带状Toeplitz行列式与积和式的对称约化与递归度

中文摘要 AI 辅助

本文研究平衡带状Toeplitz行列式与积和式的标量递归复杂度的对称诱导约化。共出现三种机制：行列式状态与谱压缩、例外积和式-行列式转换、以及积和式状态空间的对称性。对于对称行列式，拉直操作将行列状态从$\binom{2m}{m}$约化为Catalan数$C_{m+1}$；原始辛权重压缩随后留下$(3^m+1)/2$个不同的自治模式。斜对称具有平行的复合/Hodge解释：中间外表示分裂为两个维度为$\binom{2m}{m}/2$的Hodge半部，单步复合转移交换两半，其两步限制具有$3^{m-1}$个一般三元模式。因此完整的斜对称界为$2\cdot3^{m-1}$。Widom-Hankel论证证明了两个对称类中的一般标量最小性，且每个偶数斜对称Toeplitz行列式具有显式的半尺寸平方分解。对于零对角五对角支撑，Pólya-Kasteleyn符号化将每个积和式转换为行列式。在连续的零对角双侧带中，通用逐项转换——以及单独的Toeplitz逐对角转换——仅出现在Hessenberg族和此五对角情形中。配对更新恒等式恢复了所有恢复对角行列式和积和式层。对于积和式，转置给出开放对称界$(\binom{2m}{m}+2^m)/2$，而循环缺陷扇区配对给出$(4^m+\binom{2m}{m})/2$。斜对称强制奇数尺寸消失和相应的偶数子序列界。在半带宽为2时，开放对称界一般是尖锐的；更高半带宽的积和式最小性与此处证明的对称约化分开处理。

英文摘要

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 $\binom{2m}{m}$ 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 $\binom{2m}{m}/2$, 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 $2\cdot3^{m-1}$. 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 $(\binom{2m}{m}+2^m)/2$, while cyclic defect-sector pairing gives $(4^m+\binom{2m}{m})/2$. 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.

发表机构

• The George Washington University（乔治华盛顿大学）

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