Let $X$ be the Grassmannian variety $\operatorname{Gr}(k,n)$ of $k$-planes in $\mathbb{C}^n$. I'm aware of two ways to describe its $T$-equivariant cohomology:


1. (Quotient ring) $H\_T^\*(X)=\Lambda[e\_1(x|t),\dotsc,e\_k(x|t)]/(h\_{n-k+1}(x|\dotsc,h\_{n}(x|t))$
where $\Lambda=\mathbb{C}[t\_1,\dotsc,t\_n]$, and $e\_i(x|t)$ (resp. $h\_i(x|t)$) are the elementary (resp. complete homogeneous) factorial symmetric polynomials.
2. (GKM) The fixed points $X^T$ are indexed by Young diagrams $\lambda$ inside the $k\times (n-k)$ box, and $H\_T^\*(X)$ is a certain subring of the ring $\bigoplus\_{\lambda \in X^T}\Lambda$. Specifically, a tuple $(f\_\lambda)$ in this subring must satisfy certain congruences of the form $t\_i-t\_j \mid f\_{\mu} - f\_{\nu}$.



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> What is the (ring) isomorphism from 1. to 2.?
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This is definitely somewhere in the literature, but I haven't been able to find a reference stating the explicit map. A more general reference for e.g. partial flag varieties would also be greatly appreciated.


Edit: I believe the right map is the evaluation given by Hunter Spiker's comment, I'm looking for a reference stating that fact.



Explicit isomorphism between two realizations of $H^*_T(\operatorname{Gr}(k,n))$ (reference request)
$\def\Fl{\mathcal{F}\ell}$I'd been hoping someone else would answer this, since I don't have a good reference to cite and I tend to make minor errors in things like this, but some answer is better than none. As in your question, $\Lambda := H\_T^{\ast}(\text{point}) \cong \mathbb{Z}[t\_1, t\_2, \ldots, t\_n]$.


Let's do the full flag manifold first. The corresponding descriptions are


(1) $H\_T^{\ast}(\Fl) \cong \Lambda[x\_1, x\_2, \ldots, x\_n]/\langle e\_j(x) - e\_j(t) \rangle$. In other words, we take the symmetric functions in the $x$-variables and set them equal to the symmetric functions in the $t$-variables.


(2) The fixed points $\Fl^T$ are indexed by the symmetric group $S\_n$. The ring $H\_T^{\ast}(\Fl)$ is a subring of $\bigoplus\_{w \in S\_n} \Lambda$. Namely, if $f$ is a function from $S\_n$ to $\Lambda$ then we must have $f(u) \equiv f((ij) \cdot u) \bmod t\_i - t\_j$. The subring $H\_T^{\ast}(\text{point})$ is identified with the functions which take the same value at every permutation.


Given $g(x\_1, x\_2, \ldots, x\_n, t\_1, t\_2, \ldots, t\_n)$ in the first ring, we define an element of the second ring by
$$f(w) = g(t\_{w(1)}, t\_{w(2)}, \ldots, t\_{w(n)}, t\_1, t\_2, \ldots, t\_n).$$
So the generator $x\_j$ turns into the function $w \mapsto t\_{w(j)}$ and $t\_j$ is the function which is $t\_j$ on every permutation.


The projection $\Fl\_n \to G(k,n)$ induces an injection $H\_T^{\ast}(G(k,n)) \to H\_T^{\ast}(\Fl\_n)$. We can describe $H\_T^{\ast}(G(k,n))$ as a subring of both (1) and (2).


In (1), we take the ring of invariants for $S\_k \times S\_{n-k}$, permuting the $x$-variables and leaving the $t$-variables fixed. This should match your description (1).


In (2), we take the ring of functions which obey $f(u) = f(up)$ for $p \in S\_k \times S\_{n-k}$. In other words, we can think of $f$ as a function on cosets $S\_n/(S\_k \times S\_{n-k})$, which is in easy bijection with $k$-element subsets of $\{ 1,2,\ldots, n \}$, these can be identified with Young diagrams in the $k \times (n-k)$ box in a standard way, giving your (2).


