论文标题
关于对称顺序的注释
A note on symmetric orderings
论文作者
论文摘要
令$ \ hat {a} _n $为$ n $ th weyl代数的差异操作员的完成，带有发电机$ x_1，\ ldots，x_n，\ partial^1，\ ldots，\ ldots，\ partial^n $。考虑$ n $ elements $ x_1，\ ldots，x_n $ in $ \ hat {a} _n $的表单 $$ x_i = x_i + \ sum_ {k = 1}^\ infty \ sum_ {l = 1}^n \ sum_ {j = 1}^n x_l p_l p_ {ij}^ij}^{k-1，l} {k-1，l}（\ partial）\ partial^j， $$其中$ p^{k-1，l} _ {ij}（\ partial）$是$ \ \ partial^1，\ ldots，\ partial^n $，在下标$ i，j $中的抗异压。然后，对于任何天然$ k $和任何函数$ i：\ {1，\ ldots，k \} \ to \ {1，\ ldots，n \} $我们证明 $$ \ sum_ {σ\在σ（k）} x_ {i_ {σ（1）}}} \ cdots x_ {i_ {σ（k）}} \ triangleright 1 = k！ \，x_ {i_1} \ cdots x_ {i_k}，$$，其中$σ（k）$是$ k $ letters上的对称组，$ \ triangleright $表示$ \ hat {a} _n $的fock动作在（通勤）polynomials的空间上。
Let $\hat{A}_n$ be the completion by the degree of a differential operator of the $n$-th Weyl algebra with generators $x_1,\ldots,x_n,\partial^1,\ldots,\partial^n$. Consider $n$ elements $X_1,\ldots,X_n$ 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 $p^{K-1,l}_{ij}(\partial)$ is a degree $(K-1)$ homogeneous polynomial in $\partial^1,\ldots,\partial^n$, antisymmetric in subscripts $i,j$. Then for any natural $k$ and any function $i : \{1,\ldots,k\}\to\{1,\ldots,n\}$ we prove $$ \sum_{σ\in Σ(k)} X_{i_{σ(1)}}\cdots X_{i_{σ(k)}}\triangleright 1 = k! \,x_{i_1}\cdots x_{i_k}, $$ where $Σ(k)$ is the symmetric group on $k$ letters and $\triangleright$ denotes the Fock action of the $\hat{A}_n$ on the space of (commutative) polynomials.
