Secant varieties of flag varieties via Schur apolarity

We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity–fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of $\mathrm{Fl}(1,2;V_n)$ embedded by $\mathcal{O}(1,1)$: the only defective cases are ${\sigma }_2(\mathrm{Fl}(1,2;V_3))$ and ${\sigma }_3(\mathrm{Fl}(1,2;V_4))$, both of defect one.