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

Abstract

In this manuscript we consider a special complex torus, denoted $S_{{\Delta }_{2k}}$ (for each $k\in \mathbb{N},k\ge 1$) and called the Dirac spinor torus. It is an Abelian variety of complex dimension $2^k$ whose covering space is the space of Dirac spinors, ${\Delta }_{2k}$, for the Clifford algebra $Cl({\mathbb{C}}^{2k})$ associated with the vector space ${\mathbb{C}}^{2k}$. Fixing an isomorphism $\rho :Cl({\mathbb{C}}^{2k})\to End({\Delta }_{2k})$, we define Clifford multiplication on $S_{{\Delta }_{2k}}$ as the actions of those endomorphisms in the image of $\rho$ 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 $S_{{\Delta }_{2k}}$ is a subset.