The pre-Pieri rules

Abstract

Let $R$ be a commutative ring and $n\ge 1$ and $p\ge 0$ two integers. Let $h_{k,i}$ be an element of $R$ for all $k\in \mathbb{Z}$ and $i\in [n]$. For any $\alpha \in {\mathbb{Z}}^n$, 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 ${\alpha }_i$ denotes the $i$-th entry of $\alpha$). 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 $\alpha +\beta$ denotes the entrywise sum of the tuples $\alpha$ and $\beta$). Furthermore, if $p\le n$, 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 $\xi =(1,2,\dots ,n-p,n-p+2,n-p+3,\dots ,n+1)$. 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.

44 pages. Main results stated in Sections 2 and 4. v2 corrects Corollary 4.12 (assumption was insufficient; error found by GPT-5.5). Not sure how new the results are, whence no attempts at publication, but the writeup may be useful nevertheless

References in corpus (2)

• Notes on the combinatorial fundamentals of algebra
• A basis for a quotient of symmetric polynomials