Understanding the notation on Grassmanians

Question

I'm studying a book about Lie Group Actions, and there is a part where they talk about Grassmanians. A k-Grassmanian on vector space $V$ is defined as follows $$Gr_k(V)=\{W: W \text{ is a subset of } V \land \dim W=k\}$$ Then there is a result stating that $$Gr_k(\mathbb{R}^n)=SO(n)/S(O(n-k)\times O(k))$$ I'm having some trouble understatinf the notation $S(O(n-k)\times O(k))$, what does it mean? I know for example that $$A\in S(O(n)\times O(1))\Leftrightarrow A=\begin{pmatrix}B&0\\0&\pm 1\end{pmatrix},\qquad det(A)=1\qquad and\qquad B\in O(n)$$ But i am having some issues extending this idea to $S(O(n-k)\times O(k))$.

Answer 1

It's the matrices in $O(n-k)\times O(k)$ with determinant $1$, i.e. $S(O(n-k)\times O(k))=SO(n)\cap(O(n-k)\times O(k))$, the intersection taking place in $O(n)$. I do not think this notation is standard, though.