A note on symmetric orderings

Abstract

Let A^n be the completion by the degree of a differential operator of the n-th Weyl algebra with generators x1,…,xn,∂1,…,∂n. Consider n elements X1,…,Xn in A^n of the formXi=xi+∑K=1∞∑l=1n∑j=1nxlpijK−1,l(∂)∂j,where pijK−1,l(∂) is a degree (K−1) homogeneous polynomial in ∂1,…,∂n, antisymmetric in subscripts i,j. Then for any natural k and any function i:{1,…,k}→{1,…,n} we prove∑σ∈Σ(k)Xiσ(1)⋯Xiσ(k)▹1=k!xi1⋯xik,where Σ(k) is the symmetric group on k letters and ▹ denotes the Fock action of the A^n on the space of (commutative) polynomials.