Counting cosets in the Quotient of Weyl groups

Question

Let $\Sigma=G/P$ be a flag variety of type $A$,i.e. $G=SL_{n+1}$ and is a stabilizer of the partial flag $0 \subset V_{d_1}\subset \cdots\subset V_{d_r}\subset V_{n+1}$ of length $r$. Let $W$ be its Weyl group which is the symmetric group $S_n$. Then $W_P=<\sigma_i: i \notin \{d_1,\cdots,d_r\}> $ the subgroup of $W$ corresponding to $P$. Let $W^P=W/W_P$ be the set of minimal coset representatives. It is know that the set $W^P$ parametrizes the Schubert varieties in $\Sigma$.

Also recall, that if $\omega_k=\sum_{i=1}^k e_i$ denote the dominant fundamental weights of the weight lattice of $G$ then $\Sigma$ is a projective subvariety of $\mathbb P V_\lambda$. Here $V_\lambda$ is the highest weight representation with of $G$ with highest weight $$\lambda=\sum_{i=j}^r\omega_{d_j}$$.

$\textbf{Question}$: Does there exist a formula to calculate the number of elements in $W^P$ in terms of the above data or in terms of some kind of tableau associated to the highest weight representation $V_\lambda$.

To make the question more clear, we know that for the Grassmannian $Gr(k,n)$ the number of cosets $W^P$ are $n\choose{k}$. Moreover they are the number of partitions $\mu$ in $k \times (n-k)$ boxes of type $$n-k\ge \mu_1\ge \cdots \ge \mu_k\ge 0. $$ Does there exist similar descriptions in the case of general $G/P$? If yes, can we calculate the number of coset representatives of various lengths in $W^P$ combinatorially.

The answer to same question in types $B, C, D$ will indeed be a bonus.

Answer 1

Every standard parabolic subgroup can be encoded by crossing out some nodes in the Dynkin diagram of $\mathfrak{g}$. The subgroup $W_P$ is then the Weyl group of a semisimple Lie algebra whose Dynkin diagram is that of $\mathfrak{g}$ with the corssed nodes deleted. By the Lagrange theorem, the number of cosets (and hence the number of coset representative) is then just $|W|/|W_P|$ and this can be computed directly using formulas for number of elements of $W$ for any simple Lie algebra $\mathfrak{g}$. See e.g. Wikipedia. So for example for Grassmannian you get that $W_P$ is the Weyl group of a Lie algebra isomorphic to $\mathfrak{sl}(k)\oplus\mathfrak{sl}(n-k)$ and thus it has $k! (n-k)!$ elements.

Description in terms of tableaus or some such is a trickier business because if done properly, it should actually count not only $|W^P|$ but elements of $W^P$ of certain length, i.e. the Betti numbers of $\Sigma$. I think that's a difficult problem in general.