On the transposition anti-involution in real Cliffrd algebras II: Stabilizer groups of primitive idempotents

TL;DR

This paper explores the properties of the transposition anti-involution in real Clifford algebras, revealing how it induces various matrix conjugations in spinor representations depending on the algebra's signature.

Contribution

It characterizes the stabilizer groups of primitive idempotents and introduces a new spinor norm, expanding understanding of Clifford algebra automorphisms and their representations.

Findings

The anti-involution induces different conjugations based on signature mod 8.

A new spinor norm is defined, distinct from existing ones.

Classification of stabilizer groups according to algebra signature.

Abstract

In the first article of this work [... I: The transposition map] we showed that real Clifford algebras CL(V,Q) posses a unique transposition anti-involution \tp. There it was shown that the map reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the associated matrix of that element in the left regular representation of the algebra. In this paper we show that, depending on the value of (p-q) mod 8, where \ve=(p,q) is the signature of Q, the anti-involution gives rise to transposition, Hermitian complex, and Hermitian quaternionic conjugation of representation matrices in spinor representations. We realize spinors in minimal left ideals S=CL_{p,q}f generated by a primitive idempotent f. The map \tp allows us to define a dual spinor space S^\ast, and a new spinor norm on S, which is different, in general, from two spinor norms known to exist. We study a transitive action of generalized Salingaros' multiplicative vee groups G_{p,q} on complete sets of mutually annihilating primitive idempotents. Using the normal stabilizer subgroup G_{p,q}(f) we construct left transversals, spinor bases, and maps between spinor spaces for different orthogonal idempotents f_i summing up to 1. We classify the stabilizer groups according to the signature in simple and semisimple cases.