An integral regular summand in every staircase Specht tensor cube
Abstract
For every staircase partition rho=(m,m-1,...,1) of N=m(m+1)/2, we prove that the integral regular module Z[S_N] is a direct summand of (S_Z^rho)^{tensor 3}. Ordered row and column polytabloids give a tensor whose translates form a basis of that summand. A single integral coordinate functional gives an explicit equivariant retraction. The splitting survives base change to every commutative unital ring, including nonflat base change, with no restriction on the characteristic. The proof uses the separating-product mechanism of Bryant and Kovács, Ikenmeyer's triangular palette uniqueness, and a square-deficit calculation from the pinned cyclic-polytabloid construction.
Comments
This is an AI-assisted preprint with internal independent agent proof, contribution and manuscript reviews, not external peer review. Supporting Lean material is partial and does not formalize the complete theorem. Mathematical results from OpenAI/math at revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb are stipulated premises, with exact source interfaces and other credited inputs disclosed in the paper. The literature comparison is bounded and does not certify worldwide priority. Sixteen supporting Lean theorems prove conditional coordinate splitting, equivariance, injectivity, complement, scalar extension of given split maps and integer zero-deficit implication. The staircase-specific separation identity, polytabloids, palettes, support budget, unit sign, all-m instance and multiplicities remain prose proofs. Supporting package will be linked after Zenodo publication. Paper/prose CC BY 4.0; original code/Lean MIT.