# Is the Hodge dual the unique map which commutes with exterior powers of isometries?

Let $$V$$ be a real oriented $$d$$-dimensional inner product space, $$d \ge 3$$. For $$1 \le k \le d-1$$, the Hodge dual map $$\star: \bigwedge^k V \to \bigwedge^{d-k} V$$ commutes with orientation-preserving isometries:

For every $$Q \in \text{SO}(V)$$, we have $$\star \circ \bigwedge^k Q= \bigwedge^{d-k} Q \circ \star \tag{1}.$$

Is $$\star$$ the unique linear map $$\bigwedge^k V \to \bigwedge^{d-k} V$$ satisfying $$(1)$$ up to scaling?

In the language of representation theory, I ask if the space of equivariant maps w.r.t the natural representations of $$\text{SO}(V)$$ on $$\bigwedge^k V,\bigwedge^{d-k} V$$ is one dimensional.

In $$d=2$$, $$\star:V \to V$$ is of course not the unique map up to scaling which commutes with all isometries, since $$\text{SO}(2)$$ is commutative, we have additional elements... (This is why I restricted $$d \ge 3$$).

This is true unless $$d$$ is even and $$k=d-k=\frac{d}2$$. This follows directly from the representation theory description you give in the question using Schur's lemma. Unless $$d$$ is even and $$k=\frac{d}2$$ the representation $$\Lambda^kV$$ is irreducible and so the isomorphism to $$\Lambda^{d-k}V$$ is unique up to a scalar multiple.If $$d$$ is even and $$k=\frac{d}2$$, then $$\Lambda^kV$$ is the direct sum of two non-isomorphic irreducible representations (the two eigenspaces of $$*$$) and you can choose independently choose sclalar factors on the two components (so there is a two parameter family of homomorphisms).
• Thanks. Do you have a reference for these facts? (The irreducibility of $\bigwedge^k V$ for $k \neq d/2$ and the direct sum decomposition in the case $k=d/2$). Also, in the case where $d$ is not a multiple of $4$, the eigenvalues of $\star$ are $\pm i$, so the maps $\omega \to \omega^+,\omega \to \omega^-$ are not real. Doesn't this create problems, if we restrict the discussion to the real case? (which is what I asked about). Thank you for your help and patience...I am weak on representation theory. – Asaf Shachar Jan 30 at 15:04
• You are right about the real vs. complex issue, I wasn't careful enough there. In a real setting uniqueness indeed holds unless $d$ is a multiple of $4$. You can certainly find the results in the complex case in Fulton-Harris. They are fundamental for the representation theory of SO and $\mathfrak{so}$ since these are the so-called fundamental representations. So they should show up whethever the representation theory of orthogonal groups and algebras is discussed. – Andreas Cap Jan 31 at 8:23