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 $\alpha\in\mathbb Z^n$, we define \[ t_{\alpha}:=\det\begin{pmatrix} h_{\alpha_1+1,\ 1} & h_{\alpha_1+2,\ 1} & \cdots & h_{\alpha_1+n,\ 1}\\ h_{\alpha_2+1,\ 2} & h_{\alpha_2+2,\ 2} & \cdots & h_{\alpha_2+n,\ 2}\\ \vdots & \vdots & \ddots & \vdots\\ h_{\alpha_n+1,\ n} & h_{\alpha_n+2,\ n} & \cdots & h_{\alpha_n+n,\ n} \end{pmatrix} \in R \] (where $\alpha_i$ denotes the $i$-th entry of $\alpha$). Then, we have the identity \[ \sum_{\substack{\beta\in\{0,1,2,\ldots\}^n ;\\ \left|\beta \right|=p}}t_{\alpha+\beta} =\det \begin{pmatrix} h_{\alpha_1+1,\ 1} & h_{\alpha_1+2,\ 1} & \cdots & h_{\alpha_1+(n-1),\ 1} & h_{\alpha_1+(n+p),\ 1}\\ h_{\alpha_2+1,\ 2} & h_{\alpha_2+2,\ 2} & \cdots & h_{\alpha_2+(n-1),\ 2} & h_{\alpha_2+(n+p),\ 2}\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ h_{\alpha_n+1,\ n} & h_{\alpha_n+2,\ n} & \cdots & h_{\alpha_n+(n-1),\ n} & h_{\alpha_n+(n+p),\ n} \end{pmatrix} \] (where $\alpha+\beta$ denotes the entrywise sum of the tuples $\alpha$ and $\beta$). Furthermore, if $p\leq n$, then \[ \sum_{\substack{\beta\in\left\{ 0,1\right\} ^n ;\\\left| \beta \right| =p}}t_{\alpha+\beta}=\det \begin{pmatrix} h_{\alpha_1+\xi_1 ,\ 1} & h_{\alpha_1+\xi_2 ,\ 1} & \cdots & h_{\alpha_1+\xi_n ,\ 1}\\ h_{\alpha_2+\xi_1 ,\ 2} & h_{\alpha_2+\xi_2 ,\ 2} & \cdots & h_{\alpha_2+\xi_n ,\ 2}\\ \vdots & \vdots & \ddots & \vdots\\ h_{\alpha_n+\xi_1 ,\ n} & h_{\alpha_n+\xi_2 ,\ n} & \cdots & h_{\alpha_n+\xi_n ,\ n} \end{pmatrix} , \] where $\xi=(1,2,\ldots,n-p,n-p+2,n-p+3,\ldots,n+1)$. 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.
| Comments: | 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 |
| Subjects: | Combinatorics (math.CO); Rings and Algebras (math.RA) |
| MSC classes: | 15A15, 05E05, 15A24 |
| Cite as: | arXiv:2110.03108 [math.CO] |
| (or arXiv:2110.03108v2 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2110.03108 arXiv-issued DOI via DataCite |
From: Darij Grinberg [view email]
[v1]
Wed, 6 Oct 2021 23:48:04 UTC (62 KB)
[v2]
Thu, 21 May 2026 20:28:41 UTC (63 KB)
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