Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents

Abstract

This paper studies symmetry-induced reductions of scalar recurrence complexity for balanced banded Toeplitz determinants and permanents. Three mechanisms emerge: determinant state and spectral compression, exceptional permanent--determinant conversion, and symmetries of permanent state spaces. For symmetric determinants, straightening reduces the row-column states from binom2mm to the Catalan number C_m+1; primitive symplectic weight compression then leaves (3^m+1)/2 distinct autonomous modes. Skew symmetry has a parallel compound/Hodge explanation: the middle exterior representation splits into two Hodge halves of dimension binom2mm/2, the one-step compound transfer exchanges the halves, and its two-step restriction has 3^m−1 generic ternary modes. Thus the full skew bound is 2·3^m−1. Widom--Hankel arguments prove generic scalar minimality in both symmetry classes, and every even skew Toeplitz determinant admits an explicit half-size square factorization. For the zero-diagonal pentadiagonal support, a Pólya--Kasteleyn signing converts every permanent to a determinant. Among consecutive zero-diagonal two-sided bands, universal entrywise conversion---and separately Toeplitz diagonal-wise conversion---occurs only in the Hessenberg families and this pentadiagonal case. Paired renewal identities recover all restored-diagonal determinant and permanent layers. For permanents, transposition gives the open symmetric bound (binom2mm+2^m)/2, while cyclic defect-sector pairing gives (4^m+binom2mm)/2. Skew-symmetry forces odd-size vanishing and corresponding even-subsequence bounds. In semibandwidth two the open symmetric bound is generically sharp; higher-semibandwidth permanent minimality is left separate from the symmetry reductions proved here.

Constructive recurrences for determinants and permanents of banded Toeplitz matrices
Max A. Alekseyev, Dmitry I. Khomovsky
arXiv12 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Hankel, Toeplitz, Hermitian--Toeplitz Determinants and the Zalcman Functional for a Class of Biholomorphic Mappings on Complex Banach Spaces
Nabadwip Sarkar, Pradip Das
arXiv3 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
A proof of Karp's conjecture on the zeros of Toeplitz determinants of rising factorial series
Henry Zweiman
arXiv10 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Toeplitz determinants for starlike mappings of order α in several complex variables
Shantanu Panja, Sujoy Majumder, Abhijit Banerjee
arXiv11 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
One-Dimensional Symmetry Reductions of the Monge-Ampère Equation on Some Semi-Homogeneous Convex Cones in ℝ⁴
Amir Valiakhmetov, Ewsey Obzherin, Alexey Chernov +1
arXiv19 Oct 2026PreprintCC BY 4.0Full textOptimisation & controlarXiv

Cite

BibTeX RIS

BibTeX

@misc{alekseyev2026symmetry,
  title = {{Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents}},
  author = {Max A. Alekseyev and Dmitry I. Khomovsky},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23224},
  eprint = {2609.23224},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23224},
}

RIS

TY  - UNPB
TI  - Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents
AU  - Max A. Alekseyev
AU  - Dmitry I. Khomovsky
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23224
UR  - https://arxiv.org/abs/2609.23224
AB  - This paper studies symmetry-induced reductions of scalar recurrence complexity for balanced banded Toeplitz determinants and permanents. Three mechanisms emerge: determinant state and spectral compression, exceptional permanent--determinant conversion, and symmetries of permanent state spaces. For symmetric determinants, straightening reduces the row-column states from binom2mm to the Catalan number C_m+1; primitive symplectic weight compression then leaves (3^m+1)/2 distinct autonomous modes. Skew symmetry has a parallel compound/Hodge explanation: the middle exterior representation splits into two Hodge halves of dimension binom2mm/2, the one-step compound transfer exchanges the halves, and its two-step restriction has 3^m−1 generic ternary modes. Thus the full skew bound is 2·3^m−1. Widom--Hankel arguments prove generic scalar minimality in both symmetry classes, and every even skew Toeplitz determinant admits an explicit half-size square factorization. For the zero-diagonal pentadiagonal support, a Pólya--Kasteleyn signing converts every permanent to a determinant. Among consecutive zero-diagonal two-sided bands, universal entrywise conversion---and separately Toeplitz diagonal-wise conversion---occurs only in the Hessenberg families and this pentadiagonal case. Paired renewal identities recover all restored-diagonal determinant and permanent layers. For permanents, transposition gives the open symmetric bound (binom2mm+2^m)/2, while cyclic defect-sector pairing gives (4^m+binom2mm)/2. Skew-symmetry forces odd-size vanishing and corresponding even-subsequence bounds. In semibandwidth two the open symmetric bound is generically sharp; higher-semibandwidth permanent minimality is left separate from the symmetry reductions proved here.
ER  -

Abstract

Related

Cite