Center Preserving Automorphisms of Finite Heisenberg Group over $\mathbb Z_N$

TL;DR

This paper characterizes the structure of center-preserving automorphisms of the finite Heisenberg group over Z_N, revealing their composition, cocycle properties, and implications for projective Weil representations in quantum mechanics.

Contribution

It explicitly constructs the automorphism group structure, identifies conditions for non-trivial cocycles, and shows how to lift projective representations to linear ones.

Findings

Automorphism group is a semidirect product of Sp_N and Z_N^2 for certain N.

Non-trivial 2-cocycle exists when N is divisible by 2^l (l ge 2).

Projective Weil representation can be lifted to a linear representation.

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 $Sp_N$ 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.