I'm trying to understand Schubert cells. I've just seen the definition and its connection with Young diagrams. What does the closure of a Schubert cell look like? I'm having trouble how to even think about them. What do I need to do to show that one cell is contained in the closure of another?
2025-01-13 07:53:12.1736754792
closure of a Schubert cell
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If you think of Schubert cells as parametrized by the corresponding Weyl group $W$, you are asking what is the structure of the graph with vertices $w \in W$ and an arrow $w \to y$ if $X_w \subset \overline X_y$ and $\dim X_y - \dim X_w = 1$. It turns out that such a graph is exactly the Bruhat graph. For a reference I'm not sure but in type $A$ it's easy to see it by hands.
However, another aspect of $\overline X_w$ is that they carry interesting singularities. To understand these you need a more carefuly analysis. I'm not sure what can be said in general, but there is a famous result saying that closure of Schubert cells have rational singularities (it can be proved using a natural resolution namely the Samuelson-Bott resolution).
You might also be interested by the last chapter of Fulton's "Young Tableaux" that covers Schubert varieties.