The permutation group $S_n$ acts as via Weyl reflections of $A_{n-1}$ on $R^{n-1}$ und thus on the sphere $S^{n-2}$. On this sphere, we have a natural action of $SO(n-1)$ generated by the angular momentum differential operators. Its universal enveloping algebra contains differential operators of any order. I am interested in two problems:
(1) find (characterize, classify?) all Weyl invariant operators in this algebra, or at least a minimal algebraically independent set.
(2) find (characterize, classify?) all Weyl anti-invariant operators in this algebra, or at least a minimal algebraically independent set. (Anti-tinvariant means totally antisymmetric under any Weyl reflection.)
Is there a good representation-theoretic method to answer (2) ?
The first nontrivial case I have looked at is $n=4$, hence $S_4$ (anti-)invariant words in the $SO(3)$ angular momentum generators $(L_x,L_y,L_z)$. The set of invariant operators seems to be generated by the Newton polynomials in $L_*$ of order $0,2,4,6$. The set of anti-invariant operators seems to be generated by just two operators: $L_x L_y L_z +$ permutations (of order $3$) and a specific operator of order $6$. I would love to have an argument ruling out further generators.