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3. (No, 3. Silly site software!) Belkale has the best (least decategorified) proof I've seen. He takes three Schubert cycles meeting transversely, and for each point of intersection, constructs an actual invariant vector inside the corresponding triple product of representations. The set of such vectors is then a basis.

3. (No, 3. Silly site software!) Belkale has the best (least decategorified) proof I've seen (which appeared separately as _Invariant theory of $GL(n)$ and intersection theory of Grassmannians _ IMRN 2004 Issue 69 (2004) 3709–3721, https://doi.org/10.1155/S107379280414155X). He takes three Schubert cycles meeting transversely, and for each point of intersection, constructs an actual invariant vector inside the corresponding triple product of representations. The set of such vectors is then a basis.

There are several rings-with-bases to get straight here. I'll explain that, then describe three serious connections (not just Ehresmann's proof as recounted in the OP).

There are several rings-with-bases to get straight here. I'll explain that, then describe three serious connections (not just Ehresmann's Lesieur's proof as recounted in the OP).

1. Harry Tamvakis' proof is to define a natural ring homomorphism $Rep({\bf Vec}) \to H^*(Gr(d,\infty))$, applying a functor to the tautological vector bundle, then doing a Chern-Weil trick to obtain a cohomology class. (It's not just the Euler class of the resulting huge vector bundle.) The Chern-Weil theorem is essentially the statement that Harry's map takes alternating powers to special Schubert classes. So then it must do the right thing, but to know that he essentially repeats the Ehresmann proof.

2. Kostant studied $H^* (G/P)$ in general, in "Lie algebra cohomology and something something Schubert cells" (sorry!), by passing to the compact picture $H^* (K/L)$, then to de Rham cohomology, then taking $K$-invariant forms, which means $L$-invariant forms on the tangent space $Lie(K)/Lie(L)$. Then he complexifies that space to $Lie(G)/Lie(L_C)$, and identifies that with $n_+ \oplus n_-$, where $n_+$ is the nilpotent radical of $Lie(P)$. Therefore forms on that space is $Alt^* (n_+) \otimes Alt^* (n_-)$.

It's fun to see what's going on in the Grassmannian case -- $L = U(d) \times U(n-d)$, $n_+ = M_{d,n-d}$, and $Alt^* (n_+)$ contains each partition (or rather, the $U(d)$-irrep corresponging) fitting inside that rectangle tensor its transpose (or rather, the $U(n-d)$-irrep).

1. Harry Tamvakis' proof is to define a natural ring homomorphism $Rep({\bf Vec}) \to H^*(Gr(d,\infty))$, applying a functor to the tautological vector bundle, then doing a Chern-Weil trick to obtain a cohomology class. (It's not just the Euler class of the resulting huge vector bundle.) The Chern-Weil theorem is essentially the statement that Harry's map takes alternating powers to special Schubert classes. So then it must do the right thing, but to know that he essentially repeats the Ehresmann proof.

2. Kostant studied $H^* (G/P)$ in general, in "Lie algebra cohomology and generalized Schubert cells", by passing to the compact picture $H^* (K/L)$, then to de Rham cohomology, then taking $K$-invariant forms, which means $L$-invariant forms on the tangent space $Lie(K)/Lie(L)$. Then he complexifies that space to $Lie(G)/Lie(L_C)$, and identifies that with $n_+ \oplus n_-$, where $n_+$ is the nilpotent radical of $Lie(P)$. Therefore forms on that space is $Alt^* (n_+) \otimes Alt^* (n_-)$.

It's fun to see what's going on in the Grassmannian case -- $L = U(d) \times U(n-d)$, $n_+ = M_{d,n-d}$, and $Alt^* (n_+)$ contains each partition (or rather, the $U(d)$-irrep corresponding) fitting inside that rectangle tensor its transpose (or rather, the $U(n-d)$-irrep).