(Pre-seminar)  Subvarieties of the flag variety, smoothness, and patterns

Consider the set \$X\$ of all "flags" on the complex vector space \$\mathbb{C}^n\$, a "flag" being an ascending chain of nested vector subspaces of \$\mathbb{C}^n\$, one in each dimension.  \$X\$ has the structure of a smooth algebraic variety, and comes with a natural action of the group \$GL_n(\mathbb{C})\$.  Any subgroup of \$GL_n(\mathbb{C})\$ thus acts on \$X\$, sometimes with finitely many orbits.  In such cases, it is natural to ask for a set of "nice combinatorial objects" which parametrize the orbits.  For example, when we consider the subgroup of \$GL_n(\mathbb{C})\$ consisting of upper-triangular invertible matrices, the orbits are parametrized by permutations of the set \$\{1,\ldots,n\}\$.