The pre-Pieri rules
Pith reviewed 2026-05-25 08:20 UTC · model grok-4.3
The pith
Two summation identities for determinants t_α hold over any ring and imply Pieri rule variants.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Let R be a ring and h_{k,i} elements of R. Define t_α as the determinant of the matrix whose i-th row is h_{α_i +1 ,i}, h_{α_i +2,i}, ..., h_{α_i +n,i}. Then the sum of t_{α+β} over β nonnegative integers with sum p equals the determinant with last column replaced by h_{α_i +(n+p),i}. If p ≤ n the sum over β with 0,1 entries summing to p equals the determinant whose columns are shifted by the indices in ξ=(1,2,...,n-p,n-p+2,...,n+1). These identities are proved in the non-commutative setting and used to derive Pieri rule variants.
What carries the argument
The determinants t_α of the matrix with entries h_{α_i + k, i} for k=1 to n; column operations on these determinants collapse the indicated sums into single modified determinants and remain valid without commutativity.
If this is right
- The identities hold for non-commutative rings as well as commutative ones.
- Variants of the Pieri rule follow directly from these summation formulas.
- The results apply to arbitrary families of ring elements h_{k,i} without further restrictions.
Where Pith is reading between the lines
- The same technique could be applied to other summation problems involving determinants in algebra.
- Explicit checks with matrix rings would verify the non-commutative case.
- These pre-Pieri identities may simplify proofs of related combinatorial rules.
Load-bearing premise
A determinant function on matrices over the ring must satisfy the column-replacement identities used to prove the summations, even without commutativity.
What would settle it
Compute both sides of either identity in a specific non-commutative ring such as 2x2 matrices over the integers with chosen values for the h_{k,i} and check if they match.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript defines t_α as the determinant (over a ring R, not necessarily commutative) of the n×n matrix whose (i,j)-entry is h_{α_i + j, i}. It proves two summation identities: (i) the sum of t_{α+β} over all β ∈ ℕ^n with |β|=p equals the determinant obtained by replacing the last column with the entries h_{α_i + (n+p), i}; (ii) when p≤n the sum over β ∈ {0,1}^n with |β|=p equals the determinant whose columns are shifted according to the tuple ξ=(1,…,n-p,n-p+2,…,n+1). Both identities are established in the non-commutative setting and are applied to obtain variants of the Pieri rule.
Significance. The explicit verification that the two summation identities continue to hold when R is non-commutative supplies a concrete algebraic strengthening of known commutative determinant identities; this is a positive feature of the work because it directly addresses the column-replacement and multilinearity properties needed for the proofs without assuming commutativity.
minor comments (2)
- The abstract states that the identities are proved 'in a slightly more general setting' beyond non-commutativity; the precise additional generality (if any) should be stated explicitly in the introduction or in the statement of the main theorems.
- Notation for the matrix in the second displayed identity uses ξ without an explicit definition in the abstract; a parenthetical reminder of the definition of ξ would improve readability.
Simulated Author's Rebuttal
We thank the referee for their positive summary of the manuscript and for recommending acceptance. We are pleased that the algebraic strengthening in the non-commutative setting was viewed as a positive feature.
Circularity Check
No circularity; identities proved from explicit determinant definition
full rationale
The paper defines t_α directly as a determinant over the ring R (possibly non-commutative) and states two summation identities as theorems to be proved from that definition plus standard determinant expansion properties. No self-citation is invoked to justify the central claim, no parameters are fitted to data and then relabeled as predictions, and the claimed equalities are not equivalent to the input definition by construction. The non-commutative case is presented as a direct (if technically delicate) extension of the Leibniz formula, with no reduction to prior results by the same author.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math The determinant of an n by n matrix over a (possibly non-commutative) ring is well-defined via the usual alternating sum over permutations and satisfies the column-replacement identities used in the proof.
Reference graph
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discussion (0)
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