EXTREMAL NEGATIVITY IN SYMMETRIC-GROUP CHARACTER TABLES: PARITY GEOMETRY, SUPPORT BARRIERS, AND CRITICAL JUCYS-WEINGARTEN DUALITY

Abstract

This preprint investigates an open problem in algebraic combinatorics: determining whether the sign representation of the symmetric group \(S_n\) maximizes the number of negative entries in its irreducible character row. The paper develops a comprehensive framework combining structural restrictions, asymptotic support barriers, and a new algebraic duality to analyze this extremal negativity phenomenon. The first part of the work establishes exact reductions and geometric constraints on any potential counterexample. These include: • Parity‑balance identities that reformulate the extremal problem in terms of even/odd class contributions.• Hook‑support and missing‑hook barriers, showing that any counterexample must have unusually large hook sets.• Durfee‑rank localization, proving vanishing and support restrictions near the Durfee boundary.• Long‑cycle Ferrers matching and a genus‑zero boundary rank‑one structure for full‑cycle factorizations.• Primitive‑idempotent positivity forcing, which limits how negative values can appear across parity sectors. The second part introduces the paper’s main algebraic contribution: a critical Jucys–Weingarten duality at the singular parameter \(M = n - 1\). At this value, the Jucys–Murphy content polynomial vanishes exactly for the sign representation, producing a corank‑one kernel whose missing eigenvector is the sign row. The paper constructs its reciprocal‑content dual, proving: • an exact rank‑one regularization,• an invertible bordered pair, and• a rectangular complementary‑nullity theorem. These identities convert the extremal negativity problem into a uniform selected‑rank conjecture, equivalent to a bordered critical‑Weingarten rank condition and a dual‑flatness boundary‑value problem. The conjecture has been computationally verified for all irreducible rows for \(5 \le n \le 22\), though the proof remains open. Overall, the paper provides new structural theorems, a novel duality framework, and a unified reduction that connects unweighted sign‑count extremality to a single spectral rank condition.