The pre-Pieri rules

Abstract

Let $R$ be a commutative ring and $n\geq1$ and $p\geq0$ two integers. Let $h_{k,\ i}$ be an element of $R$ for all $k\in\mathbb Z$ and $i\in [n]$. For any $α\in\mathbb Z^n$, we define \[ t_α:=\det\begin{pmatrix} h_{α_1+1,\ 1} & h_{α_1+2,\ 1} & \cdots & h_{α_1+n,\ 1}\\ h_{α_2+1,\ 2} & h_{α_2+2,\ 2} & \cdots & h_{α_2+n,\ 2}\\ \vdots & \vdots & \ddots & \vdots\\ h_{α_n+1,\ n} & h_{α_n+2,\ n} & \cdots & h_{α_n+n,\ n} \end{pmatrix} \in R \] (where $α_i$ denotes the $i$-th entry of $α$). Then, we have the identity \[ \sum_{\substack{β\in\{0,1,2,\ldots\}^n ;\\ \left|β\right|=p}}t_{α+β} =\det \begin{pmatrix} h_{α_1+1,\ 1} & h_{α_1+2,\ 1} & \cdots & h_{α_1+(n-1),\ 1} & h_{α_1+(n+p),\ 1}\\ h_{α_2+1,\ 2} & h_{α_2+2,\ 2} & \cdots & h_{α_2+(n-1),\ 2} & h_{α_2+(n+p),\ 2}\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ h_{α_n+1,\ n} & h_{α_n+2,\ n} & \cdots & h_{α_n+(n-1),\ n} & h_{α_n+(n+p),\ n} \end{pmatrix} \] (where $α+β$ denotes the entrywise sum of the tuples $α$ and $β$). Furthermore, if $p\leq n$, then \[ \sum_{\substack{β\in\left\{ 0,1\right\} ^n ;\\\left| β\right| =p}}t_{α+β}=\det \begin{pmatrix} h_{α_1+ξ_1 ,\ 1} & h_{α_1+ξ_2 ,\ 1} & \cdots & h_{α_1+ξ_n ,\ 1}\\ h_{α_2+ξ_1 ,\ 2} & h_{α_2+ξ_2 ,\ 2} & \cdots & h_{α_2+ξ_n ,\ 2}\\ \vdots & \vdots & \ddots & \vdots\\ h_{α_n+ξ_1 ,\ n} & h_{α_n+ξ_2 ,\ n} & \cdots & h_{α_n+ξ_n ,\ n} \end{pmatrix} , \] where $ξ=(1,2,\ldots,n-p,n-p+2,n-p+3,\ldots,n+1)$. We prove these two identities (in a slightly more general setting, where $R$ is not assumed commutative) and use them to derive some variants of the Pieri rule found in the literature.