The dimension of the Grassmannian cohomology ring $H^*(\mathrm{Gr}_{n,d})$ and the fundamental $\frak{sl}_n$-representation $V_{\pi_d}$

Question

The vector space dimension of the cohomology group of the $2$-plane Grassmannian $\mathrm{Gr}_{2,n}$ is given by the number of tuples $(\lambda_1,\lambda_2)$ satisfying $$ n - 2 \geq \lambda_1 \geq \lambda_2 \geq 0. $$ Explicitly this is given by $$ \binom{n}{2}. $$ This also happens to be the dimension of $V_{\pi_2}$ the second fundamental representation of $\frak{sl}_n$. I am guessing this is not an accident, especially since the $2$-plane Grassmannian corresponds (in the usual way) to $V_{\pi_2}$.

Does this extend to the general identity $$ \mathrm{dim}(H^{*}(\mathrm{Gr}_{d,n})) = \mathrm{dim}(V_{\pi_d})? $$ If it does, then what is a conceptual explanation for this?

EDIT: Since $V_{\pi_d}$ is isomorphic to the exterior power $$ \Lambda^d(V_{\pi_1}) $$ and $V_{\pi_1}$ is of dimension of $n$, we see that the RHS of the claimed identity is the binomial coefficient $$ \binom{n}{d}. $$ It follows from the general formula given in this answer that the LHS is the same binomial coefficient. Thus the identity does indeed extend from $2$-planes to $d$-planes. So the question is if there is a conceptual reason for this . . .

Answer 1

This is very standard. For a compact complex variety admitting a cell decomposition, the (co)homology is the free Abelian group generated by the cells (over $\mathbb{C}$ there is no room for the differential in the cellular complex). In the cellular decomposition of the Grassmannian, the (Schubert) cells are indexed by possible reduced row echelon forms of a $d\times n$-matrix, that is by possible positions of pivots. To the cell having pivots in positions $i_1,\ldots,i_d$ you can assign the basis element $e_{i_1}\wedge\cdots\wedge e_{i_d}$ in $\Lambda^d(V_{\pi_1})$.

Answer 2

Harry Tamvakis gives an explanation of the relation between Grassmannian cohomology and representations of SL_n in The connection between representation theory and Schubert calculus.

One fundamental obstacle for explaining this phenomenon is that there is not an analogous relationship between the cohomology rings of other flag varieties (Grassmannian-analogues for other Lie groups) and the representations of those Lie groups. For Tamvakis, the key fact that makes his explanation go is that GL_n is dense in its own Lie algebra, which is not generally true for other Lie groups. (At least this is what I remember; it's been a while since I read the paper carefully.)

Answer 3

This is a special case of geometric Satake. I have to admit, I'm really struggling to find a place where this theorem is stated in an elementary-ish statement that makes this clear and am totally failing. By Mirkovic-Vilonen, Thm. 7.3 (and the discussion below), the irreps $V_{\lambda}$ of any complex algebraic group can be written as the intersection cohomology of the closure $\overline{\check{G}[[t]] \cdot t^{\lambda}\cdot \check{G}[[t]]}/G[[t]]\subset G((t))/G[[t]]$ in the affine Grassmannian. If $\lambda$ is minuscule, then this orbit is already closed and smooth, so the intersection cohomology is usual cohomology. Furthermore, it is of the form $\check{G}/\check{P}$ for $\check{P}$ the parabolic corresponding to the stabilizer of the minuscule weight in the Weyl group.

For $G=\check{G}=GL_n$, these are all Grassmannians; more generally, every minuscule representation is isomorphic to the cohomology of the corresponding cominuscule flag variety of the dual group.

Answer 4

I accidentally (looking for something else) came across another paper where a very elegant explanation is given:

Dan Laksov, Anders Thorup: Schubert Calculus on Grassmannians and Exterior Powers

Indiana University Mathematics Journal, Vol. 58, No. 1 (2009), pp. 283-300

They introduce the "$d$-th factorization algebra" that "controls" the ways to factor a polynomial $p(x)$ of degree $n$ into a product of factors of degrees $d$ and $n-d$, relate the $d$-th exterior power of the quotient by $p(x)$ to the $d$-th factorization algebra (Main Theorem, see Sec.0.6), and also relate the $d$-th factorization algebra to the Schubert calculus (Sections 4 and 5).