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).

  • $\begingroup$ 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. $\endgroup$ – Asaf Shachar Jan 30 at 15:04
  • $\begingroup$ 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. $\endgroup$ – Andreas Cap Jan 31 at 8:23

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