Consider the Complete Flag Variety $Fl_4(\mathbb C)$ and its Schubert Variety given by the permutation $(1234) \to (3142)$, i.e. its highest dimension cell can be parametrized by $$\begin{bmatrix} * & 1 & 0 & 0 \\ * & 0 & * & 1 \\ 1 & 0 &0& 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$
We want to show that it is a singular variety in two ways:
$1)$ Showing that Poincaré Duality fails,
$2)$ Using the standard equations defining the Schubert varieties: $$X=\{V_{\bullet}\ | dim (E_i \cap V_j) \geq w_{i,j}\ \forall i,j \in(1,4)\} $$ where $E_1 \subset E_2 \subset E_3 \subset E_4$ is the standard flag of a fixed base, and $w_{i,j}$is the rank of the principal (i,j) minor of the $4 \times 4$ permutation matrix $w$ associated to our permutation.
Thoughts: $1)$ Since there are one $0$-cell, three $2$-cells, three $4$-cells and one $6$-cell I don't get why Poincaré Duality should fail.
$2)$ The only thing I can think of is to take the several equations defining the flag and the several planes defining the Schubert Variety and intersect everything, then derive and find a singularity, but I guess this is much much complicated! There must be something simpler I'm not thinking of.
Thanks!