Let \(\mathbb{R}\) denote the set of all real numbers and let \(i = \sqrt{-1}\). Consider the matrices
\[
S = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\quad \text{and} \quad
T = \begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}.
\]
Let \(a, b, c, d\) be real numbers such that
\[
ST =
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}.
\]
Let
\[
H = \{x + iy : x, y \in \mathbb{R} \text{ and } y>0\}.
\]
Then which of the following statements is (are) TRUE?