[proofplan]
We realize the special Schubert variety as a rank-drop locus for the bundle map from the universal subbundle $S$ to a fixed quotient [vector space](/page/Vector%20Space) of $\mathbb C^n$. The chosen index $F_{n-k+1-j}$ makes the expected complex codimension equal to $j$, and the Thom-Porteous degeneracy-locus formula identifies the Poincare dual of this locus with the $j$-th Chern class of the virtual bundle $(\mathbb C^n/F_{n-k+1-j})-S$, which is the same Chern class as $Q$ by the universal exact sequence. Finally, Whitney multiplicativity applied to the universal exact sequence gives $c(S)c(Q)=1$, and substituting the Schubert identifications gives the inverse formula for $c(S)$.
[/proofplan]
[step:Express the Schubert condition as a rank-drop condition]
Fix $j\in\{1,\dots,n-k\}$ and define
\begin{align*}
a:=n-k+1-j.
\end{align*}
Then $1\le a\le n-k$. Let
\begin{align*}
\rho_a:\mathbb C^n\longrightarrow \mathbb C^n/F_a
\end{align*}
be the quotient map. Pulling $\rho_a$ back over $\operatorname{Gr}(k,n)$ and restricting it to the universal subbundle gives a morphism of complex vector bundles
\begin{align*}
\varphi_a:S\longrightarrow (\mathbb C^n/F_a)\times \operatorname{Gr}(k,n),
\end{align*}
whose fiber over $V\in\operatorname{Gr}(k,n)$ is the [linear map](/page/Linear%20Map)
\begin{align*}
(\varphi_a)_V:V\longrightarrow \mathbb C^n/F_a
\end{align*}
defined by $(\varphi_a)_V(v)=v+F_a$ for every $v\in V$. The kernel of $(\varphi_a)_V$ is $V\cap F_a$. Therefore
\begin{align*}
\dim_{\mathbb C}(V\cap F_a)\ge 1
\end{align*}
if and only if $(\varphi_a)_V$ is not injective, which is equivalent to
\begin{align*}
\operatorname{rank}(\varphi_a)_V\le k-1.
\end{align*}
Thus
\begin{align*}
\Sigma_j(F_\bullet)=D_{k-1}(\varphi_a),
\end{align*}
where $D_{k-1}(\varphi_a)$ denotes the degeneracy locus on which $\varphi_a$ has rank at most $k-1$.
[/step]
[step:Check the expected codimension and apply Thom-Porteous]
The bundle $S$ has rank $k$, and the product bundle $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$ has rank
\begin{align*}
n-a=n-(n-k+1-j)=k-1+j.
\end{align*}
For the rank condition $\operatorname{rank}(\varphi_a)\le k-1$, the expected complex codimension is
\begin{align*}
(k-(k-1))((k-1+j)-(k-1))=j.
\end{align*}
The locus $D_{k-1}(\varphi_a)$ is precisely the special Schubert variety indexed by the one-row partition $(j)$ for the quotient-bundle convention determined in the statement. The Schubert cell dimension formula gives complex codimension $j$ for this one-row Schubert variety, so the degeneracy locus has the expected codimension. In the standard Schubert determinantal description, the local defining equations for this one-row Schubert variety are exactly the maximal minors of the map obtained from $\varphi_a$ after choosing local frames for $S$ and for $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$. These determinantal charts are reduced for Schubert varieties in the Grassmannian, hence the determinantal scheme structure induced by $\varphi_a$ agrees with the reduced complex subvariety structure on $\Sigma_j(F_\bullet)$. Therefore its cycle class is the integral fundamental class used to define $\sigma_j$. Since $S$ and $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$ are complex vector bundles over the smooth complex Grassmannian, and $D_{k-1}(\varphi_a)$ is the corresponding complex algebraic degeneracy locus of the expected codimension with this natural integral fundamental class, the integral Thom-Porteous degeneracy-locus formula applies.
By the integral Thom-Porteous degeneracy-locus formula for the bundle map
\begin{align*}
\varphi_a:S\longrightarrow (\mathbb C^n/F_a)\times\operatorname{Gr}(k,n),
\end{align*}
the Poincare dual of the integral fundamental class of $D_{k-1}(\varphi_a)$ is
\begin{align*}
c_j\bigl((\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)-S\bigr).
\end{align*}
Since $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$ is a product complex vector bundle with constant fiber $\mathbb C^n/F_a$, its total Chern class is $1$. Hence
\begin{align*}
c\bigl((\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)-S\bigr)=c(S)^{-1}.
\end{align*}
[guided]
Fix $j\in\{1,\dots,n-k\}$ and define $a:=n-k+1-j$, so $1\le a\le n-k$. Let $F_a$ be the $a$-dimensional member of the fixed complete flag, and let $\rho_a:\mathbb C^n\to \mathbb C^n/F_a$ be the quotient map. Pulling $\rho_a$ back to $\operatorname{Gr}(k,n)$ and restricting to the universal subbundle defines the complex vector bundle morphism
\begin{align*}
\varphi_a:S\longrightarrow (\mathbb C^n/F_a)\times\operatorname{Gr}(k,n).
\end{align*}
The point of introducing $\varphi_a$ is that the Schubert condition is exactly a failure of injectivity. For each $V\in\operatorname{Gr}(k,n)$, the fiber map
\begin{align*}
(\varphi_a)_V:V\longrightarrow \mathbb C^n/F_a
\end{align*}
is defined by $(\varphi_a)_V(v)=v+F_a$ for every $v\in V$. Its kernel is precisely the set of vectors of $V$ that already lie in $F_a$, namely
\begin{align*}
\ker(\varphi_a)_V=V\cap F_a.
\end{align*}
Thus the condition $\dim_{\mathbb C}(V\cap F_a)\ge 1$ says exactly that $(\varphi_a)_V$ has nonzero kernel. Since the domain has dimension $k$, this is equivalent to
\begin{align*}
\operatorname{rank}(\varphi_a)_V\le k-1.
\end{align*}
We next verify the codimension required by the Thom-Porteous formula. The domain bundle $S$ has rank $k$. The target bundle is the product bundle with fiber $\mathbb C^n/F_a$, whose rank is
\begin{align*}
\dim_{\mathbb C}(\mathbb C^n/F_a)=n-a=n-(n-k+1-j)=k-1+j.
\end{align*}
For a map from a rank-$k$ bundle to a rank-$(k-1+j)$ bundle, the degeneracy locus where the rank is at most $k-1$ has expected complex codimension
\begin{align*}
(k-(k-1))((k-1+j)-(k-1))=j.
\end{align*}
This is the standard Schubert variety of one-row shape $(j)$ in the convention where special Schubert classes are represented by the quotient-bundle degeneracy condition. The Schubert cell dimension formula says that this one-row Schubert variety has complex codimension $j$, matching the expected codimension computed above. Locally on the Grassmannian, choose frames for $S$ and for $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$. In those frames, the rank condition on $\varphi_a$ is imposed by the maximal minors of the representing matrix. These are exactly the usual determinantal equations for this one-row Schubert variety, and the corresponding Schubert determinantal charts are reduced. Thus the determinantal scheme defined by the rank condition on $\varphi_a$ is the reduced Schubert variety $\Sigma_j(F_\bullet)$, so the cycle class appearing in Thom-Porteous is the integral fundamental class used in the definition of $\sigma_j$. The bundles involved are complex vector bundles over the smooth complex Grassmannian, and the degeneracy locus is a complex algebraic Schubert variety of the expected codimension equipped with this natural integral fundamental class. These are precisely the geometric hypotheses needed to apply the integral Thom-Porteous degeneracy-locus formula.
The integral Thom-Porteous degeneracy-locus formula now applies to the morphism
\begin{align*}
\varphi_a:S\longrightarrow (\mathbb C^n/F_a)\times\operatorname{Gr}(k,n),
\end{align*}
because the degeneracy locus has the expected codimension $j$. It gives that the Poincare dual of the integral fundamental class of this codimension-$j$ locus is the $j$-th Chern class of the virtual bundle
\begin{align*}
(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)-S.
\end{align*}
The target bundle is a product bundle with constant fiber $\mathbb C^n/F_a$, so its total Chern class is $1$, and therefore the total Chern class of this virtual bundle is
\begin{align*}
c\bigl((\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)-S\bigr)=c(S)^{-1}.
\end{align*}
[/guided]
[/step]
[step:Identify the resulting virtual Chern class with $c_j(Q)$]
The universal exact sequence
\begin{align*}
0\longrightarrow S\longrightarrow \mathbb C^n\times\operatorname{Gr}(k,n)\longrightarrow Q\longrightarrow 0
\end{align*}
and the Chern relation for tautological and quotient bundles [citetheorem:9802] give
\begin{align*}
c(S)c(Q)=1.
\end{align*}
Hence
\begin{align*}
c(Q)=c(S)^{-1}.
\end{align*}
Combining this identity with the Thom-Porteous computation from the previous step yields
\begin{align*}
[\Sigma_j(F_\bullet)]^\vee=c_j(Q),
\end{align*}
where $[\Sigma_j(F_\bullet)]^\vee\in H^{2j}(\operatorname{Gr}(k,n);\mathbb Z)$ denotes the Poincare dual of the integral fundamental class of $\Sigma_j(F_\bullet)$. By the definition of $\sigma_j$, this says
\begin{align*}
\sigma_j=c_j(Q).
\end{align*}
[/step]
[step:Invert the total quotient Chern class]
Since $Q$ has rank $n-k$, its total Chern class is
\begin{align*}
c(Q)=1+c_1(Q)+\cdots+c_{n-k}(Q).
\end{align*}
Using the identifications $c_j(Q)=\sigma_j$ for $1\le j\le n-k$, we obtain
\begin{align*}
c(Q)=1+\sigma_1+\cdots+\sigma_{n-k}.
\end{align*}
The identity $c(S)c(Q)=1$ in $H^*(\operatorname{Gr}(k,n);\mathbb Z)$ therefore gives
\begin{align*}
c(S)=c(Q)^{-1}=\bigl(1+\sigma_1+\cdots+\sigma_{n-k}\bigr)^{-1}.
\end{align*}
This proves both the Schubert compatibility statement and the asserted formula for the total Chern class of $S$.
[/step]
