determining symplecticity (if that's a word)

2012-05-23T17:50:02Z

Suppose you have a matrix $M$ in $SL(n, \mathbb{Z}).$ Question: is there a necessary and sufficient condition for $M$ to be conjugate to $N \in Sp(n, \mathbb{Z}).$ It is clearly necessary that the characteristic polynomial of $M$ be palindromic, but I would assume that this is not sufficient.

Answer by Mark Sapir for determining symplecticity (if that's a word)

2012-05-24T00:01:11Z

You forgot the condition that $n$ is even. I do not think there is a better criterion than the tautology.

The unipotent matrix

$$\left[ \begin {array}{cccc} 1&1&2&3\\ 0&1&4&5\\ 0&0&1&1\\0&0&0&1\end{array} \right] $$ is not conjugate to a symplectic matrix in $SL(4,\mathbb{Z})$ (although it is conjugate to a symplectic matrix in $SL(4,\mathbb{C})$.

On the other hand the matrix

$$\left[ \begin {array}{cccc} 1&0&0&1\\ 0&1&0&0\\ 0&0&1&0\\0&0&0&1\end{array} \right] $$

is conjugate to a symplectic matrix.

Both facts can be easily verified by using Maple.

determining symplecticity (if that's a word) - MathOverflow

determining symplecticity (if that's a word)

Answer by Mark Sapir for determining symplecticity (if that's a word)