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Grothendieck rings of near-Tambara--Yamagami fusion categories

This paper determines and categorifies the Grothendieck rings of fusion categories containing a nonpointed Tambara–Yamagami subcategory and exactly one additional simple object, identifying three specific cases based on the coefficient $\kappa \in \{0, 2, 7\}$ and providing explicit realizations for each.

Original authors: Jingcheng Dong, Sébastien Palcoux, Kai Wang

Published 2026-10-27

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CC BY 4.0

Original authors: Jingcheng Dong, Sébastien Palcoux, Kai Wang

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 modern mathematics, there is a field dedicated to understanding how complex systems can be built from simple, indivisible parts. Imagine a universe where the basic building blocks are not atoms or particles, but abstract "objects" that can be combined or fused together. When two of these objects meet, they do not simply merge; they transform into a new collection of objects, following strict, unbreakable rules. Mathematicians call these systems fusion categories. They are not just theoretical curiosities; they appear in the study of quantum physics, the behavior of exotic particles called anyons, and the deep structures of space and time. To make sense of these systems, researchers use a tool called a Grothendieck ring, which acts like a ledger. It records the rules of combination in a way that turns fluid, complex interactions into a discrete, countable list of possibilities. However, having the list of rules is not the same as proving the system actually exists. Just because a set of numbers adds up correctly on paper does not mean a real mathematical structure can be built to match them. The true challenge lies in finding the "associator," the hidden rule that dictates how these objects group together when three or more are fused at once. Without this, the system is merely a shadow of a potential reality.

A specific family of these systems, known as Tambara–Yamagami categories, has long served as a testing ground for these ideas. These are special because almost all their building blocks are "invertible," meaning they can be undone, with only one unique object that behaves differently and carries a larger, more complex weight. Recently, mathematicians have been exploring a slightly more complex variation: systems that contain one of these Tambara–Yamagami structures plus exactly one additional, new building block. This small addition creates a sharp test for the limits of what is possible. In a recent study, Jingcheng Dong, Sébastien Palcoux, and Kai Wang set out to map every possible version of this specific setup. They asked a fundamental question: given the rules for how these objects combine, which numerical possibilities can actually be realized as a working mathematical universe, and which are merely illusions that look correct on paper but collapse under scrutiny?

The researchers began by listing every numerical candidate that seemed to fit the basic requirements. They found that the size of the underlying group of invertible objects had to be four, forming a specific shape known as the Klein four-group. This fixed the size of the system to six distinct types of building blocks. The critical variable was a single number, which they called the coefficient, representing how much of the new object appeared when it was fused with itself. Their initial analysis suggested four possible values for this number: zero, two, seven, and eight. The team then embarked on a rigorous journey to determine which of these four numbers could actually support a real, functioning system.

The investigation first tackled the case where the number was eight. This scenario involved irrational numbers, meaning the dimensions of the objects could not be expressed as simple fractions. The researchers used a powerful method involving symmetry and the properties of roots of unity to test this possibility. They constructed a detailed argument showing that if such a system existed, it would require a specific arrangement of mathematical "twists" that simply could not coexist without contradiction. The numbers demanded by the rules of the system were too large to fit within the available space. Consequently, they proved with certainty that the case with the coefficient of eight is impossible. No such fusion category exists.

Next, they turned their attention to the case where the coefficient was zero. Here, the system behaves in a very specific way, effectively splitting into two distinct layers. The researchers showed that any system with this property must be mathematically equivalent to one of eight known solutions that had been identified in previous work. These solutions are tied to a specific algebraic structure known as the Kac–Paljutkin Hopf algebra. In this scenario, the researchers did not need to invent a new system; they simply confirmed that every possible realization of this zero-coefficient case falls neatly into one of these eight existing categories.

The remaining two possibilities, the coefficients of two and seven, were found to be not only possible but realizable. For the number seven, the researchers pointed to a known example from the representation theory of a specific group called PSU(3, 2). This group, which describes symmetries in a complex geometric space, naturally produces a fusion category that matches the rules perfectly. For the number two, the situation was more subtle. The team did not find a pre-existing example; instead, they constructed one from scratch. They built a new, explicit mathematical universe using a group of symmetries related to a finite field of numbers. By carefully defining how the objects interact and ensuring the rules of combination were consistent, they created a unitary fusion category that perfectly matches the requirements for the coefficient of two.

The final result is a complete classification of this specific family of systems. The researchers have determined that there are exactly three valid scenarios for a fusion category containing a Tambara–Yamagami subcategory and one extra object. The first corresponds to the coefficient of zero, which connects to a known list of eight solutions. The second corresponds to the coefficient of two, which the authors have explicitly constructed. The third corresponds to the coefficient of seven, which is realized by the representation of the group PSU(3, 2). The fourth candidate, the coefficient of eight, has been definitively ruled out. This work closes the book on this particular class of problems, providing a clear map of what is possible and what is not in this corner of mathematical reality. It demonstrates that while many numerical patterns look promising, only a few can sustain the complex architecture required to exist as a true fusion category.

Technical Summary: Grothendieck Rings of Near-Tambara–Yamagami Fusion Categories

Problem Statement
The classification of fusion categories remains a central structural problem at the intersection of representation theory, quantum topology, and mathematical physics. While the Grothendieck ring (fusion ring) renders fusion rules discrete and computable, it does not uniquely determine the associator; the pentagon equations impose highly restrictive constraints that often prevent a valid categorification of a given ring. This paper addresses the classification of "near-Tambara–Yamagami" fusion categories: fusion categories $\mathcal{C}$ over $\mathbb{C}$ that contain a non-pointed Tambara–Yamagami fusion subcategory $\mathcal{D}$ and exactly one additional simple object $\rho$ . The goal is to determine the possible Grothendieck rings for such categories and to establish which of these rings admit a categorification.

Methodology
The authors employ a combination of numerical constraints, Galois theory, and categorical reconstruction techniques:

Numerical Analysis and Formal Codegrees: The study begins by deriving necessary conditions on the parameters $n$ (where $|G|=n^2$ and $FPdim(t)=n$) and $\kappa$ (the coefficient of $\rho$ in $\rho^2$ ). Using the formal codegree theorem and the Drinfeld center $Z(\mathcal{C})$ , the authors analyze the dimensions of induced objects $I(X)$ .
Galois Reflections and Trace Bounds: For the irrational case (where the Frobenius–Perron dimension $d = FPdim(\rho)$ is not an integer), the authors utilize Galois conjugation to map the category to a pseudo-unitary one. They apply Galois reflections on the Drinfeld center and analyze Frobenius–Schur indicators ( $\nu_j$ ) to derive simultaneous constraints on sums of roots of unity. A quadratic trace bound and cyclotomic moment arguments are used to eliminate impossible parameter values.
Group-Theoretical Realizations: For the integral case, the authors analyze the structure of the subcategory generated by the invertible objects and the additional object. They utilize Tannakian reconstruction and the theory of sharply doubly transitive groups to constrain the group $G$ and the parameter $\kappa$ .
Explicit Construction: For surviving candidates, the authors construct explicit realizations. For $\kappa=7$ , they identify the category with $\text{Rep}(PSU(3, 2))$ . For $\kappa=2$ , they construct a unitary group-theoretical fusion category using the group $L = PGL_2(\mathbb{F}_3)$ , a specific 3-cocycle, and a subgroup $H \cong C_2 \times C_2$ .

Key Contributions and Results

Rank Reduction and Parameter Constraints: The authors prove that any such category must have rank 6. The group of invertible objects $G$ is necessarily isomorphic to the Klein four-group ( $C_2 \times C_2$ ), implying $n=2$ .
Classification of Candidates: Numerical analysis initially yields four candidate values for $\kappa$ : $0, 2, 7, 8$.
- $\kappa = 8$ (Non-existence): The paper proves that the candidate $\kappa=8$ (which corresponds to an irrational dimension $d = 4 + 2\sqrt{6}$ ) is impossible. This is achieved by constructing a multiset of roots of unity derived from the twists of the Drinfeld center and showing that the required moment conditions lead to a contradiction ( $46 \ge 48$ ).
- $\kappa = 0$ (Irrational Case): The case $\kappa=0$ (with $d=2\sqrt{2}$ ) is shown to be realizable. Every such category is tensor equivalent to an entry in the list of rank-one odd-component extensions of $\text{Rep}(H_8)$ (where $H_8$ is the Kac–Paljutkin Hopf algebra), as classified by Choi–Lu–Sun.
- $\kappa = 2$ (Integral Case): The authors construct an explicit unitary group-theoretical categorification for the ring $R_2$ . This involves the group $PGL_2(\mathbb{F}_3)$ and specific cohomological data.
- $\kappa = 7$ (Integral Case): The ring $R_7$ is realized by the representation category $\text{Rep}(PSU(3, 2))$ .
Main Theorem (Theorem 1.2): The paper establishes that for a fusion category $\mathcal{C}$ containing a non-pointed Tambara–Yamagami subcategory $\mathcal{D}$ and exactly one extra simple object, the Grothendieck ring $K_0(\mathcal{C})$ is isomorphic to $R_\kappa$ with $\kappa \in \{0, 2, 7\}$ . Each of these three based rings is categorifiable.

Significance and Claims
The paper claims to provide a complete classification of the based rings (Grothendieck rings) for this specific family of near-integral fusion categories. It demonstrates that while numerical conditions (such as formal codegrees and Frobenius–Perron dimensions) narrow the possibilities, the existence of a categorification is a much stricter condition that eliminates the $\kappa=8$ candidate.

The authors emphasize that their results serve the broader classification program by providing "reliable algebraic models" for topological and physical constructions. They explicitly state that for $\kappa=2$ and $\kappa=7$ , the results concern the existence of a categorification and the isomorphism of the based rings, but do not claim to classify all possible tensor equivalences (associators) for these rings, only that specific realizations exist. The work highlights the utility of the Drinfeld center and Galois symmetry in ruling out potential fusion rings that satisfy all elementary numerical constraints.

Technical Summary: Grothendieck Rings of Near-Tambara–Yamagami Fusion Categories

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