Group theoretic properties of Clifford multiplication on 2-torsion points on the Dirac Spinor Abelian Variety

In this manuscript we consider a special complex torus, denoted S_Δ₂ₖ (for each k ∈ N,\, k ≥ 1) and called the Dirac spinor torus. It is an Abelian variety of complex dimension 2ᵏ whose covering space is the space of Dirac spinors, Δ₂ₖ, for the Clifford algebra Cl(C²ᵏ) associated with the vector space C²ᵏ. Fixing an isomorphism ρ:Cl(C²ᵏ)→ End (Δ₂ₖ), we define Clifford multiplication on S_Δ₂ₖ as the actions of those endomorphisms in the image of ρ that preserve the full rank lattice. We analyze the properties of that Clifford multiplication on the 2-torsion points of the Dirac spinor torus. We identify the Clifford actions with permutation maps that represent all isomorphism classes of these actions on the group of 2-torsion points. We provide a structure theorem describing these isomorphism classes of Clifford actions in a way that is independent of the choice of representatives. We conclude by extending the scope of our analysis to the group of n-torsion points and analyzing the fixed points and translation constants of entry-permuting maps, a broader class of actions of which the Clifford actions on the 2-torsion points of S_Δ₂ₖ is a subset.