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

• URL: http://arxiv.org/abs/2307.00874v3
• Date: Mon, 2 Oct 2023 08:29:20 GMT
• Title: Center Preserving Automorphisms of Finite Heisenberg Group over $\mathbb Z_N$
• Authors: T.Hashimoto, M.Horibe, A.Hayashi
• Abstract summary: We investigate the group structure of automorphisms of the finite Heisenberg group over $mathbb Z_N$ with $U(1)$ extension. By utilizing the splitting, it is demonstrated that the corresponding projective Weil representation can be lifted to linear representation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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