Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :

1. $G/H = SO(n)\big/S\big({\prod_{r=1}^s (O_{t_r})}_\Delta ^{d_r}\big). \quad $
2. $G/H = SU(n)\big/S\big(\prod_{r=1}^s (U_{t_r})_\Delta ^{d_r}\big)$.

Note that in both the cases $H$ may NOT be connected. At least some reference is also helpful.

Notations: For a group $H$, we denote by $H_\Delta^n$ the diagonally embeded copy of $H$ in the $n$-fold direct product $H^n$. Let $H_i \subset GL_{l_i}$ be a matrix subgroup, for $1 \leq i \leq m$, then $ S(\prod_i H_i) :=\{ (h_1,\dotsc, h_m ) \in \prod H_i \bigm| \prod_i \det h_i = 1 \}.$

Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :
$$ G/H = SO(n)\big/S\big({(O_{t_1})}_\Delta ^{d_1} \times \cdots \times {(O_{t_s})}_\Delta ^{d_s} \big), $$
$$ G/H = SU(n)\big/S\big( (U_{t_1})_\Delta ^{d_1}\times \cdots \times (U_{t_s})_\Delta ^{d_s} \big). $$ Note that in both the cases $H$ may NOT be connected. At least some reference is also helpful.

Notations: For a group $H$, we denote by $H_\Delta^n$ the diagonally embedded copy of $H$ in the $n$-fold direct product $H^n$.

Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :

1. $G/H = SO(n)\big/S\big({(\prod_{r=1}^s O_{t_r})}_\Delta ^{d_r}\big). \quad $
2. $G/H = SU(n)\big/S\big((\prod_{r=1}^s U_{t_r})_\Delta ^{d_r}\big)$.

Note that in both the cases $H$ may NOT be connected. At least some reference is also helpful.

For a group $H$, we denote by $H_\Delta^n$ the diagonally embeded copy of $H$ in the $n$-fold direct product $H^n$. Let $H_i \subset GL_{l_i}$ be a matrix subgroup, for $1 \leq i \leq m$, then $ S(\prod_i H_i) :=\{ (h_1,\dotsc, h_m ) \in \prod H_i \bigm| \prod_i \det h_i = 1 \}.$

Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :

1. $G/H = SO(n)\big/S\big({\prod_{r=1}^s (O_{t_r})}_\Delta ^{d_r}\big). \quad $
2. $G/H = SU(n)\big/S\big(\prod_{r=1}^s (U_{t_r})_\Delta ^{d_r}\big)$.

Note that in both the cases $H$ may NOT be connected. At least some reference is also helpful.

Notations: For a group $H$, we denote by $H_\Delta^n$ the diagonally embeded copy of $H$ in the $n$-fold direct product $H^n$. Let $H_i \subset GL_{l_i}$ be a matrix subgroup, for $1 \leq i \leq m$, then $ S(\prod_i H_i) :=\{ (h_1,\dotsc, h_m ) \in \prod H_i \bigm| \prod_i \det h_i = 1 \}.$