Grade reversion, intuitively, is reversing the multiplication order of basis vectors.
Formally, it's an algebra homomorphism \(\tau : \mathcal {C}\kern -2pt\ell (Q) \to _{a} \mathcal {C}\kern -2pt\ell (Q)^{\mathtt {op}}\), satisfying:
- \(\tau (m_1 m_2) = \tau (m_2) \tau (m_1)\)
- \(\tau \circ \tau = \operatorname {id}\)
- \(\tau (\iota (m)) = \iota (m)\)
That is, the following diagram commutes:
It's called anti-involution \(\tau \) in [jadczyk2019notes], the canonical anti-automorphism in [gallier2008clifford],
also called transpose/transposition in some literature, following tensor algebra or matrix.
It's denoted \(\tilde {m}\) in [lounesto2001clifford], \(m^\tau \) in [jadczyk2019notes] (with variants like \(m^t\) or \(m^\top \) in other literatures), \(m^\dagger \) in [chisolm2012geometric].