determining symplecticity (if that's a word) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:00:36Z http://mathoverflow.net/feeds/question/97778 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97778/determining-symplecticity-if-thats-a-word determining symplecticity (if that's a word) Igor Rivin 2012-05-23T17:50:02Z 2012-06-02T09:25:39Z <p>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.</p> http://mathoverflow.net/questions/97778/determining-symplecticity-if-thats-a-word/97801#97801 Answer by Mark Sapir for determining symplecticity (if that's a word) Mark Sapir 2012-05-24T00:01:11Z 2012-05-24T00:01:11Z <p>You forgot the condition that $n$ is even. I do not think there is a better criterion than the tautology.</p> <p>The unipotent matrix</p> <p>$$\left[ \begin {array}{cccc} 1&amp;1&amp;2&amp;3\\ 0&amp;1&amp;4&amp;5\\ 0&amp;0&amp;1&amp;1\\0&amp;0&amp;0&amp;1\end{array} \right]$$ is not conjugate to a symplectic matrix in $SL(4,\mathbb{Z})$ (although it is conjugate to a symplectic matrix in $SL(4,\mathbb{C})$. </p> <p>On the other hand the matrix</p> <p>$$\left[ \begin {array}{cccc} 1&amp;0&amp;0&amp;1\\ 0&amp;1&amp;0&amp;0\\ 0&amp;0&amp;1&amp;0\\0&amp;0&amp;0&amp;1\end{array} \right]$$</p> <p>is conjugate to a symplectic matrix. </p> <p>Both facts can be easily verified by using Maple. </p>