Explicit isomorphism between two realizations of $H^*_T(\operatorname{Gr}(k,n))$ (reference request)

Question

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 $t_i-t_j \mid f_{\mu} - f_{\nu}$.

What is the (ring) isomorphism from 1. to 2.?

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.

Answer 1

$\def\Fl{\mathcal{F}\ell}$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).