Related papers: On the classification of almost square-free modula…

We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group $G$ as induced from module categories over fusion…

For a positive integer $N$, let $\mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $\mathscr C(N)(\mathbb Q):=\mathscr C(N)\cap J_0(N)(\mathbb Q)$ be its $\mathbb…

In this paper, we present an explicit structure for the Baer invariant of a free nilpotent group (the $n$-th nilpotent product of the infinite cyclic group, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*... \st{n}*\textbf{Z}$) with respect to the…

Let $G$ be a group and write $\mbox{Perm}(G)$ for its symmetric group. Define $\mbox{Hol}(G)$ to be the holomorph of $G$, regarded as a subgroup of $\mbox{Perm}(G)$, and let $\mbox{NHol}(G)$ denote its normalizer. The quotient $T(G) =…

We construct a series of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and…

For the cyclic group $C_2$ we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring $\underline{\mathbb{Z}/\ell}$, for $\ell$ a prime. This is fairly simple for $\ell$ odd, but for…

Given a negative $D>-(\log X)^{\log 2-\delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of…

We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra $A$ generated by an irreducible representation of such a group has…

An algebraic category $\mathcal{C}$ is called balanced if the cotriple cohomology of any object of $\mathcal{C}$ vanishes in positive dimensions on injective coefficient modules. Important examples of balanced and of non-balanced categories…

Given an oligomorphic group $G$ and a measure $\mu$ for $G$ (in a sense that we introduce), we define a rigid tensor category $\underline{\mathrm{Perm}}(G; \mu)$ of "permutation modules," and, in certain cases, an abelian envelope…

Suppose that a metacyclic Frobenius group $FH$, with kernel $F$ and complement $H$, acts by automorphisms on a finite group $G$, in such a way that $C_G(F)$ is trivial and $C_G(H)$ is nilpotent. It is known that $G$ is nilpotent and its…

The purpose of this paper is to investigate the finite Frobenius groups with "perfect order classes"; that is, those for which the number of elements of each order is a divisor of the order of the group. If a finite Frobenius group has…

Let $k$ and $n$ be positive integers. Let $c\phi_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$ and $\mathrm{C}\Phi_k(q)$ be the generating function of $c\phi_{k}(n)$. In this article, we study…

Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number $\ell$ not dividing…

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the…

Let $\mathcal{C}\subseteq \mathbb{N}^p$ be an integer cone. A $\mathcal{C}$-semigroup $S\subseteq \mathcal{C}$ is an affine semigroup such that the set $\mathcal{C}\setminus S$ is finite. Such $\mathcal{C}$-semigroups are central to our…

It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We…

Let $H$ be a subgroup of $\text{Sym}_n$, the symmetric group of degree $n$. For a fixed integer $l \geq 2$, the group $G$ presented with generators $x_1, x_2, \ldots ,x_n$ and with relations $x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma (i_1)}…

We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the…

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon >…