Speaker
Description
Let $G$ be a matrix group over a field $\mathbb{F}$ and $\phi: M_n(\mathbb{F}) \rightarrow M_n(\mathbb{F})$ be a map such that $\phi(A) \in G$ for all $A \in G.$
An element $A \in G$ is said to be $\phi-$reversible if there exists $P \in G$ such that $PAP^{-1}=\phi(A).$ If $P$ can be chosen to be an involution (i.e., $P^2=I),$ then $A$ is said to be strongly $\phi-$reversible. The strongly $\phi-$reversible elements of the complex symplectic group
$$\operatorname{Sp}(2n, \mathbb{C}):=\left\{A \in GL_{2n}(\mathbb{C}) \, \Bigg\lvert \, A^T\begin{bmatrix}
0 & I_n \\
-I_n & 0
\end{bmatrix}A= \begin{bmatrix}
0 & I_n \\
-I_n & 0
\end{bmatrix}\right\}$$ for the map $\phi: A \mapsto -A$ have been completely classified. In this talk, we consider the real case. We show that several results from the complex setting no longer hold over $\mathbb{R}$. In particular, for each $n$, we construct a real symplectic matrix that is strongly $\phi-$reversible as an element of $\operatorname{Sp}(4n,\mathbb{C})$, but not as an element of $\operatorname{Sp}(4n,\mathbb{R})$. Using a suitable canonical form for real symplectic matrices, we classify the strongly $\phi-$reversible elements of the real symplectic group $\operatorname{Sp}(4,\mathbb{R})$.