AI 中文总结
本文研究单位根扭曲的Schur多项式与互反对的因式分解，给出其求值公式、消失准则，推广Ayyer-Kumari的独立性准则，解读t=2时的平面分拆枚举，并证明部分逆命题。
AI 中文摘要
设μ_t为全体t次单位根构成的集合，(z,z⁻¹)为自由互反对。我们确定分拆s_λ(μ_t,z,z⁻¹)能体现的内容：三个整数的多重集与一个符号，别无其他，因此所有在该多重集上一致的任意大小分拆都具有相同的取值。对于每个t≥2和每个分拆λ（对其形状无任何假设），该求值结果要么是带符号的乘积，该乘积恰好包含三个因子且分母固定，要么为零，这三个参数可从t-商中读出。证明过程是沿双交替式的t个冻结行进行拉普拉斯展开，结合对称群中的一个消去引理，得到的符号与Littlewood在μ_t处的求值结果中的符号一致。由此得到三个推论：1. 一个消失准则：当模t的一个剩余类为空，或两个特定类作为区间同心时，该多项式恰好消失，后者仅在t为偶数时成立；2. Ayyer-Kumari近期独立性准则的推广：对于互反轨迹上的两行形状，该准则恰好新增一个由核与商分类的族；3. 一个枚举解读：当t=2时，对盒内平面分拆的(-1)-枚举由一个保持自由的参数精细化。该因式分解是孤立的：对字母表的四种变形，每一种都会导致其失效，且原因相同。最后一节讨论零轨迹，该轨迹可存在于更多对中。使Ψ_r=s_λ(1,-1,z₁^±¹,…,z_r^±¹)消失的两个条件为：β集具有恒定奇偶性，且λ为宽度奇数的自互补分拆。该方向是Ayyer与Behrend选定的索引族上互补恒等式的推论，新内容是其逆命题，即无其他情况会导致其消失，该逆命题已在一对参数、Littlewood范围内的每个r，以及|λ|≤2r+2时的每个r处得到证明，其余部分为猜想，已在表格范围内的所有形状上验证。
英文摘要
Let $μ_t$ be the full set of $t$-th roots of unity. Adjoining $r$ free reciprocal pairs gives a two-parameter family of alphabets; we settle three parts of it. At $r=1$, for every $t\ge2$ and every $λ$ with at most $t+2$ parts, $s_λ(μ_t,z,z^{-1})$ is a signed product of exactly three factors over a fixed denominator, or zero, the arguments read off core and quotient. What it sees of $λ$ is a multiset of three integers and a sign, and exactly that: two partitions of any sizes share a nonzero value if and only if they agree on that datum. The proof is a Laplace expansion along the $t$ frozen rows with one cancellation lemma, and delivers the sign, of which Littlewood's is one factor. At $t=2$ and every $r$, $s_λ(1,-1,z_1^{\pm1},\dots,z_r^{\pm1})$ vanishes exactly when the beta set has constant parity or $λ$ is self-complementary of odd width; that direction is a corollary of complementation over an index family of Ayyer and Behrend, the converse an extremal argument in the degree filtration, modulo one rigidity theorem for Schur products. Equivalently: exactly those $V_λ$ restrict to $O(N,\mathbb{C})$ $\det$-stably. At odd $t$ and every $r$ it vanishes exactly when a residue class is absent, at no external cost. And for every $t$ and $r$, a reflection of the beta set's excess part with one increment hitting its centre forces vanishing. Three consequences of the first. A vanishing criterion: an empty residue class, or two distinguished classes concentric as intervals, the second only for even $t$. An extension of Ayyer-Kumari's independence criterion: on the reciprocal locus it acquires one further family, classified by core and quotient. And at $t=2$ a $(-1)$-enumeration of plane partitions in a box refined by a parameter that stays free. The factorization is isolated: it fails under each of four deformations of the alphabet, for one reason.
Comments76 pages, 21 figures; ancillary scripts and their archived output included
