Lieb's Permanental Dominance Conjecture for Ordinary Immanants through Order Fifteen
Abstract
Pate proved ordinary irreducible-immanant permanental dominance through order $13$ and identified $(4,4,3,3)$ as the sole remaining order-$14$ case, with $(5,4,3,3)$ and (35) forming the order-$15$ frontier. These three cases are settled here; consequently dλ(A)/fλ≤ per(A) for every partition λ⊢ n with n≤15 and every complex Hermitian positive-semidefinite matrix A. The argument also yields results beyond this finite frontier: an exact four-term bridge for $(4,4,3,3)$, the uniform family $(m,4,3,3)$, a two-parameter family $(a,b,3,3)$ for a≥ b≥4 and 5a≥8b, and a long-first-row criterion for arbitrary fixed tails. These results arise from explicit specializations of Pate's W-function positivity framework using partial swaps, Young projectors, Pieri--content identities, and branching data. For (35), an exact Farkas certificate shows that the central-projector partial-swap cone is insufficient; a branching-refined one-swap construction escapes this obstruction and yields a positive 106+19-witness certificate. Boundary-compression and node-moving results further describe the reach and limitations of the local-filter method. All finite certificates are checked by exact integer or rational arithmetic and are supplied as ancillary material. The order-$14$ bridge is additionally formalized and kernel-checked in Lean 4 for all complex Hermitian positive-semidefinite matrices, including the exact coefficient normalization and the deduction of $(4,4,3,3)$ permanental dominance from four explicitly stated Pate inequalities.
Related
- Haiman's conjecture for n <= 10: Kazhdan-Lusztig positivity of monomial immanants
- The Tate Conjecture for Ordinary Calabi-Yau Threefolds
- Counterexamples to the ordinary Fried conjecture near hyperbolic metrics
- Kolyvagin's conjecture at non-ordinary primes
- An RKHS Framework for Fixed Effects in Permanental Process Models
BibTeX
@misc{li2026liebs,
title = {{Lieb's Permanental Dominance Conjecture for Ordinary Immanants through Order Fifteen}},
author = {Yinjie Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13412},
eprint = {2609.13412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13412},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lieb's Permanental Dominance Conjecture for Ordinary Immanants through Order Fifteen
AU - Yinjie Li
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13412
UR - https://arxiv.org/abs/2609.13412
AB - Pate proved ordinary irreducible-immanant permanental dominance through order $13$ and identified $(4,4,3,3)$ as the sole remaining order-$14$ case, with $(5,4,3,3)$ and (3^5) forming the order-$15$ frontier. These three cases are settled here; consequently d_λ(A)/f^λ≤ per(A) for every partition λ⊢ n with n≤15 and every complex Hermitian positive-semidefinite matrix A. The argument also yields results beyond this finite frontier: an exact four-term bridge for $(4,4,3,3)$, the uniform family $(m,4,3,3)$, a two-parameter family $(a,b,3,3)$ for a≥ b≥4 and 5a≥8b, and a long-first-row criterion for arbitrary fixed tails. These results arise from explicit specializations of Pate's W-function positivity framework using partial swaps, Young projectors, Pieri--content identities, and branching data. For (3^5), an exact Farkas certificate shows that the central-projector partial-swap cone is insufficient; a branching-refined one-swap construction escapes this obstruction and yields a positive 106+19-witness certificate. Boundary-compression and node-moving results further describe the reach and limitations of the local-filter method. All finite certificates are checked by exact integer or rational arithmetic and are supplied as ancillary material. The order-$14$ bridge is additionally formalized and kernel-checked in Lean 4 for all complex Hermitian positive-semidefinite matrices, including the exact coefficient normalization and the deduction of $(4,4,3,3)$ permanental dominance from four explicitly stated Pate inequalities.
ER -