Related papers: Representation functions of additive bases for abe…

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions.…

For a finite group $G$, we consider the zeta function $\zeta_G(s) = \sum_{H} \abs{H}^{-s}$, where $H$ runs over the subgroups of $G$. First we give simple examples of abelian $p$-group $G$ and non-abelian $p$-group $G'$ of order $p^m, \; m…

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…

In this paper, we prove some results on the asymptotic behavior arising in modular representation theory over abelian $p$-groups. First, we embed the representation ring of a cyclic $p$-group into a real algebra of functions. Second, we…

Filling functions are asymptotic invariants of finitely presentable groups; the seminal work on the subject is by M.Gromov. They record features of combinatorial homotopy discs (van Kampen diagrams) filling loops in Cayley 2-complexes.…

We give a proof, using so-called fusion rings and q-deformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring Gr(O(\infinity)). This is obtained here as a…

Let $G$ be an infinite abelian group with $|2G|=|G|$. We show that if $G$ is not the direct sum of a group of exponent 3 and the group of order 2, then $G$ possesses a perfect additive basis; that is, there is a subset $S\subseteq G$ such…

Let $G$ be a connected semisimple Lie group with finite centre and $K$ be a maximal compact subgroup thereof. Given a function $u$ on $G$, we define $\mathcal{A} u$ to be the root mean square average over $K$, acting both on the left and…

Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $\zeta_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $\zeta_G(s)$ has…

Given an abstract group $G$, we study the function $ab_n(G) := \sup_{|G:H| \leq n} |H/[H,H]|$. If $G$ has no abelian composition factors, then $ab_n(G)$ is bounded by a polynomial: as a consequence, we find a sharp upper bound for the…

In this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group G. We show that the set SAT(G,k) of all equations in k > 1 variables over G which are satisfiable in G has an intermediate…

Let $G$ be a real Lie group with Lie algebra $\mathfrak g$. Given a unitary representation $\pi$ of $G$, one obtains by differentiation a representation $d\pi$ of $\mathfrak g$ by unbounded, skew-adjoint operators. Representations of…

For a finite group $G$, the representation dimension is the smallest integer realizable as the degree of a complex faithful representation of $G$. In this article, we compute representation dimension for some $p$-groups, their direct…

In this article, we investigate the representability of actions of the category $\mathsf{Nil}_2(\mathsf{Grp})$ of $2$-nilpotent groups. We first provide an algebraic characterisation of derived actions in $\mathsf{Nil}_2(\mathsf{Grp})$ by…

Let $G$ be a finitely generated abelian group, and let $S = A[x_1, ..., x_n]$ be a $G$-graded polynomial ring over a commutative ring $A$. Let $I_1, ..., I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded…

For a finite group $G$ and a non-negative integer $d$, denote by $\Omega_d(G)$ the number of functions $G^d\rightarrow G$ that are induced by substitution into a word with variables among $X_1,\ldots,X_d$. In this note, we show that…

Consider an arithmetic group $\mathbf{G}(O_S)$, where $\mathbf{G}$ is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of $S$-integers $O_S$ of a number field $K$ with…

The notion of a shift-compact set in an abelian topological group $X$ plays a significant role in functional equations and inequalities, especially so since each Borel set that is not Haar-meagre, alternatively not Haar-null, is necessarily…

A global representation is a compatible collection of representations of the outer automorphism groups of the finite groups belonging to a family $\mathscr{U}$. These arise in classical representation theory, in the study of representation…

We demonstrate that any function $f$ from a finite set $Y$ to itself can be represented linearly. Specifically, we prove the existence of an injective map $j$ from $Y$ into a modular ring $\mathbb{Z}/m\mathbb{Z}$ and a constant $a \in…