Let $n\ge 0$ and $0\le k\le n$ be integers, and let $X:=\operatorname{Gr}(k,n)$ be the complex Grassmannian of $k$-dimensional complex linear subspaces of $\mathbb C^n$. Let $S\to X$ be the tautological rank-$k$ complex vector bundle, whose fiber over $V\in X$ is $S_V:=V$, and let $\underline{\mathbb C}^n:=X\times \mathbb C^n\to X$ be the product rank-$n$ complex vector bundle. Let $Q:=\underline{\mathbb C}^n/S$ be the tautological quotient bundle, so that there is a short exact sequence of complex vector bundles over $X$

\begin{align*} 0\longrightarrow S\longrightarrow \underline{\mathbb C}^n\longrightarrow Q\longrightarrow 0. \end{align*}

Then, in the integral [cohomology ring](/theorems/2271) $H^*(X;\mathbb Z)$,

\begin{align*} c(S)c(Q)=1. \end{align*}

Equivalently, if

\begin{align*} c(S)=1+s_1+\cdots+s_k \end{align*}

and

\begin{align*} c(Q)=1+q_1+\cdots+q_{n-k}, \end{align*}

where $s_i:=c_i(S)\in H^{2i}(X;\mathbb Z)$ and $q_j:=c_j(Q)\in H^{2j}(X;\mathbb Z)$, then for every integer $d\ge 1$,

\begin{align*} \sum_{i+j=d}s_iq_j=0 \end{align*}

in $H^{2d}(X;\mathbb Z)$, with the conventions $s_0=q_0=1$, $s_i=0$ for $i>k$, and $q_j=0$ for $j>n-k$.