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Let $G=SL_n$, it is proven that $R:=H_*(Gr_G)\cong \mathbb{C}[\sigma_1,...,\sigma_{n-1}]$ where $\sigma_i$ are of degree $2i$ as a polynomial ring generated by $n-1$ variables and the ring structure comes from the fact the $Gr_G$ is homotopic to $\Omega K$ ($K$ is the maximal compact subgroup of $G$) and $\Omega K$ has a group structure. It is known that $R$ can be viewed as a subring of symmetric function $Sym$ by mapping each $\sigma_i$ to $h_i$ ($i$-th complete homogeneous symmetric function of degree $i$ ).

My question is: Why should we view $R$ as a subring in $Sym$ in such way? What is the geometric reason behind it?

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You should look at the book of Lam, Lapointe, Morse, Schilling, Shimozono and Zabrocki. More specifically, under k-Schur functions and how/why they constitute a basis of $H_*(Gr_{SL_k})$. They mainly work in K-theory, but one of the main results in loc. cit originally proved by Lam is that there is a natural Hopf algebra isomorphism between the Hopf subalgebra of k-Schur functions $\Lambda_{(k)}:=\mathbb{Q}[h_1,\ldots,h_{k-1}]\subset \Lambda$ and $H_*(Gr_{SL_k})$. See the references therein for the Hopf algebra structures.

Geometrically the k-Schurs are just the Schubert basis in the (K-)homology of the affine Grassmannian. Multiplication by $h_i$ is more interesting in this basis, it's given by the k-Pieri rule.

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