Exterior-Model Spinors in Split Rank: Exact Levi Images and Square-Determinant Obstructions

Abstract

Let $K$ be a field with $2 \in K^\times$, and let $H_W$ denote the standard hyperbolic form on $W \oplus W^*$. We study the exterior spinor model $S = \bigwedge V(W)$ together with the spin-to-orthogonal map for this split form, keeping the chosen hyperbolic presentation explicit. The main results determine the field-sensitive part of the split Levi image. In positive split rank the kernel of $\mathrm{Spin}(V,Q) \to SO(V,Q)$ is $\{\pm 1\}$; therefore the exterior spinor action descends to the orthogonal image only projectively. For the split line the image of $\mathrm{Spin}(H_K) \to SO(H_K)$ is precisely the square-scaling subgroup. In arbitrary split rank we construct explicit Clifford representatives for hyperbolic transvections and chosen-line square scalings, prove the weight-2 torus conjugation law, and show that any split Levi lift acts on $\bigwedge V(W)$ as a scalar multiple of the natural exterior action. If $\det(g) \in u^2$, the transported Levi element $\hat{g} = (g, g^{-\top})$ admits an explicit even unitary Clifford lift acting as $u^{-1} \bigwedge(g)$ on $S$. In finite split rank at least three, if $H_W \in \mathrm{im}(\mathrm{Spin}(H_W) \to SO(H_W))$, then $g_{H_W} \in \det(g) \cdot K^{\times 2}$. Equivalently, the spin image meets the split Levi subgroup exactly in its square-determinant subgroup. This recovers, by direct Clifford calculation, the determinant-modulo-squares spinor-norm criterion on the split Levi.