EXTREMAL NEGATIVITY IN SYMMETRIC-GROUP CHARACTER TABLES: SOLVED SECTORS AND AN ARITHMETIC SQUARE-CORE REDUCTION
Abstract
For an irreducible character χλ of Sn, let N− n (λ) = #{µ⊢n: χ λ(µ) <0}, with one entry counted for each conjugacy class. OPAC-038 asks whether the sign representation eventually maximizes N− n . We show that the sign row has exactly (p(n)− q(n))/2 negative entries and reduce fixed-degree extremality to the parity-balance inequality E−(λ) ≤O−(λ) + O0(λ). We prove this inequality for every self-conjugate row in every degree. For two-row characters we obtain a staircase-core positivity criterion and a uniform solved range k<(√6/(4π)−ε)√nlog n. The pairs (n−1,1),(2,1n−2) and (n−2,2),(2,2,1n−4) are settled in every degree. A uniform character-polynomial argument further shows that, for every c < 1/2, OPAC holds eventually whenever either n−λ1 or n−λ′ 1 is at most clog n/log log n. For the remaining non-self-conjugate rows we isolate an arithmetic square core. After removing the q(n) distinct-odd split columns and the q(n) self-conjugate rows, the nor- malized nonsplit-even block is square and invertible, as is the odd block. The split-even columns form a q(n)-dimensional graph correction. After Frobenius scaling this correction is integral and satisfies a determinant-one Gram identity. Both square cores are invertible over every Fℓ with ℓ>n, the canonical odd-to-nonsplit-even bridge is an isometry outside at most q(n) directions, and the two cores admit an orthogonal phase normal form together with a canonical mod-2 congruence. An explicit real square-core countermodel proves that these real metric identities alone cannot force the required sign inequality. Finally, the parity deficit is exactly an integral pairing of a virtual character with the conjugation character. Exact computation verifies OPAC through n = 29. The global conjecture remains open.Certificates
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