Let $U$ be the maximal upper unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Then $$ U(w) = \{(a_{ij}) \in U: a_{ij} = 0, \text{ if } i<j, w^{-1}(i) < w^{-1}(j) \}. $$
My question is: are there some similar characterization of $U/U(w)$ as the above formula? Thank you very much.