Related papers: On Frobenius-Perron Dimension

We introduce the notion of the Frobenius-Perron dimension of an integral $\Bbb Z_+$-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of…

From the perspective of $\tau$-tilting theory, we study Frobenius--Perron dimensions of finite-dimensional algebras. First, we evaluate the Frobenius--Perron dimensions of $\tau$-tilting finite algebras by a combinatorial method in…

The Frobenius-Perron dimension for an abelian category was recently introduced. We apply this theory to the category of representations of the finite-dimensional radical squared zero algebras associated to certain modified ADE graphs. In…

The spectral radius of matrix, also known as Frobenius-Perron dimension, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study…

Here we study bounds on the Frobenius-Schur exponent of spherical fusion categories based on their global dimension generalizing bounds from the representation theory of finite-dimensional quasi-Hopf algebras. Our main result is that if the…

We establish relations between representation dimensions of two algebras connected by a Frobenius bimodule or extension. Consequently, upper bounds and equality formulas for representation dimensions of group algebras, symmetric separably…

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius-Perron dimension $p^nm$, where $p$ is a prime number, $m$ is a square-free natural number and ${\rm gcd}(p,m)=1$. We prove that if…

The integral group rings $\mathbb{Z}G$ for finite groups $G$ are precisely those fusion rings whose basis elements have Frobenius-Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a…

Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite…

Let $R=\mathbb{F}_p[x_1,\ldots,x_n]$ and let $\mathbf{F}$ be the ring of Frobenius operators over $R$. We introduce a notion of Bernstein dimension and multiplicity for the class of finitely generated $\mathbf{F}$-modules whose structure…

From a unifying lemma concerning fusion rings, we prove a collection of number-theoretic results about fusion, braided, and modular tensor categories. First, we prove that every fusion ring has a dimensional grading by an elementary abelian…

We study the Frobenius-Perron dimension of representation-directed algebras and quotient algebras of canonical algebras of type ADE, prove that the Frobenius-Perron dimension of a representation-directed algebra is always zero and the…

We introduce the Frobenius-Perron dimension of an endofunctor of a k-linear category and provide some applications.

Let k be an algebraically closed field of characteristic zero. In this paper we prove that fusion categories of Frobenius-Perron dimensions 84 and 90 are of Frobenius type. Combining this with previous results in the literature, we obtain…

Integral modular categories of Frobenius-Perron dimension $pq^n$, where $p$ and $q$ are primes, are considered. It is already known that such categories are group-theoretical in the cases of $0 \leq n \leq 4$. In the general case we…

We use the Perron-Frobenius Theorem to define, study and, in some sense, classify special simple modules over arbitrary finite dimensional positively based algebras. For group algebras of finite Weyl groups with respect to the…

This article is divided into two parts. In the first part we work over a field $\mathbb{k}$ and prove that the Frobenius space associated to a Frobenius algebra is generated as left A-module by the Frobenius coproduct. In particular, we…

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion…

In this paper we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps. In particular, a few classical results of Steinberg and Deligne & Lusztig on complex representations of finite…

The Frobenius-Perron theory of an endofunctor of a $\Bbbk$-linear category (recently introduced in [CG]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories…