# A Hodge Dual as Sequence of Grade Reducing Steps

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1. Aug 2, 2016

### 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 $G$?

2. Aug 3, 2016

### Lucas SV

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?

3. Aug 3, 2016

### MisterX

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.