Eric C. Rowell
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View article: Egyptian fractions for few primes
Egyptian fractions for few primes Open
We study solutions to the Egyptian fractions equation with the prime factors of the denominators constrained to lie in a fixed set of primes. We evaluate the effectiveness of the greedy algorithm in establishing bounds on such solutions. A…
View article: A Categorical Perspective on Braid Representations
A Categorical Perspective on Braid Representations Open
We study categories whose objects are the braid representations, i.e. strict monoidal functors $F\colon B\rightarrow Mat$ from the braid category $B$ to the category of matrices $Mat$. Braid representations are equivalent to solutions to t…
View article: Low-dimensional indecomposable representations of the braid group $B_3$
Low-dimensional indecomposable representations of the braid group $B_3$ Open
In this note we give a complete classification of all indecomposable yet reducible representations of $B_3$ for dimensions $2$ and $3$ over an algebraically closed field $K$ with characteristic $0$, up to equivalence. We illustrate their u…
View article: The Condensed Fiber Product and Zesting
The Condensed Fiber Product and Zesting Open
We introduce the condensed fiber product of two $G$-crossed braided fusion categories, generalizing existing constructions in the literature. We show that this product is closely related to the cohomological construction known as zesting. …
View article: Braided Zestings of Verlinde Modular Categories and Their Modular Data
Braided Zestings of Verlinde Modular Categories and Their Modular Data Open
Zesting of braided fusion categories is a procedure that can be used to obtain new modular categories from a modular category with non-trivial invertible objects. In this paper, we classify and construct all possible braided zesting data f…
View article: Integral non-group-theoretical modular categories of dimension $p^2q^2$
Integral non-group-theoretical modular categories of dimension $p^2q^2$ Open
We construct all integral non-group-theoretical modular categories of dimension $p^2q^2$, where $p$ and $q$ are distinct prime numbers, establishing that a necessary and sufficient condition for their existence is that $p \mid q+1$, and th…
View article: Braided Zestings of Verlinde Modular Categories and Their Modular Data
Braided Zestings of Verlinde Modular Categories and Their Modular Data Open
Zesting of braided fusion categories is a procedure that can be used to obtain new modular categories from a modular category with non-trivial invertible objects. In this paper, we classify and construct all possible braided zesting data f…
View article: Solutions to the constant Yang-Baxter equation: additive charge conservation in three dimensions
Solutions to the constant Yang-Baxter equation: additive charge conservation in three dimensions Open
We find all solutions to the constant Yang--Baxter equation $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$ in three dimensions, subject to an additive charge-conservation ansatz. This ansatz is a generalisation of (strict) charge-conservation, fo…
View article: Classification of modular data up to rank 12
Classification of modular data up to rank 12 Open
We use the computer algebra system GAP to classify modular data up to rank 12. This extends the previously obtained classification of modular data up to rank 6. Our classification includes all the modular data from modular tensor categorie…
View article: Generalisations of Hecke algebras from loop braid groups
Generalisations of Hecke algebras from loop braid groups Open
We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation…
View article: On near-group centers and super-modular categories
On near-group centers and super-modular categories Open
The construction and classification of super-modular categories is an ongoing project, of interest in algebra, topology and physics. In a recent paper, Cho, Kim, Seo and You produced two mysterious families of super-modular data, with no k…
View article: Classification of charge-conserving loop braid representations
Classification of charge-conserving loop braid representations Open
Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conservin…
View article: $G$-crossed braided zesting
$G$-crossed braided zesting Open
For a finite group $G$, a $G$-crossed braided fusion category is $G$-graded fusion category with additional structures, namely a $G$-action and a $G$-braiding. We develop the notion of $G$-crossed braided zesting: an explicit method for co…
View article: Braids, Motions and Topological Quantum Computing
Braids, Motions and Topological Quantum Computing Open
The topological model for quantum computation is an inherently fault-tolerant model built on anyons in topological phases of matter. A key role is played by the braid group, and in this survey we focus on a selection of ways that the mathe…
View article: The Witt classes of so(2r)2r
The Witt classes of so(2r)2r Open
We study the Witt classes of the modular categories $SO(2r)_{2r}$ associated with quantum groups of type $D_r$ at $4r-2$th roots of unity. From these classes we derive infinitely many Witt classes of order 2 that are linearly independent m…
View article: Reconstructing Braided Subcategories of $SU(N)_k$
Reconstructing Braided Subcategories of $SU(N)_k$ Open
Ocneanu rigidity implies that there are finitely many (braided) fusion categories with a given set of fusion rules. While there is no method for determining all such categories up to equivalence, there are a few cases for which can. For ex…
View article: Classification of spin-chain braid representations
Classification of spin-chain braid representations Open
A braid representation is a monoidal functor from the braid category $\mathsf{B}$, for example given by a solution to the constant Yang-Baxter equation. Given a monoidal category $\mathsf{C}$ with $ob(\mathsf{C})=\mathbb{N}$, a rank-$N$ ch…
View article: Metaplectic categories, gauging and property $F$
Metaplectic categories, gauging and property $F$ Open
$N$-Metaplectic categories, unitary modular categories with the same fusion rules as $SO(N)_2$, are prototypical examples of weakly integral modular categories generalizing the model for the Ising anyons, i.e. metaplectic anyons. A conject…
View article: Symplectic level-rank duality via tensor categories
Symplectic level-rank duality via tensor categories Open
We give two proofs of a level-rank duality for braided fusion categories obtained from quantum groups of type $C$ at roots of unity. The first proof uses conformal embeddings, while the second uses a classification of braided fusion catego…
View article: On Realizing Modular Data
On Realizing Modular Data Open
We use zesting and symmetry gauging of modular tensor categories to analyze some previously unrealized modular data obtained by Grossman and Izumi. In one case we find all realizations and in the other we determine the form of possible rea…
View article: Classification of super-modular categories
Classification of super-modular categories Open
We develop categorical and number theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank $8$. In particular we find three distinct f…
View article: Braid group representations from twisted tensor products of algebras
Braid group representations from twisted tensor products of algebras Open
We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed mo…
View article: Integral Metaplectic Modular Categories
Integral Metaplectic Modular Categories Open
A braided fusion category is said to have Property $\textbf{F}$ if the associated braid group representations factor over a finite group. We verify integral metaplectic modular categories have property $\textbf{F}$ by showing these categor…
View article: Modular categories of dimension $p^3m$ with $m$ square-free
Modular categories of dimension $p^3m$ with $m$ square-free Open
We give a complete classification of modular categories of dimension $p^3m$ where $p$ is prime and $m$ is a square-free integer. When $p$ is odd, all such categories are pointed. For $p=2$ one encounters modular categories with the same fu…
View article: Metaplectic Categories, Gauging and Property F
Metaplectic Categories, Gauging and Property F Open
$N$-Metaplectic categories, unitary modular categories with the same fusion rules as $SO(N)_2$, are prototypical examples of weakly integral modular categories. As such, a conjecture of the second author would imply that images of the brai…
View article: On invariants of Modular categories beyond modular data
On invariants of Modular categories beyond modular data Open
We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix--the quantum invariant of a colored framed Whi…
View article: Mathematics of topological quantum computing
Mathematics of topological quantum computing Open
In topological quantum computing, information is encoded in “knotted” quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by exp…