Schur module ($D_{\lambda }(V)$)

A Schur module is the value of a Schur functor on a vector space. Over the complex numbers it is either zero or an irreducible polynomial representation of the general linear group. Its character is the Schur polynomial $s_{\lambda }$ in the eigenvalues, and its highest weight is $\lambda$ padded with zeros.

A column of length greater than $m$ antisymmetrizes more than $m$ vectors, giving zero. When there are at most $m$ rows, place the $i$th basis vector in every tensor position of row $i$. Row symmetrization multiplies by a nonzero factorial product, and column antisymmetrization is nonzero because the vectors in every column are distinct. This proves the precise nonvanishing criterion for a Schur module.