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?