Let $X$ be a smooth projective variety and $E\longrightarrow X$ a vector bundle of rank $n$. For any $0\leq k\leq n$ the associated Grassmann bundle $G\_k(E)\longrightarrow X$ yields and we have the so-called "basis theorem" (see Fulton "Intersection theory", Proposition 14.6.5) which asserts that for any $s\geq 0$,
$$CH\_s(G\_k(E))=\bigoplus\_{\lambda}CH\_{s-k(n-k)+|\lambda|}(X)$$
where $\lambda$ runs over all partitions $\lambda=(\lambda\_1,...,\lambda\_k)$ with $n-k\geq \lambda\_1\geq...\geq\lambda\_k\geq 0$.


I would like to know if there is some similar result for fiber product of Grassman bundles, i.e. consider a vector bundle $E\longrightarrow X$ of rank $n$ and $0\leq k\_1\leq k\_2\leq n$ two integer : does the $s$-dimensional Chow group $CH\_s(G\_{k\_1}(E)\times\_{X} G\_{k\_2}(E))$ is isomorphic to a direct sum of $\bigoplus\_{p\in \mathcal{P}}CH\_p(X)$ for some set $\mathcal{P}$ ?


I tried to look a an answer considering the immersion $G\_{k\_1}(E)\times\_X G\_{k\_2}(E)\longrightarrow G\_{k\_1k\_2}(E\times\_X E)$ without success.


This question studying the case where $X=\mathbb{P}^1$ and $E$ the tautological vector bundle.



Chow group of a fiber product of grassmann bundles
You just need to repeat the same procedure twice. Indeed, let $E'$ be the pullback of $E$ from $X$ to $G\_{k\_1}(E)$. Then $G\_{k\_1}(E)\times\_X G\_{k\_2}(E) \cong G\_{k\_2}(E')$.


