Eulerian insertion operators and an Eulerian form of the Pieri rule

We study the operators obtained by inserting copies of a new largest letter into multiset permutations. Let $G_r$ denote the operator which inserts $r$ copies of a new largest letter. After the change of variables $δ=y-x$, $u=x/y$, and $E=u\partial_u$, we find that $$G_r=\frac{δ^r}{r!}E(E+1)\cdots(E+r-1).$$ Its generating series acts by a rational substitution, which yields the composition law. Our main result gives a common symmetric-function explanation for the ordinary and major-index operators. For $N\geq 0$, define $Φ_N(F_{N,S})=x^{|S|+1}y^{N-|S|}$. We prove that multiplication by the complete homogeneous symmetric function $h_r$ becomes the ordinary insertion operator: $Φ_{N+r}(h_r f)=G_rΦ_N(f)$, where $f\in\mathrm{QSym}_N$. There is a parallel specialization for the major index. A reverse finite principal specialization sends multiplication by $h_r$ to an operator $Q_r$, which is a polynomial in the $q$-shift $Θ_qf(t)=f(qt)$. Thus the ordinary and major-index operators arise from the same multiplication operator $f\mapsto h_r f$. Since the functions $h_r$ freely generate the ring of symmetric functions, the assignment $h_r\mapsto G_r$ extends to an algebra homomorphism. We determine the kernel of this homomorphism and the image of every homogeneous component. The images of Schur functions satisfy the Littlewood--Richardson multiplication identities, and the one-row case gives an Eulerian form of the Pieri rule.