In Nakajima's [Geometric construction of algebras](http://www.kurims.kyoto-u.ac.jp/~nakajima/TeX/HongKong/hongkong.pdf)(pages 3-7), he uses the subalgebra of the convolution algebra of $Gr(k^N)\times Gr(k^N)$ invariant under $GL\_N$ action to construct $U\_q(sl\_2)$. To do this, he finds imposes relations to form some special operators, which he shows are isomorphic to the generators of $U\_q(sl\_2)$. In particular, he uses characteristic functions on certain subsets of $k^N$ with dimensionality conditions. These conditions smell like Schubert conditions.



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> Can we see these as the Schubert conditions in some way?
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I have compared them, but I don't see the link.


EDIT: Read the comments below for a related paper.



Can we see the geometric realization of $U_q(sl_2)$'s relations as Schubert Conditions?
Well, it depends what you mean; there's certainly a close link.


Any $GL\_N$ invariant closed subset of the product of two Grassmannians can be described as a family of Schubert varieties. Consider the projection to the first factor. This must be surjective, since $GL\_n$ acts transitively on the Grassmannian. The fiber over a subspace $V\subset \mathbb{C}^N$ must be a closed subset of the other Grassmannian which is preserved by the subgroup stabilizing $V$; that is, it must be a Schubert variety of the form $\{U\subset \mathbb{C}^N | \dim(U)=m, \dim(U\cap V)\geq h\}$.


The same argument can easily be applied to any product of two partial flag varieties with the same conclusion (unfortunately, it completely breaks down for more factors).


