Hi,
I tried to build J following @cgunn3 but I’m unable to check the duality/polarity formula e^SI = e^{S\bot} using \textbf{P}( \mathbb{R}_{n, 0, 0}). Actually, I get a -1 factor for grade 2 and 3. Any idea ?
Thanks
S = \{\}
S^\bot = \{0,1,2,3\}
SS^\bot = \{0,1,2,3\} is even partition
J(1) = \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}
e^SI = \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}
S = \{0\}
S^\bot = \{1,2,3\}
SS^\bot = \{0,1,2,3\} is even partition
J(\boldsymbol{e}_{0}) = \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}
e^SI = \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}
S = \{1\}
S^\bot = \{0,2,3\}
SS^\bot = \{1,0,2,3\} is odd partition
J(\boldsymbol{e}_{1}) = - \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}
e^SI = - \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}
S = \{2\}
S^\bot = \{0,1,3\}
SS^\bot = \{2,0,1,3\} is even partition
J(\boldsymbol{e}_{2}) = \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}
e^SI = \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}
S = \{3\}
S^\bot = \{0,1,2\}
SS^\bot = \{3,0,1,2\} is odd partition
J(\boldsymbol{e}_{3}) = - \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}
e^SI = - \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}
S = \{0,1\}
S^\bot = \{2,3\}
SS^\bot = \{0,1,2,3\} is even partition
J(\boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}) = \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}
e^SI = - \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}
S = \{0,2\}
S^\bot = \{1,3\}
SS^\bot = \{0,2,1,3\} is odd partition
J(\boldsymbol{e}_{0}\wedge \boldsymbol{e}_{2}) = - \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}
e^SI = \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}
S = \{0,3\}
S^\bot = \{1,2\}
SS^\bot = \{0,3,1,2\} is even partition
J(\boldsymbol{e}_{0}\wedge \boldsymbol{e}_{3}) = \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}
e^SI = - \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}
S = \{1,2\}
S^\bot = \{0,3\}
SS^\bot = \{1,2,0,3\} is even partition
J(\boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}) = \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{3}
e^SI = - \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{3}
S = \{1,3\}
S^\bot = \{0,2\}
SS^\bot = \{1,3,0,2\} is odd partition
J(\boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}) = - \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{2}
e^SI = \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{2}
S = \{2,3\}
S^\bot = \{0,1\}
SS^\bot = \{2,3,0,1\} is even partition
J(\boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}) = \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}
e^SI = - \boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}
S = \{0,1,2\}
S^\bot = \{3\}
SS^\bot = \{0,1,2,3\} is even partition
J(\boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}) = \boldsymbol{e}_{3}
e^SI = - \boldsymbol{e}_{3}
S = \{0,1,3\}
S^\bot = \{2\}
SS^\bot = \{0,1,3,2\} is odd partition
J(\boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}) = - \boldsymbol{e}_{2}
e^SI = \boldsymbol{e}_{2}
S = \{0,2,3\}
S^\bot = \{1\}
SS^\bot = \{0,2,3,1\} is even partition
J(\boldsymbol{e}_{0}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}) = \boldsymbol{e}_{1}
e^SI = - \boldsymbol{e}_{1}
S = \{1,2,3\}
S^\bot = \{0\}
SS^\bot = \{1,2,3,0\} is odd partition
J(\boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}) = - \boldsymbol{e}_{0}
e^SI = \boldsymbol{e}_{0}
S = \{0,1,2,3\}
S^\bot = \{\}
SS^\bot = \{0,1,2,3\} is even partition
J(\boldsymbol{e}_{0}\wedge \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}) = 1
e^SI = 1