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

TL;DR

This paper investigates the algebraic structure of Clifford multiplication on 2-torsion points of a special complex torus called the Dirac spinor torus, revealing permutation-based characterizations and classifying their isomorphism types.

Contribution

It provides a structure theorem for Clifford actions on 2-torsion points, independent of choices, and extends analysis to n-torsion points and broader permutation actions.

Findings

Clifford actions correspond to permutation maps on 2-torsion points.

A classification of isomorphism classes of Clifford actions is established.

Extension of analysis to n-torsion points and fixed point properties.

Abstract

In this manuscript we consider a special complex torus, denoted $S_{\Delta_{2k}}$ (for each $k \in \mathbb{N},\, k \geq 1$) 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 $\rho:Cl(\mathbb{C}^{2k})\rightarrow End (\Delta_{2k})$, 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.