We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.

In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset that is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras.

We give a description of the atoms in the Boolean algebra generated by the integral subsets of a finite group.

In this paper we show that, in the stable case, when m≥2n−1, the cohomology ring H*(Repn(m)B) of the representation variety with Borel mold Repn(m)B and are isomorphic as algebras. Here the degree of si is 2m−3 when 1≤i<n. In the unstable cases, when m≤2n−2, we also calculate the cohomology group H*(Repn(m)B) when n=3,4 . In the most exotic case, when m=2 , Rep n (2)B is homotopy equivalent to Fn (ℂ2)×PGL n (ℂ) , where Fn (ℂ2) is the configuration space of n distinct points in ℂ2. We regard Rep n (2)B as a scheme over ℤ, and show that the Picard group Pic (Rep n (2)B) of Rep n (2)B is isomorphic to ℤ/nℤ. We give an explicit generator of the Picard group.

We derive the spectrum of the Laplace–Beltrami operator on the quotient orbifold of the nonhyperbolic triangle groups.

Let $\mathbbm{k}$ be a (topological) field of characteristic 0. To any representation of a given Hopf algebra $\mathfrak{B}_n(\mathbbm{k})$, one can associate (using a Drinfeld associator) a representation $\widehat{\Phi}(\rho)$ of the braid group over the field $\mathbbm{k}((h))$ of Laurent series. We investigate the dependence on $\Phi$ of $\widehat{\Phi}(\rho)$ for a certain class of representations (so-called GT-rigid representations) and from this dependence deduce (continuous) projective representations of the Grothendieck–Teichmüller group $GT_1(\mathbbm{k})$; in particular, for $\mathbbm{k} = \mathbbm{Q}_l$ we obtain representations of the absolute Galois group of $\mathbbm{Q}(\mu_{l^{\infty}})$. In most situations, these projective representations can be decomposed into linear characters, as we do for the representations of the Iwahori–Hecke algebra of type A. In this case, moreover, we express $\widehat{\Phi}(\rho)$ when $\Phi$ is even and obtain unitary matrix models for the representations of the Iwahori–Hecke algebra. The representations of this algebra corresponding to hook diagrams have noteworthy properties under the action of $GT_1(\mathbbm{k})$.

The integers coprime to $n$ are called the totatives of $n$. D. H. Lehmer and Paul Erdős were interested in understanding when the number of totatives between $in/k$ and $\left( i\,+1 \right)n/k$ are $1/k\text{th}$ of the total number of totatives up to $n$. They provided criteria in various cases. Here we give an “if and only if” criterion which allows us to recover most of the previous results in this literature and to go beyond, as well to reformulate the problem in terms of combinatorial group theory. Our criterion is that the above holds if and only if for every odd character $\chi \,\left( \bmod \,\kappa \right)\,\left( \text{where}\,\kappa :=k/\gcd \left( k,\,n/{{\Pi }_{p|n}}p \right) \right)$ there exists a prime $p={{p}_{\chi }}$ dividing $n$ for which $\chi \left( p \right)=1.$

The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types.

Using naive algebraic geometric methods a new proof of the following celebrated theorem of Magnus is given: Let G be a group with a presentation having n generators and m relations. If G also has a presentation on n − m generators, then G is free of rank n − m.

We give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.