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The pre-Pieri rules

This paper establishes two general determinant identities involving sums of specific determinants over integer tuples and demonstrates how these "pre-Pieri rules" can be used to derive various known variants of the Pieri rule.

Original authors: Darij Grinberg

Published 2026-05-25

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: Darij Grinberg

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Mathematical "Recipe Book"

Imagine you are a chef working in a very strange kitchen. In this kitchen, you don't just mix ingredients; you arrange them in specific grids (matrices) and calculate a special "flavor score" for each grid called a determinant.

In the world of mathematics, there are famous recipes called Pieri rules. These rules tell you what happens when you multiply a specific type of mathematical object (a Schur function) by a simple ingredient (like a "complete" or "elementary" symmetric function). Usually, the result is a sum of many different new objects.

This paper is about discovering the "pre-Pieri rules." Think of these as the raw, uncooked ingredients before you apply the final garnish. The author, Darij Grinberg, has found two powerful, general formulas that explain how these grids of numbers behave when you shift them around. Once you have these general formulas, you can derive all the famous, specific recipes (the actual Pieri rules) just by plugging in specific numbers.

The Two Main Rules

The paper presents two main identities (formulas). Let's break them down using a metaphor of moving furniture in a room.

1. The First Pre-Pieri Rule: The "Infinite Expansion"

Imagine you have a row of $n$ shelves. On each shelf, you have a stack of boxes labeled with numbers.

The Setup: You have a specific arrangement of boxes.
The Action: You want to add exactly $p$ new boxes to the room, but you can distribute them however you like among the $n$ shelves. You can put all $p$ on one shelf, or spread them out one by one.
The Result: The paper proves that if you take the "flavor score" (determinant) of every possible way you could add these $p$ boxes, and add all those scores together, it equals the flavor score of one single, special arrangement.
The Trick: In this special arrangement, the last shelf gets a massive jump in its box count (it gets $p$ extra boxes added to its label), while the others stay mostly the same.

Why it's cool: This rule works even if your "kitchen" is chaotic. The paper proves this works even if the order in which you multiply your ingredients matters (non-commutative rings). It's a universal law of how these grids shift.

2. The Second Pre-Pieri Rule: The "Binary Switch"

This is the "twin" of the first rule, but with a different constraint.

The Setup: Same $n$ shelves.
The Action: This time, you have $p$ switches. For each shelf, you can either flip the switch (add 1 box) or leave it alone (add 0 boxes). You must flip exactly $p$ switches in total. You cannot add 2 boxes to one shelf; it's all or nothing.
The Result: If you sum up the flavor scores of all possible ways to flip exactly $p$ switches, it equals the flavor score of one single, special arrangement.
The Trick: In this special arrangement, the shelves are rearranged in a specific "staircase" pattern where the last $p$ shelves get bumped up by 1, and one specific shelf is skipped.

The Connection: The first rule is like adding a "multiset" of items (you can add as many as you want to one spot). The second rule is like adding a "set" of items (you can only add one to a spot, or none). They are mathematical opposites, much like how "choosing with replacement" is different from "choosing without replacement."

The "Magic" Behind the Scenes

How does the author prove this? He uses a technique called combinatorial cancellation.

Imagine you are trying to count the number of ways to arrange a deck of cards. You list every single possibility. Some arrangements are "good," and some are "bad."

The author sets up a system where every "bad" arrangement has a twin "bad" arrangement that cancels it out (like a positive and a negative number summing to zero).
The only things left standing after all the canceling are the "good" arrangements, which magically line up to form the single, special arrangement on the right side of the equation.

He does this using a tool called the row-determinant. In a normal math class, you learn that the determinant of a matrix is a single number. In this paper, the author uses a slightly different version (the row-determinant) that works even if your numbers don't play nice with each other (non-commutative). This allows the rules to apply to a much wider universe of math problems.

What Does This Give Us? (The Corollaries)

The paper doesn't just stop at the big, abstract rules. It shows how to "cook" specific dishes from these raw ingredients:

Recovering Old Recipes: By setting the variables to specific values (like making the ring commutative, which is the standard math world), the author shows that these new rules instantly turn into the famous Pieri rules for Schur functions (used in geometry and physics) and Immaculate functions (a newer, more complex type of function).
Fun's Rule: It proves a specific rule discovered by another mathematician named Fun, showing that Fun's rule was just a special case of this bigger picture.
The "Pre-LR" Speculation: At the end, the author wonders: "Is there a 'pre-Littlewood-Richardson' rule?" The Littlewood-Richardson rule is the "super-recipe" that combines two complex shapes. The author suggests that since these two pre-Pieri rules are the "antipodes" (opposites) of each other, there might be an even bigger, hidden rule that contains both of them, just as the Littlewood-Richardson rule contains the Pieri rules.

Summary in One Sentence

This paper discovers two universal "master formulas" that describe how complex grids of numbers behave when you add a fixed amount of value to them; these formulas act as a foundation that explains and unifies many famous, specific rules in the theory of symmetric functions, working even in the most chaotic mathematical environments.

Technical Summary of "The pre-Pieri rules" by Darij Grinberg

Problem Statement
The paper addresses the problem of generalizing classical Pieri rules within the theory of symmetric functions. Standard Pieri rules describe the product of a Schur function $s_\lambda$ with a complete homogeneous symmetric function $h_p$ or an elementary symmetric function $e_p$ as a sum of Schur functions $s_\mu$ over specific combinatorial constraints (horizontal and vertical strips, respectively).

The author focuses on "alternative" or "uncancelled" versions of these rules. Specifically, the paper investigates identities where the sum ranges over all integer $n$ -tuples $\mu$ satisfying entrywise inequalities and a fixed sum condition, rather than restricting the sum to partitions or specific strip conditions. In these alternative formulations, terms that do not correspond to valid partitions or strip conditions often vanish or cancel out. The goal is to establish these identities in a highly general setting: over arbitrary (potentially noncommutative) rings, using row-determinants, and for generalized Schur-like functions defined by determinants of matrices with arbitrary entries $h_{k,i}$ .

Methodology
The paper employs a combinatorial-algebraic approach centered on the manipulation of row-determinants (rowdet) over noncommutative rings. The methodology proceeds as follows:

Definitions: The author defines a generalized Schur-like element $t_\alpha$ for an integer tuple $\alpha \in \mathbb{Z}^n$ as the row-determinant of a matrix with entries $h_{\alpha_i+j, i}$ .
Combinatorial Lemmas: The proofs rely on a series of lemmas involving:
- The action of the symmetric group $S_n$ on integer tuples.
- Properties of Iverson brackets $[A]$ (1 if true, 0 if false).
- Determinantal identities involving matrices of the form $([ \nu_i \ge j ])_{i,j}$ or $([ \nu_i \trianglerighteq j ])_{i,j}$ , where $\trianglerighteq$ denotes the relation $u-v \in \{0, 1\}$ .
- Pigeonhole principle arguments to establish when a tuple is a permutation of another.
Summation Manipulation: The core technique involves expanding the row-determinant definition, interchanging the order of summation over permutations $\sigma \in S_n$ and integer tuples $\beta$ , and utilizing bijections to re-index the sums.
Generalization to Noncommutative Rings: The proofs are constructed to hold for any unital associative ring $R$ , not necessarily commutative, by carefully maintaining the order of factors in products (using row-determinants rather than standard determinants).

Key Contributions and Results

The paper establishes two primary identities, termed the "pre-Pieri rules," and derives several corollaries.

1. The First Pre-Pieri Rule (Theorem 2.1)
This identity generalizes the alternative first Pieri rule (multiplication by $h_p$ ).

Statement: For a ring $R$ , integers $n \ge 1, p \ge 0$ , and elements $h_{k,i} \in R$ , let $t_\alpha = \text{rowdet}(h_{\alpha_i+j, i})$ . Let $\eta = (1, 2, \dots, n-1, n+p)$ . Then:
$\sum_{\substack{\beta \in \mathbb{N}^n \\ |\beta|=p}} t_{\alpha+\beta} = \text{rowdet}(h_{\alpha_i+\eta_j, i})$
The right-hand side is a single row-determinant where the last column of the standard matrix is shifted by $p$ .
Proof Strategy: The proof expands the sum of row-determinants, re-indexes the summation variables, and utilizes Lemma 2.9 to show that the coefficient of each term in the expansion corresponds to the determinant of a matrix of Iverson brackets, which evaluates to the sign of the unique permutation mapping a specific tuple $\eta$ to the summation index.

2. The Second Pre-Pieri Rule (Theorem 4.1)
This identity is the analogue for multiplication by elementary symmetric functions ( $e_p$ ), involving a sum over tuples with entries in $\{0, 1\}$ .

Statement: For $p \in \{0, 1, \dots, n\}$ , let $\xi = (1, 2, \dots, n-p, n-p+2, \dots, n+1)$ . Then:
$\sum_{\substack{\beta \in \{0,1\}^n \\ |\beta|=p}} t_{\alpha+\beta} = \text{rowdet}(h_{\alpha_i+\xi_j, i})$
Proof Strategy: Similar to the first rule but utilizes a relation $\trianglerighteq$ (where $u \trianglerighteq v \iff u-v \in \{0,1\}$ ) and Lemma 4.6 to handle the restricted summation domain.

3. Corollaries and Recoveries
The paper demonstrates that these general identities recover and generalize several known results:

Commutative Case: When $R$ is commutative, the row-determinant becomes the standard determinant, recovering classical identities.
Schur Functions: By setting $h_{k,i} = h_k$ (complete homogeneous symmetric functions) and appropriate boundary conditions, the paper recovers the "alternative first Pieri rule" (Equation 5 in the introduction) where $s_\lambda h_p = \sum s_{\lambda+\beta}$ .
Immaculate Functions: The paper recovers the "right-Pieri rule" for immaculate functions (Berg et al., [BBSSZ13, Theorem 3.5]) as a special case of the first pre-Pieri rule.
Macdonald's 9th Variation: The paper recovers a Pieri-like rule for Macdonald's 9th-variation Schur functions (Fun, [Fun12, Proposition 3.9]).
Vanishing Conditions: The paper provides specific corollaries (e.g., Corollary 3.1, 4.9) detailing how the identities simplify when certain $h_{k,i}$ vanish (e.g., for negative $k$ ), leading to factorizations of the resulting determinants.

Significance and Claims
The paper claims significance primarily in the unification and generalization of existing combinatorial identities:

Generality: The identities hold over noncommutative rings, extending the scope of Pieri rules beyond symmetric functions to noncommutative polynomial algebras (such as the algebra of noncommutative symmetric functions, NSym).
Unification: The two pre-Pieri rules serve as a "pre" (preliminary or underlying) framework for various known Pieri rules. The paper notes that the standard Pieri rules (summing over partitions with strip conditions) can be derived from these pre-Pieri rules by removing vanishing terms or observing cancellations, though the pre-Pieri rules themselves do not require such cancellations to be valid identities.
Speculation on LR Rules: The author speculates on the existence of a "pre-Littlewood-Richardson rule" that might generalize both pre-Pieri rules, potentially encompassing the "immaculate Littlewood-Richardson rule" as a particular case. Proposition 5.2 is presented as a step in this direction, showing that sums of row-determinants over invariant sets of tuples can be expressed as linear combinations of row-determinants.

The paper does not propose new experimental applications or future computational algorithms but focuses on establishing these algebraic identities and their logical consequences within the existing literature of symmetric and noncommutative symmetric functions.