Related papers: Non-induced modular representations of cyclic grou…

A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain…

For a set of nonnegative integers $A$, denote by $R_{A}(n)$ the number of unordered representations of the integer $n$ as the sum of two different terms from $A$. In this paper we partially describe the structure of the sets, which have…

Let $n$ be a positive integer and $G(n)$ denote the number of non-isomorphic finite groups of order $n$. It is well-known that $G(n) = 1$ if and only if $(n,\phi(n)) = 1$, where $\phi(n)$ and $(a, b)$ denote the Euler's totient function and…

We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \]…

A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…

Prunescu and Sauras-Altuzarra showed that all C-recursive sequences of natural numbers have an arithmetic div-mod representation that can be derived from their generating function. This representation consists of computing the quotient of…

Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. For $\pi$ an admissible smooth representation of $\operatorname{GL}_2(K)$ over $\mathbb{F}$ satisfying…

In this paper, we compute the $RO(C_n)$-graded coefficient ring of equivariant cohomology for cyclic groups $C_n$, in the case of Burnside ring coefficients, and in the case of constant coefficients. We use the invertible Mackey functors…

Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with the maximal ideal $\wp$ and the finite residue field of characteristic $p.$ Let $\mathbf{G}$ be the General Linear or Special Linear group with entries from…

Motivated by recent progress in operator representation of frames, we investigate the frames of the form $ \{T^n \varphi\}_{n\in I}$ for $ I=\mathbb{N}, \mathbb{Z} $, and answer questions about representations, perturbations and frames…

Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point…

Let p be an odd prime number. We show that there exists a finite group of order p^{p+3} whose the mod p cycle map from the mod p Chow ring of its classifying space to its ordinary mod p cohomology is not injective.

We revisit the construction of Castella and Do of an anticyclotomic Euler system for the $p$-adic Galois representation of a modular form, using diagonal classes. Combining this construction and some previous results of ours, we obtain new…

Let q be an odd positive integer and P \in F2[z] be of order q and such that P(0) = 1. We denote by A = A(P) the unique set of positive integers satisfying \sum_{n=0}^\infty p(A, n) z^n \equiv P(z) (mod 2), where p(A,n) is the number of…

The universal covering of SO(3) is modelled as a reflection group G_R in a representation independent fashion. For relativistic quantum fields, the Unruh effect of vacuum states is known to imply an intrinsic form of reflection symmetry,…

We prove that for any integer $n\geq 12$, and for every $r$ in the interval $[3, \ldots, \lfloor (n-1)/2\rfloor]$, the group $A_n$ has a string C-group representation of rank $r$ therefore showing that the only alternating group whose set…

We present an oracle factorisation algorithm which finds a nontrivial factor of almost all positive integers $n$ based on the knowledge of the number of points on certain elliptic curves in residue rings modulo $n$.

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper we derive the formulas describing linear representations of objects in the category of formal loops and apply them to obtain…

For a function $f\colon \mathbb{N}\to\mathbb{N}$, let $$ N^+_f(x)=\{n\leq x: n=k+f(k) \mbox{ for some } k\}. $$ Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…

We explore computational tools that allow to compute the class on the Grothendieck ring of varieties of finite cyclic quotients in some interesting examples. As an main application, we determine the motive of low rank representation…