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Being antisymmetric means the wedge product is invariant to reflection. Of graded algebras in general, sure, you just need the set of grades to be a monoid. Not for Clifford algebras though. You're starting with the scalars, then building vectors, the bivectors, ... (or tessarines, biquanterions, ...), so negative grades would only make sense if scalars were composed of something else (and constructable from that something else with the wedge product).Also, on a slightly unrelated note, can there be a negative grade? And if so, how would that change the dot and wedge products (if at all)?
Formerly_Me · 3429 days ago · link ·
Okay! That actually makes much more sense. My notes talked about antisymmetry and reflection separately, so I was very confused. Thanks for answering!