Related papers: $\mathrm{GE}_2$-rings and a graph of unimodular ro…

In this paper, we build on the work from our previous paper (arXiv:2002.01556) to show that periodic rational $G$-equivariant topological $K$-theory has a unique genuine-commutative ring structure for $G$ a finite abelian group. This means…

For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for…

In this paper we completely describe the unital compressed commuting graph of the ring $\mathcal{M}_3(\mathrm{GF}(p))$ of $3 \times 3$ matrices over the finite prime field $\mathrm{GF}(p)$. To achieve this we combine methods from linear…

A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs G and H are 2-isomorphic, or equivalently, their cycle matroids are isomorphic, if and only if G can be transformed into H by a series of operations called Whitney…

Let $R$ be a commutative ring with $1\not = 0$, $Z(R)$ be the set of all zero-divisors of $R$, and $n \geq 1$. This paper introduces the $n$-total graph of a commutative ring $R$. The $n$-total graph of a commutative ring $R$, denoted by…

Johnson's characterization of amenable groups states that a discrete group $\Gamma$ is amenable if and only if $H_b^{n \geq 1}(\Gamma; V) = 0$ for all dual normed $\mathbb{R}[\Gamma]$-modules V. In this paper, we extend the previous result…

We show that if G is any nilpotent, finite group, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle.

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order…

In this note, starting with any group homomorphism $f\colon\Gamma\to G$, which is surjective upon abelianization, we construct a universal central extension $u\colon U\twoheadrightarrow G,$ UNDER $\Gamma$ with the same surjective property,…

Let $\Gamma$ denote a finite, connected graph with vertex set $X$. Fix $x \in X$ and let $\varepsilon \ge 3$ denote the eccentricity of $x$. For mutually distinct scalars $\{\theta^*_i\}_{i=0}^\varepsilon$ define a diagonal matrix…

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating…

Let $A$ be an algebra over a commutative ring $k$. We prove that braidings on the category of $A$-bimodules are in bijective correspondence to canonical R-matrices, these are elements in $A\ot A\ot A$ satisfying certain axioms. We show that…

In \cite{1}, a class of gyrogroups $(G,\odot)$ has been associated to certain groups $(G,\cdot)$. We give a necessary and sufficient condition for $(G,\odot)$ to be gyrocommutative. We also prove that under a suitable assumption two finite…

In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module $A\to 1$. The cohomology of the Lie 2-groups corresponding to the…

The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $\Gamma$ is located between…

The power graph of a group $G$ is a graph with vertex set $G$, in which two vertices are adjacent if one is some power of the other. In the commuting graph, with $G$ as the vertex set, two vertices are joined by an edge if they commute in…

Let $\mathcal{A}$ be a laterally complete commutative regular algebra and $X$ be a laterally complete $\mathcal{A}$-module. In this paper we introduce a notion of passport $\Gamma(X)$ for $X$, which consist of uniquely defined partition of…

Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$ on $X$, denoted by ${\cal C}(G,X)$, is the graph that has $X$ as its vertex set with $x,y\in X$ joined by an edge whenever $x\neq y$ and $xy=yx$. Let $T$ be a…

Let $G=GL_n(K)$ be the general linear group defined over an infinite field $K$ of positive characteristic $p$ and let $\Delta(\lambda)$ be the Weyl module of $G$ which corresponds to a partition $\lambda$. In this paper we classify all…

We prove that $\mathrm{St}(n, A)$ is a crossed module over $\mathrm{GL}(n, A)$ under a local stable rank condition on an algebra $A$ over a commutative ring. Our proof uses only elementary localization techniques in terms of pro-groups and…