Steenrod operations on cohomology of grassmannians

Question

Let $G_k(\mathbb{R}^n)$, $n\geq k$ and $G_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of universal Stiefel-Whitney classes $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,w_2,\cdots,w_k],$$ $$ H^*(G_k(\mathbb{R}^n);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,w_2,\cdots,w_k]/(\bar w_{n-k+1},\bar w_{n-k+2}\cdots,\bar w_n).$$

What are the Steenrod operations $Sq^i$ on these cohomology rings? I want to know the expressions of $Sq^i w_j$ for all possible $i,j$.

Answer 1

As in Prasit's comment, the action of the Steenrod squares on the Stiefel-Whitney classes of any vector bundle are given by Wu's formula $$ Sq^i(w_j) = \sum_{t=0}^i \binom{j+t-i-1}{t} w_{i-t} w_{j+t}. $$ Here $w_k$ is of course $0$ if $k$ exceeds the dimension of the bundle. Together with the Cartan formula, this completely answers your question.