The sharp minimum multiplicity in a staircase tensor square

For the staircase partition ρ_m = (m,m−1,…,1), assuming the specified pinned cyclic-polytabloid source theorems, every irreducible complex representation of S_{m(m+1)/2} other than the trivial and sign representations occurs in the ordinary diagonal tensor square of S^{ρ_m} with multiplicity at least m−1, for every m ≥ 2. Equality holds for the partition (m(m+1)/2−1,1). The proof evaluates an inherited recursion for a specified cyclic submodule using an asymmetric invariant, predecessor diagrams, explicit rigged configurations and exact finite initial calculations. The estimate is presented as new to this project; literature priority is not claimed.

AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review or worldwide-priority certification. Identified OpenAI source inputs at revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb are stipulated true. Lean verifies scoped supporting results only, not the full theorem. Original prose: CC BY 4.0; original code and Lean: MIT; third-party rights preserved. See the supporting archive for precise assumptions, coverage, provenance and corrections.