James's submodule theorem over the complex numbers

Given: $n\ge 0$, $\lambda ⊢n$, and an $S_n$-submodule $U\subseteq M^{\lambda }$.

The tabloids form a basis of $M^{\lambda }$, and the $S_n$-action extends linearly from the tabloid action (Young subgroups, tabloids, and permutation modules).

An $S_n$-submodule is a linear subspace stable under each element of $S_n$ (Subrepresentations, direct sums of representations, and irreducibility).

The column antisymmetrizer, polytabloid, and Specht space are

for each $\lambda$-tableau $t$ (Column antisymmetrizers, polytabloids, and Specht modules).

For every $t$, ${\kappa }_t{M}^{\lambda }=\mathbb{C}{e}_t$ and $e_t\mathrm{\ne }0$ (The antisymmetrizer image in its own tabloid module is one-dimensional).

The Specht space is generated as an $S_n$-module by any one polytabloid $e_t$ (Polytabloid covariance and the column sign rule).

The tabloid product is conjugate-linear in its first argument and linear in its second (Invariant Hermitian product on a tabloid module).

Each ${\kappa }_t$ is self-adjoint for the tabloid product (Invariant Hermitian product on a tabloid module).

This Hermitian product is positive definite: if $x\mathrm{\ne }0$, then $⟨x,x⟩>0$ (Invariant Hermitian product on a tabloid module).

The orthogonal complement is $(S^{\lambda }{)}^{\perp }=\{v:⟨v,s⟩=0\text{ for every }s\in S^{\lambda }\}$ (Orthogonality and the orthogonal complement).

Every shape has a canonical standard row-filled tableau $t_0$; for $n=0$ it is the empty tableau (Young subgroups, tabloids, and permutation modules).

No form of the Axiom of Choice is used. The first branch uses only witnesses to one existential statement, and all group-algebra sums are finite.