Center Preserving Automorphisms of Finite Heisenberg Group over ${\mathbb{Z}}_N$

Abstract

We investigate the group structure of center-preserving automorphisms of the finite Heisenberg group over ${\mathbb{Z}}_N$ with $U(1)$ extension, which arises in finite-dimensional quantum mechanics on a discrete phase space. Constructing an explicit splitting, it is shown that, for $N=2(2k+1)$, the group is isomorphic to the semidirect product of $S{p}_N$ and ${\mathbb{Z}}_N^2$. Moreover, when N is divisible by $2^l(l\ge 2)$, the group has a non-trivial 2-cocycle, and its explicit form is provided. By utilizing the splitting, it is demonstrated that the corresponding projective Weil representation can be lifted to linear representation.

23 pages, 1 figure