I have a question about Example 6.19 (p. 70) in Harris' Algebraic Geometry: A First Course.
There is stated that the Fano variety
$$F_k(X) := \{\Lambda \in \mathbb{G}(k,n) \ \vert \ \Lambda \subset X \} \subset \mathbb{G}(k,n)$$
(... the later space is the Grassmannian of $k$-planes; we can consider $G(k+1,n+1)$ as usual Grassmannian and $\mathbb{G}(k,n) $ as it's projective version) associated to a variety $X \subset \mathbb{P}^n$ is a variety; ie given by a vanishing set of polynomials.
The text states:
To see that $F_k(X)$ is indeed a variety, observe first that it is enough to do this in case $X$ is the hypersurface given by a single polynomial $G(Z)$. Otherwise consider the intersection. Also, we can work locally: we restrict our attention to the affine open subset $U \subset G(k+1,n+1)$ of $(k+1)$-planes $\Lambda \subset K^{n+1}$ complementary to a given fix $(n - k)$-plane $\Lambda_0$ and exhibit explicitly equations for $F_k(X) \cap U \subset U \cong K^{(k+l)(n-k)}$. (later was showed before).
We start by choosing a basis $v_0(\Lambda), ... , v_k(\Lambda)$ for each $\Lambda \in U$ by taking vectors $v_0,..., v_k \in V$ that, together with v_0(\Lambda), span all of $K^{n+1}$, and setting
$$ v_i(\Lambda):= \Lambda \cap (\Lambda_0 +v_i) $$
Earlier was shown that the coordinates of these vectors are regular functions on $U$. Now, we can view the homogeneous polynomial $G$ of degree $d$ as an element of $Sym^d(K^{n+1}) \subset (K^{n+1})^{\otimes d}$ and set, for each multi-index $I=\{i_1,..., i_d \}$,
$$ a_I(\Lambda) :=G(v_{i_1}(\Lambda),..., v_{i_d}(\Lambda)) $$
Below it is also remarked an "alternative" interpretation of this strange term on the right hand side:
(to put it another way, the $a_I$ are the coefficients of the restriction of $G$ to $\Lambda$, written in terms of the basis for $\Lambda$ dual to the basis $\{ v_1(\Lambda), ... , v_k(\Lambda)\}$).
Question: I not understand how to "read" and interpret the notation $G(v_{i_1}(\Lambda),..., v_{i_d}(\Lambda)) $. Does it mean that one "inserts" somehow the $v_{i_1}(\Lambda),..., v_{i_d}(\Lambda)$ in $G$? If yes, but it make no any sense to insert these $v_{i_j}(\Lambda)$ simultaneously as "arguments" in polynomial $G$. Which meaning does this strange notation has else?
(Note that in "another way ..." remark is given an alternative interpretation which I understand so far. But what mean the author originally in the "initial way" and not the "another way" by this expression $G(v_{i_1}(\Lambda),..., v_{i_d}(\Lambda)) $? It seems that this "another way ..." part provides an alternative interpretation of $G(v_{i_1}(\Lambda),..., v_{i_d}(\Lambda)) $, but what is the first one Harris here presents?