I am sorry if this question is not for mathoverflow. I asked the same question on stackexchange (https://math.stackexchange.com/questions/4203667/cohomology-ring-of-grassmannian-and-pieri-rule), but I didn't get an answer:
Let $X=OG(n,2n+1)$, where $OG(n,2n+1)$ denotes the variety of $n$-dimensional isotropic subspaces of a vector space $\mathbb{C}^{2n+1}$ with a nondegenerate symmetric bilinear form.
According to Theorem 2.2 a) (Page 17, Anders Skovsted Buch, Andrew Kresch, Harry Tamvakis, Quantum Pieri rules for isotropic grassmannians, https://arxiv.org/pdf/0809.4966.pdf), the cohomology ring of X is given by $$ H^{*}(X,\mathbb{Z})=\mathbb{Z}[\tau_{1},\ldots, \tau_{n}]/I,$$
where $I$ is the ideal generated by $$ \tau_{r}^{2}-2\tau_{r+1}\tau_{r-1}+2\tau_{r+2}\tau_{r-2}+\cdots +(-1)^{r}\tau_{2r}$$ for $1\leq r\leq n$.
In particular, if $n=4$, then the ideal $I$ is generated by the following four elements
$$ \tau_{1}^{2}-\tau_{2},\quad \tau_{2}^{2}-2\tau_{3}\tau_{1}+\tau_{4}, \quad \tau_{3}^{2}-2\tau_{4}\tau_{2},\quad \tau_{4}^{2}.\tag{*}\label{*}$$ But if I apply Pieri rule for X (Theorem 2.1, Page 16, Anders Skovsted Buch, Andrew Kresch, Harry Tamvakis, Quantum Pieri rules for isotropic grassmannians) to $\tau_{2}\cdot \tau_{2}$, I get the following relation
$$\tau_{2}^{2}-2\tau_{3}\tau_{1}-\tau_{4} \tag{**}\label{**}$$
Therefore, combining (\ref{*}) and (\ref{**}), I get $2\tau_{4}=0$ in $H^{*}(X,\mathbb{Z})$. I seems that some computation is wrong, but I don't know where I made a mistake.
Thank you.
