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A new antisymmetrizer-to-determinant formula and a conjecture of Colomo and Pronko

This paper establishes a new identity expressing an antisymmetrizer as a determinant, thereby proving a conjecture by Lukas Riegler and one of the authors, while also proposing a tailored version of a conjecture by Colomo and Pronko to aid in its resolution.

Original authors: Ilse Fischer, Markus Reibnegger

Published 2026-08-26

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CC BY 4.0

Original authors: Ilse Fischer, Markus Reibnegger

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

In the vast landscape of mathematics, there is a branch dedicated to counting things that seem impossible to count. Imagine a grid of squares, like a chessboard, where you are allowed to place numbers in the cells. The rules are strict: every row and every column must add up to exactly one, and the non-zero numbers must alternate between positive and negative. These are called alternating sign matrices. For decades, mathematicians have been fascinated by how many different ways these grids can be filled while obeying the rules. The answers often turn out to be surprisingly elegant, but finding them requires navigating a labyrinth of complex patterns. To solve these puzzles, researchers often rely on a powerful tool called an antisymmetrizer. Think of this tool as a machine that takes a messy, complicated expression and rearranges it by swapping its parts in every possible order, adding and subtracting the results to cancel out the noise. When this machine works perfectly, the chaotic mess collapses into a single, clean structure known as a determinant, which is a specific kind of mathematical calculation that is much easier to solve.

The paper at hand, written by Ilse Fischer and Markus Reibnegger, introduces a new and powerful version of this collapsing machine. The authors have discovered a fresh formula that allows certain complex expressions to be rewritten as determinants, a feat that was previously unknown. This discovery is not just a theoretical curiosity; it settles a long-standing guess made by other mathematicians regarding the counting of a specific type of these grids. By proving this new formula, the authors have unlocked a door that confirms a precise way to count these patterns, specifically those that are symmetric in a vertical direction. Furthermore, the researchers point toward a related, even more difficult puzzle proposed by Colomo and Pronko. While they have not yet solved that second puzzle, they have mapped out a new path that connects it to the same kind of mathematical machinery, suggesting that the solution might be within reach if one can just find the right key.

The journey begins with the concept of the alternating sign matrix, a grid where the numbers in each row and column sum to one, and the non-zero entries flip signs as you move across them. These grids are more than just abstract exercises; they appear in the study of physical systems and have deep connections to the geometry of shapes. For a long time, mathematicians knew how to count the total number of these grids for any given size, but the path to that answer was tortuous. It relied on a specific identity, a rule that showed how a complicated sum of rearranged terms could be simplified into a determinant. This rule was the engine behind the first successful proofs of the counting formulas. However, as mathematicians tried to refine their counts, looking at grids with extra symmetries or specific constraints, the old engine began to sputter. They encountered new expressions that looked similar to the old ones but refused to collapse into a simple determinant. One such stubborn expression was the subject of a conjecture by Lukas Riegler and one of the paper's authors, which remained unproven for years.

Fischer and Reibnegger tackled this problem by constructing a new engine. They focused on a specific type of expression involving rational functions, which are essentially fractions made of polynomials. Their goal was to show that when you apply the antisymmetrizer machine to these specific fractions, the result is always a determinant. They did not just guess the answer; they built a rigorous proof using a method called induction. This approach is like climbing a ladder: they first proved the statement was true for the smallest possible case, and then showed that if it is true for a grid of a certain size, it must also be true for the next size up. The core of their work involved a clever manipulation of the terms inside the expression, showing that they could be rearranged to match the structure of a determinant. This was a delicate operation, requiring them to handle the interactions between different parts of the formula with extreme precision.

The result of their labor is a new formula that acts as a bridge between the messy world of antisymmetrizers and the clean world of determinants. This formula is significant because it directly proves the conjecture that had been open since 2014. The conjecture was about a specific symmetry in the counting of these grids, known as vertically symmetric alternating sign matrices. These are grids that look the same if you flip them over a vertical line down the middle. The conjecture predicted a specific, refined way to count these grids based on where the number one appears in the second row. By proving their new formula, the authors confirmed that this prediction is correct. They showed that the complex expression describing these grids does indeed simplify into a determinant, validating the refined counting formula that had been proposed by others.

Beyond solving this specific puzzle, the authors turned their attention to an even more ambitious challenge. They discussed a conjecture by Colomo and Pronko regarding grids that have a block of zeros in the bottom-left corner. This problem is notoriously difficult because the pattern of zeros breaks the symmetry that usually makes these problems solvable. The authors proposed a new way to look at this problem. They suggested that if one could find a similar collapsing formula for a different, related expression, the Colomo and Pronko conjecture would fall into place. They did not find this formula yet, but they demonstrated that the expression in question shares deep structural similarities with the one they just solved. They even reformulated the conjecture in terms of matrices that arise from a specific type of decomposition, offering a concrete roadmap for future researchers. This reformulation is a crucial step, as it translates a vague hope into a specific mathematical target.

The paper also touches on a broader theme in mathematics: the search for generalizations. The authors showed that their new formula is part of a larger family of identities that includes a famous result known as the Cauchy determinant. This older result is a cornerstone of the field, and the authors' work extends it to a more complex setting. They posed a question for the future: can this new, more complex formula be generalized in the same way? This question highlights the ongoing nature of mathematical discovery. Just as solving one puzzle often reveals the outline of a larger one, proving this new identity opens up new avenues for exploration. The authors acknowledge that their understanding is still incomplete, noting that while they have found the key for one door, there are many others that remain locked.

In the end, this work is a testament to the power of persistence and the beauty of mathematical structure. The authors took a problem that had resisted solution for a decade, built a new tool to attack it, and succeeded in proving a conjecture that had eluded the community. They did not just find a number; they found a pattern, a rule that governs how these complex grids behave. By connecting the dots between antisymmetrizers, determinants, and symmetry classes, they have provided a clearer view of the landscape of alternating sign matrices. Their work serves as a reminder that even in the most abstract corners of mathematics, there are hidden connections waiting to be discovered, and that sometimes, the key to a difficult problem is simply finding the right way to rearrange the pieces.

Technical Summary of "A New Antisymmetrizer-to-Determinant Formula and a Conjecture of Colomo and Pronko"

Problem Statement
The paper addresses the challenge of expressing antisymmetrizers of specific rational functions as determinants. Such identities are central to the enumeration of Alternating Sign Matrices (ASMs) and related combinatorial structures like monotone triangles. While Schur polynomials and Hall–Littlewood polynomials admit determinantal expressions via antisymmetrizers, many other antisymmetrizers arising in ASM theory do not yet have known determinantal forms. Specifically, the authors aim to prove a conjecture by Lukas Riegler and Markus Reibnegger (Conjecture 2) regarding the symmetry of a specific symmetrized rational function $R_{s,t}$ . Proving this conjecture is significant because it implies a refined enumeration formula for vertically symmetric ASMs. Additionally, the paper investigates a related, more complex antisymmetrizer proposed by Colomo and Pronko, which counts ASMs with a zero block in the bottom-left corner, for which no determinantal formula has yet been found.

Methodology
The authors employ a combination of algebraic manipulation, induction, and combinatorial identities involving symmetric functions.

Inductive Proofs: The main result (Theorem 3) is proven by induction on the number of variables $n$ . The base cases ( $n = 2s-1$ ) are established using Lemma 7, which handles the specific case where the number of variables relates directly to the parameter $s$ .
Recursion and Laplace Expansion: The inductive step relies on expanding the antisymmetrizer $D_s(X_1, \dots, X_n)$ by isolating one variable and utilizing the recursive structure of the antisymmetrizer. This reduces the problem to verifying that a specific column in a determinant can be expressed as a linear combination of other columns.
Symmetric Function Identities: To handle the linear combination verification, the authors utilize elementary symmetric functions $e_k$ and specific combinatorial identities involving binomial coefficients and the functions $p(X) = X(X-1)$ and $q(X) = -p(X)X^{-3}$ . Lemmas 8 and 9 provide the necessary algebraic machinery to equate coefficients of these symmetric functions.
Generalization: In Section 4, the authors generalize Lemma 1 and the Cauchy determinant evaluation (Theorem 10) by introducing an auxiliary set of variables $Y_i$ and a rational function $f(X)$ . This is achieved by establishing a recursion for the antisymmetrizer and showing that a proposed determinant satisfies the same recursion, utilizing Lagrange interpolation arguments.

Key Contributions and Results

Proof of Conjecture 2: The primary result is the proof of Conjecture 2 (from [10]), which asserts that a specific symmetrized rational function $R_{s,t}$ is invariant under the transformation $X_i \to X_i^{-1}$ . This is achieved by establishing Theorem 3, a new antisymmetrizer-to-determinant formula for the function $D_s(X_1, \dots, X_n)$ .
- Theorem 3: For non-negative integers $n, s$ with $0 \le s \le \lfloor \frac{n+1}{2} \rfloor$ , the antisymmetrizer of a specific product involving $p(X)$ and $q(X)$ equals a determinant of a matrix with entries involving powers of $p(X_i)$ and $q(X_i)$ .
- Implication: The authors demonstrate that the determinantal expression in Theorem 3 shares the necessary invariance properties to imply Conjecture 2, thereby confirming the refined enumeration of vertically symmetric ASMs.
Analysis of the Colomo–Pronko Conjecture: The paper elaborates on a conjecture by Colomo and Pronko concerning ASMs with a zero $s \times s$ corner.
- The authors define a related antisymmetrizer $F_s$ and note its structural similarity to the function $D_s$ in Theorem 3.
- While a determinantal formula for $F_s$ remains elusive, the authors reformulate the Colomo–Pronko conjecture in terms of the LU-decomposition of specific matrices ( $B_n$ and $M_n$ ) derived from the case $s=0$ . They propose Conjecture 6, which expresses the count $F_{n,s}$ using determinants of submatrices of $H_n$ and $G_n$ , matrices constructed from the LU-decomposition of the underlying structures.
Generalization of Lemma 1: Theorem 10 presents a common generalization of Lemma 1 (the standard antisymmetrizer-to-determinant lemma) and the Cauchy determinant. This result introduces a parameter $Y_i$ and a function $f(X)$ , unifying previous results under a broader framework.

Significance
The paper claims significance primarily in the resolution of a specific open conjecture (Conjecture 2) that had remained open despite the refined enumeration of vertically symmetric ASMs being proven by other means. The authors emphasize that their contribution is the proof of the conjecture via a new determinantal identity, rather than the discovery of the enumeration formula itself.

Furthermore, the paper highlights that the identification of rational functions admitting determinantal expressions remains an incomplete understanding in the field. The authors note that while they successfully identified the formula for $D_s$ , the analogous formula for the Colomo–Pronko antisymmetrizer $F_s$ has not yet been found, despite evidence suggesting one exists. The paper posits that the reformulation of the Colomo–Pronko conjecture (Conjecture 6) may provide a pathway to resolving this open problem. Finally, the generalization in Theorem 10 opens new questions regarding potential applications and further generalizations of Theorem 3, which the authors leave as open problems.

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