On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces
Abstract
A signature epsilon=(p,q) dependent transposition anti-involution T of real Clifford algebras Clp,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 Gpqp,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 Clp,q is seen as a twisted group ring kt[(Z2)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.