The basis of Schubert classes for the cohomology ring $H^*(\text{Gr}(m,N))$ of the Grassmannian of $m$dimensional subspaces of $\mathbb{C}^N$ is indexed by $L(m,Nm)$, the poset of all partitions fitting inside a $m \times (Nm)$ box. This is the quotient of the powerset $2^{m(Nm)}$ by the action of the wreath product $S(m) \wr S(Nm)$. How does this come from the fact that $\text{Gr}(m,N) \cong U(N) / \big (U(m) \times U(Nm) \big )$? Can this be extended to other homogeneous spaces?

The Schubert classes on G/P are the classes of the Schubert varieties, which are the closures of the Schubert cells, each of which contains a unique Tfixed point. The Tfixed points on G/P are the images of Tfixed points on G/B (since T acts on the fiber, which is a projective variety, hence itself has a Tfixed point by Borel's theorem). Up on G/B, the Tfixed points are exactly of the form N_G(T)B/B, so indexed by the Weyl group W_G = N_G(T)/T. Down on G/P, they group together by the Weyl group W_P = N_P(T)/T, so they're indexed by W_G/W_P. Which is exactly what you observed in the G/P = Grassmannian case. (Actually you asked about compact groups, so K/L where K is compact and L is compact of the same rank, which includes some cases like S^4 = SO(5)/SO(4) that is not of the form G/P for G complex and P a parabolic. Then there's still a basis of "Schubert classes", indexed by W_K/W_L similarly.) 


"SchurWeyl duality" says that the permutation group S(k) is dual to the unitary group U(n). Following Wikipedia, Both of these groups act on the space $\mathbb{C}^n \otimes \mathbb{C}^n \otimes \dots \otimes \mathbb{C}^n$, with k factors.
SchurWeyl duality is the decomposition of this doublerepresentation $$ \mathbb{C}^n \otimes \mathbb{C}^n \otimes \dots \otimes \mathbb{C}^n = \sum_D \pi_k^D \otimes \rho_n^D$$ into tensor products of irreducible representations of S(k) and U(n). D runs over young diagrams with k boxes and up to n rows. Each of the young diagrams appears once. Your questions clearly involved the duality between the permutation and unitary groups, though I couldn't explain to you how. You might also see Fulton's book, Young Tableaux. 

