I was trying to understand how special the $\omega$ involution on the ring of symmetric functions $\Lambda$ or $\Lambda^n$ (restriction to $n$ variables, just in case if by some magic, the situation is different there) is. I know how it corresponds to tensoring by the sign-alternating irrep. However, it's still a somewhat arbitrary choice to me. The inner product seemed more natural arising from character orthonormality of irreps of $GL_n(\mathbb{C})$. So I started looking for candidates $\omega'$ which can replace $\omega$, which means:
- $\omega'$ is a algebra automorphism of $\Lambda$
- $\omega'$ is an isometry under the usual inner product defined by declaring Schur polynomials orthonormal.
The two obvious candidates which fulfill the roles are the identity and $\omega$. But are there others? I feel there aren't, but I don't know how to go about proving this. The tactic I had in mind was that I want to make sure my automorphism respects the Littlewood-Richardson rule (trying to exploit the obvious niceness of the inner product w.r.t Schur polynomials), but the coefficients are pretty difficult to deal with.
