\begin{aligned} & \text { A } \mathbb{Z} \text {-BASIS OF THE } S_n \text {-INVARIANT SUBRING } \\ & \text { OF } \mathbb{Z}\left[a_i-a_j\right] \end{aligned}

Abstract

Let $\mathbb{Z}\left[a_i-a_j\right]$ be the subring of the polynomial ring $\mathbb{Z}\left[a_1, \ldots, a_n\right]$ of $n$ independent variables $a_1, \ldots, a_n$ generated by the differences $a_i-a_j \quad(1 \leq i<j \leq n)$. Let $R(n)$ be the invariant subring of $\mathbb{Z}\left[a_i-a_j\right]$ under the action of the symmetric group $S_n$ of degree $n$ defined by $\sigma\left(a_i-a_j\right)=a_{\sigma(i)}-a_{\sigma(j)}\left(\sigma \in S_n\right)$. We investigate and construct a $\mathbb{Z}$-basis of the submodules $R(d, n)$ of $R(n)$ of degree $d$ under certain conditions. Furthermore, $R(n)$ is expressed as the kernel of the differential operator $\nabla=\sum_{k=1}^n \partial / \partial a_k$ of the $S_n$-invariant subring $\mathbb{Z}\left[\Lambda_1, \ldots, \Lambda_n\right]$ for elementary symmetric polynomials $\Lambda_t$ of degree $t$ of $a_1, \ldots, a_n$. With respect to an ordered $\mathbb{Z}$-basis of $\mathbb{Z}\left[\Lambda_1, \ldots, \Lambda_n\right]$, we determine the exponents $n$ on the minimal terms of the $\mathbb{Z}$-basis vectors of $R(d, n)$ (Theorem A).

Received: December 10, 2024
Accepted: March 7, 2025