= Semistandard homomorphism theorem
A filling $T$ of shape $\lambda$ and content $\alpha$ defines a <module homomorphism> $\Theta_T:M^\lambda\to M^\alpha$: given a numbered tableau, sum over distinct assignments obtained by permuting $T$ within its rows, and put each numbered entry into output row $j$ when its assigned label is $j$. Restricting to $S^\lambda$ gives $\widehat\Theta_T$. For <Semistandard Young tableaux> $T$, these restrictions are linearly independent over every <field>. They form a basis of $\operatorname{Hom}_{FS_n}(S^\lambda,M^\alpha)$ if the <characteristic of a field> is not two, or if $\lambda$ is a two-regular <partition of an integer>. For two-singular shapes in <characteristic two>, they need not span. Column antisymmetrization and triangular leading <tabloids> establish independence; straightening the images of a cyclic <polytabloid> establishes spanning under the stated hypotheses.
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