# Symplectic group over integers and finite fields

For $H=\left( \begin{smallmatrix} 0 & I_n \\ -I_n & 0 \end{smallmatrix} \right)$ and a commutative ring $F$, the symplectic group $Sp(2n,F)$ is the set of all matrices $M\in F^{2n\times 2n}$ such that $MHM^T = H$. (Here, $M^T$ means the transposed matrix.)

My question is: for $\mathbb{F}_p$ a finite field of prime order, is $Sp(2n,\mathbb{F}_p) = Sp(2n,\mathbb{Z})\otimes\mathbb F_p$, the reduction mod $p$ of $Sp(2n,\mathbb{Z})$?

• I'm not sure what you mean by your tensor product notation between a group and a field.
– YCor
Feb 11, 2016 at 18:08
• I mean reduction of the coefficients mod p. Fixed. Feb 11, 2016 at 18:09
• The problem was not so much the notation (which is fine), but the fact that you define $\operatorname{Sp}(2n,R)$ as a group, rather than an $R$-scheme. I'm sure this is what you meant, but it isn't what you're writing.
– R.P.
Feb 11, 2016 at 18:17
• @René: I see. And of course, the fact that it is a group does not matter for the question. Feb 11, 2016 at 18:20
• Are you asking whether ${\rm Sp}(2n, \mathbb{Z}) \to {\rm Sp}(2n, \mathbb{F}_p)$ is surjective? Feb 11, 2016 at 18:31

Yes: $$\operatorname{Sp}(2n, \mathbb F_p)$$ is generated by its root subgroups. Each root subgroup is cyclic, and generated by an element that lifts in an obvious way to $$\operatorname{Sp}(2n, \mathbb Z)$$.
More concretely, if I haven't messed up the calculations in your form (I'm used to a different one), then, writing $$E_{i j}$$ for the matrix that has a 1 in its $$(i, j)$$th entry and 0 elsewhere, we have that $$\operatorname{Sp}(2n, \mathbb F_p)$$ is generated by:
• $$\{1 + t(E_{i j} - E_{(j + n)(i + n)}) : t \in \mathbb F_p\}$$ (corresponding to the root $$e_i - e_j$$) for $$1 \le i < j \le n$$,
• $$\{1 + t(E_{i(j + n)} + E_{j(i + n)}) : t \in \mathbb F_p\}$$ (corresponding to the root $$e_i + e_j$$) for $$1 \le i \ne j \le n$$,
• $$\{1 + t E_{i(i + n)} : t \in \mathbb F_p\}$$ (corresponding to the root $$2e_i$$) for $$1 \le i \le n$$, and
The $$e_i - e_j$$ generators correspond to an elementary row operation in the upper left block, combined with its inverse transpose in the lower right block. The $$e_i + e_j$$ and $$2e_i$$ generators generate symmetric matrices in the upper right block.