Center Preserving Automorphisms of Finite Heisenberg Group over \mathbb Z_N

2023/07/03 by Hashimoto, T., Horibe, M., Hayashi, A.
#FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2307.00874

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_N². Moreover, when N is divisible by 2^l (l ≥ 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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