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The Hodge star is an isometry and satisfies $⋆^{2} = (- 1)^{k (n - k)}$ on $Λ^{k} V$

Statement

On $Λ^{k} V$ of an oriented $n$ -dimensional real inner product space, the Hodge star preserves the Gram pairing,

$⟨ ⋆ α, ⋆ β ⟩ = ⟨ α, β ⟩,$

and satisfies

$⋆ \circ ⋆ = (- 1)^{k (n - k)} {id}_{Λ^{k} V} .$

Facts & Assumptions

Given: An oriented $n$ -dimensional real inner product space $V$ , a degree $k$ , and a positively oriented orthonormal basis $(e_{1}, \dots, e_{n})$ .

[L1]

In a positively oriented orthonormal basis, $⋆ e_{I} = ε_{I} e_{I^{c}}$ , where $ε_{I}$ is the sign of the permutation listing $I$ followed by its complement (The Hodge star exists uniquely and is given by the complementary-basis formula in an oriented orthonormal basis).

Proof

technique · direct

1.1L1algebra

By [L1], $⋆$ sends the orthonormal wedge basis $(e_{I})_{I}$ to the family $(ε_{I} e_{I^{c}})_{I}$ , which is the complementary orthonormal wedge basis with signs attached; hence $⟨ ⋆ e_{I}, ⋆ e_{J} ⟩ = δ_{I, J} = ⟨ e_{I}, e_{J} ⟩$ , and bilinearity gives the isometry on all of $Λ^{k} V$ .

1.2L1algebra

For each $I$ , apply [L1] twice: $⋆ (⋆ e_{I}) = ε_{I} ⋆ e_{I^{c}} = ε_{I} ε_{I^{c}} e_{I}$ , where $ε_{I^{c}}$ is the sign of the permutation listing $I^{c}$ followed by $I$ . The two permutations differ by interchanging a block of size $k$ with a block of size $n - k$ , whose sign is $(- 1)^{k (n - k)}$ , so $ε_{I^{c}} = (- 1)^{k (n - k)} ε_{I}$ and $⋆^{2} e_{I} = (- 1)^{k (n - k)} e_{I}$ .

2.1step 1.2∎

Since the $e_{I}$ form a basis, step 1.2 gives $⋆ \circ ⋆ = (- 1)^{k (n - k)} id$ on $Λ^{k} V$ .

Depends on

The Hodge star exists uniquely and is given by the complementary-basis formula in an oriented orthonormal basis

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Reyer Sjamaar, Manifolds and Differential Forms, §2.4 (standard reference, not scraped)

The Hodge star is an isometry and satisfies ⋆2=(−1)k(n−k)⋆2=(−1)k(n−k) on ΛkVΛkV