty only for $n \le 4$, which provides a direct algebraic characterization of the $q$-permanent's zero locus in low dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24349 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Fahssi, Nour-Eddine Combinatorics Algebraic Geometry Quantum Algebra 15A15, 05E15, 05A05 This paper studies generalized Pólya conversion problems for the $q$-permanent. We establish a sharp threshold governing the transition from low-dimensional algebraic flexibility to higher-dimensional combinatorial rigidity. For $n \ge 3$ and $q \ne \pm 1$, we prove that the $q$-permanent is not linearly convertible to the determinant or permanent. Conversely, we completely classify the space of Schur multiplier preservers for $n=2$. Focusing on Schur multipliers, we characterize the preserver exponents as a $(2n-2)$-dimensional space of additive matrices. We show that for lower Hessenberg matrices, the general geometric obstruction disappears, yielding an explicit determinantal reduction and an $\mathcal{O}(n^3)$ evaluation algorithm. Furthermore, we classify permutational converter exponents, proving that for $n \ge 4$, admissible symmetries are strictly constrained to the dihedral group. Finally, we resolve a mixed conversion problem, showing the solution space is nonempty only for $n \le 4$, which provides a direct algebraic characterization of the $q$-permanent's zero locus in low dimensions. |
| title | The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function |
| topic | Combinatorics Algebraic Geometry Quantum Algebra 15A15, 05E15, 05A05 |
| url | https://arxiv.org/abs/2605.24349 |