A note on symmetric orderings

Abstract

Let ${\hat{A}}_n$ be the completion by the degree of a differential operator of the $n$-th Weyl algebra with generators $x_1,\dots ,x_n,{\partial }^1,\dots ,{\partial }^n$. Consider $n$ elements $X_1,\dots ,X_n$ in ${\hat{A}}_n$ of the form $$X_i=x_i+\sum _{K=1}^{\infty }\sum _{l=1}^n\sum _{j=1}^n{x}_l{p}_{ij}^{K-1,l}(\partial ){\partial }^j,$$ where $p_{ij}^{K-1,l}(\partial )$ is a degree $(K-1)$ homogeneous polynomial in ${\partial }^1,\dots ,{\partial }^n$, antisymmetric in subscripts $i,j$. Then for any natural $k$ and any function $i:\{1,\dots ,k\}\to \{1,\dots ,n\}$ we prove $$\sum _{\sigma \in \Sigma (k)}{X}_{i_{\sigma (1)}}\cdots X_{i_{\sigma (k)}}▹1=k!x_{i_1}\cdots x_{i_k},$$ where $\Sigma (k)$ is the symmetric group on $k$ letters and $▹$ denotes the Fock action of the ${\hat{A}}_n$ on the space of (commutative) polynomials.

8 pages, v2: expositional improvements and corrections; the second half of the main theorem's proof written much more explicitly