as follows.

Consider the chain of subspaces $$F_V = \bigg(
{\{0\}}=F_0 \cap V \subset F_1 \cap V \subset \cdots \subset F_{n-1} \cap V
\subset F_n \cap V = V \bigg).$$ Since $\dim V$ may be less than or equal to $\dim W$, this will no longer be a flag, as some of the $F_{i-1} \cap V \subset F_i \cap V$ may become equalities. However by eliminating all repeated elements, one gets a full flag on $V$. It is easy to check that the elements we keep correspond precisely to the [[‘$1$’]{}s ]{}in the Schubert position of $V$ with respect to $F$.

Similarly we can construct a full flag on $W/V$. Let $\Pi:W \to W/V$ denote the quotient map. Then $$F_{W/V}=
\bigg(
{\{0\}}=\Pi(F_0) \subset \Pi(F_1) \subset \cdots \subset \Pi(F_{n-1})
\subset \Pi(F_n) = W/V \bigg).$$ Again by eliminating repeated elements we obtain a full flag. In this case, we keep the elements corresponding to the [[‘$0$’]{}s ]{}in the Schubert position of $V$.

The generalised Horn conjecture
-------------------------------

We can now state Horn’s conjecture in the language of Schubert calculus. Actually we will give the slightly more general statement which appears in [@B].

To simplify the notation in the statement a little, any 