On the cohomology ring of the Grassmannian

Question

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 $L(m,N-m)$, the poset of all partitions fitting inside a $m \times (N-m)$ box. This is the quotient of the powerset $2^{m(N-m)}$ by the action of the wreath product $S(m) \wr S(N-m)$. How does this come from the fact that $\text{Gr}(m,N) \cong U(N) / \big (U(m) \times U(N-m) \big )$? Can this be extended to other homogeneous spaces?

Answer 1

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 $T$-fixed point. The $T$-fixed points on $G/P$ are the images of $T$-fixed points on $G/B$ (since $T$ acts on the fiber, which is a projective variety, hence itself has a $T$-fixed point by Borel's theorem).

Up on $G/B$, the $T$-fixed 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 $P$ parabolic. Then there's still a basis of "Schubert classes", indexed by $W_K/W_L$ similarly.)

Answer 2

"Schur-Weyl 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.

• S(k) acts by permuting the factors $\sigma(v_1 \otimes v_2 \otimes \dots \otimes v_n) = v_{\sigma(1)} \otimes v_{\sigma(2)} \otimes \dots \otimes v_{\sigma(n)}$
• U(n) acts like an n x n matrix on the $\mathbb{C}^n$, $g(v_1 \otimes v_2 \otimes \dots \otimes v_n) = g(v_1) \otimes g(v_2) \otimes \dots \otimes g(v_n) $

Schur-Weyl duality is the decomposition of this double-representation $$ \mathbb{C}^n \otimes \mathbb{C}^n \otimes \dots \otimes \mathbb{C}^n = \sum_D \pi_k^D \otimes \rho_n^D$$ 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.