On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces

TL;DR

This paper classifies the automorphism groups of a signature-dependent transposition scalar product in real Clifford algebras, revealing its relation to known conjugation types and providing a detailed group structure analysis for low-dimensional cases.

Contribution

It offers a comprehensive classification of invariance groups of a new scalar product in Clifford algebras, extending understanding of their automorphism groups and their relation to the star map and spinor space generators.

Findings

Classification of invariance groups Gpq_{p,q} for p+q <= 9

Identification of the transposition anti-involution T as the star map

Analysis of subgroups and transversals related to spinor generators

Abstract

A signature epsilon=(p,q) dependent transposition anti-involution T of real Clifford algebras Cl_{p,q} for non-degenerate quadratic forms was introduced in [arXiv.1005.3554v1]. In [arXiv.1005.3558v1] we showed that, depending on the value of (p-q) mod 8, the map T gives rise to transposition, complex Hermitian, or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [Lounesto, Clifford algebras and Spinors 2001]. We provide a full signature (p,q) dependent classification of the invariance groups Gpq_{p,q} of this product for p+q <= 9. The map T is identified as the "star" map known [Passmann, The Algebraic Structure of Group Rings 1985] from the theory of (twisted) group algebras, where the Clifford algebra Cl_{p,q} is seen as a twisted group ring k^t[(Z_2)^n], n=p+q. We discuss and list important subgroups of stabilizer groups Gpq(f)_{p,q} and their transversals in relation to generators of spinor spaces.