Positive-characteristic Specht head and cyclotomic crystal labels agree

Now let J(lambda) be the reduction of the characteristic-zero normalized simple of crystal label lambda, constructed using a rational Specht-generator lattice. Its graded word character equals that of the characteristic-zero simple. Choose an extremal word of this common character. In characteristic zero the reconstruction label is lambda; in positive characteristic the same word gives the same canonical B(Lambda) label. The displayed quotient is1 for the characteristic-zero simple, hence [J(lambda):D_crystal(lambda)]_q=1 in positive characteristic.

Independently, the corrected canonical Fock comparison and the upper expansion identify [J(lambda)] with the inverse image of the characteristic-zero upper vector. The restricted upper-transition minor is dominance-unitriangular. Invert it over Z[q,q^{-1}] and use the ACTUAL positive-characteristic cellular-head triangular expansion of restricted Specht classes. This gives [J(lambda)]=[D_cell(lambda)]+sum_{nu<lambda}a_nu(q)[D_cell(nu)]. These coefficients are the actual composition multiplicities because J(lambda) is a module; there is no asserted positivity from matrix inversion alone. Induct over a finite dominance refinement in the fixed root degree. Assume D_cell(nu)=D_crystal(nu) for the smaller labels. Since different crystal labels give different simple orbits, D_crystal(lambda) cannot be any smaller D_cell(nu). Its occurrence in J(lambda) therefore forces D_cell(lambda)=D_crystal(lambda). The coefficient1 forces shift0 as both chosen representatives are self-dual. This proves the CP label comparison before defining adjustment polynomials in the crystal-labelled basis.

The characteristic-zero simple D(lambda) is normalized for finite vector duality, so each word component e(nu)D(lambda) has a symmetric graded dimension: its Laurent polynomial is unchanged by v -> v^{-1}. The constructed J(lambda) has exactly the same graded word character. The descended generator-fixing anti-involution fixes every word idempotent; hence e(nu)D_psi J(lambda) is the graded vector dual of e(nu)J(lambda), and its graded dimension is the same symmetric polynomial. Therefore D_psi J(lambda) and J(lambda) have identical word characters. The completed local extremal-word character detection proves G0 character injectivity over the actual positive-characteristic field, so [D_psi J(lambda)]=[J(lambda)]. This proves bar invariance of its CLASS; no self-dual stable integral lattice is asserted. The Axiom of Choice is inherited through the canonical crystal labels of [F1] and the global-basis comparison of [F2]; the finite character and dominance arguments use no additional choice.