Factorization and vanishing of Schur polynomials twisted by roots of unity and reciprocal pairs

Abstract

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, already Littlewood's. 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 Ayyer and Behrend single out, 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 an independence criterion of Ayyer-Kumari: 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.

Disclosure

“t answers for which claim. The same scripts and the same archived output are also kept at https://github.com/karlesmarin/ schur-orbit-and-reciprocal-pair, which is a convenience: the ancillary files carry ev- erything the paper appeals to. Generative AI (Claude, Anthropic) was used throughout as a research assistant — for literature search, for writing and checking the ancillary code, and on the prose. The mathematics, and the responsibility for it, are the author’s.”

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2026-08-23T05:07:51.365568Z

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