The parallels between the formulas in Schubert calculus and in the theory of the representations of symmetric groups (par Geissinger-Zelevinsky) are so apparent (e.g. Giambelli formula), that one must wonder how to directly define the co-multiplication on Schubert (co)cells.



How to make Schubert calculus into a Hopf (actually a PCH) algbera?
If you're talking about Grassmannians, then you can use the direct sum maps


$Grass(k,n) \times Grass(k',n') \to Grass(k+k', n+n')$


to get the comultiplication. This induces a map on cohomology the other way. Then take the limit as $n,n' \to \infty$ and then take the limit $k,k' \to \infty$. This fixes two problems: 1) the Grassmannians aren't the same in the finite case, and 2) the values of $k,n$, etc. give truncations of the ring of symmetric functions, so you need to remove that restriction.


According to [Symmetric polynoms are Hopf algebra ? What for one needs co-product ?](https://mathoverflow.net/questions/85985/symmetric-polynoms-are-hopf-algebra-what-for-one-needs-co-product) the bialgebra is enough to get the whole Hopf structure.


Positivity (to get the PSH algebra structure) follows from geometric considerations (i.e., all structure coefficients are intersection numbers).


