Hodge Dual as Sequence of Grade Reducing Steps

[MisterX]

If we seek a bijection $$\wedge^p V \to \wedge^{n-p} V$$ for some inner product space ##V##, we might think of starting with the unit ##n##-vector and removing dimensions associated with the original vector in ##\wedge^p V ##. Might this be expressed as a sequence of steps by some binary function ##G##,
$$\star \left( \mathbf{x} \wedge \mathbf{y} \right) = G\Big(\mathbf{x}, G\big( \mathbf{y}, \mathbf{e}_1 \wedge \dots \wedge \mathbf{e}_n\big)\Big) $$
in which case how might we express ##G##?

 

[Lucas SV]

If we seek a bijection $$\wedge^p V \to \wedge^{n-p} V$$ for some inner product space ##V##, we might think of starting with the unit ##n##-vector and removing dimensions associated with the original vector in ##\wedge^p V ##. Might this be expressed as a sequence of steps by some binary function ##G##,
$$\star \left( \mathbf{x} \wedge \mathbf{y} \right) = G\Big(\mathbf{x}, G\big( \mathbf{y}, \mathbf{e}_1 \wedge \dots \wedge \mathbf{e}_n\big)\Big) $$
in which case how might we express ##G##?

May I ask why seek a bijection this way? As you probably know, the dimension of ##\wedge^p V## and ##\wedge^{n-p} V## is ##\binom {n} {p}##. Since they already have the same dimension, there is a vector space isomorphism, which you can find in proofs like http://math.stackexchange.com/quest...ctor-spaces-of-equal-dimension-are-isomorphic. A more interesting question, is whether there is a map which shows that ##\wedge^p V## and ##\wedge^{n-p} V## have the same graded structure. I don't know what grade preserving maps are called. Maybe that is what you are trying to find?

 

[MisterX]

May I ask why seek a bijection this way? As you probably know, the dimension of ##\wedge^p V## and ##\wedge^{n-p} V## is ##\binom {n} {p}##. Since they already have the same dimension, there is a vector space isomorphism, which you can find in proofs like http://math.stackexchange.com/quest...ctor-spaces-of-equal-dimension-are-isomorphic. A more interesting question, is whether there is a map which shows that ##\wedge^p V## and ##\wedge^{n-p} V## have the same graded structure. I don't know what grade preserving maps are called. Maybe that is what you are trying to find?

Well I was trying to actually express the map (in a coordinate free way), not just prove it exists. But I was also curious if I could find other uses for this operation.