Related papers: Minimal Faithful Permutation Degrees for Irreducib…

We define $\Psi$-autoreducible sets given an autoreduction procedure $\Psi$. Then, we show that for any $\Psi$, a measurable class of $\Psi$-autoreducible sets has measure zero. Using this, we show that classes of cototal, uniformly…

This work gives a classification of imprimitive irreducible finite subgroups of the orthogonal group O(7) plus the number of conjugate classes for each group.

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of…

Let R be a commutative noetherian ring. We give criteria for a complex of cotorsion flat R-modules to be minimal, in the sense that every self homotopy equivalence is an isomorphism. To do this, we exploit Enochs' description of the…

This paper establishes Noether's classical degree bound $\beta(G) \le |G|$ for finite and linearly reductive group schemes. On the other hand, we provide examples of infinitesimal group schemes where $\beta(G)$ is unbounded. We also…

The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $E_{8}$ or its highest weight is minuscule. In this paper, we prove…

Given a finite group $G$, we introduce the \textit{permutability degree} of $G$, as $$pd(G)=\frac{1}{|G| \ |\mathcal{L}(G)|} {\underset{X \in \mathcal{L}(G)}\sum}|P_G(X)|,$$ where $\mathcal{L}(G)$ is the subgroup lattice of $G$ and $P_G(X)$…

We introduce a discrete deformation of Rieffel type for finite (quantum) groups. Using this, we give a non-trivial example of a finite quantum group of order 18. We also give a deformation of finite groups of Lie type by using their maximal…

We calculate the fundamental groups $\pi=\pi_1(P^2\setminus B)$ for all irreducible plane sextics $B\subset\P^2$ with simple singularities for which $\pi$ is known to admit a dihedral quotient $D_{10}$. All groups found are shown to be…

Let $V,W$ be representations of a cyclic group $G$ of prime order $p$ over a field $k$ of characteristic $p$. The module of covariants $k[V,W]^G$ is the set of $G$-equivariant polynomial maps $V \rightarrow W$, and is a module over…

Let $G$ be a finite permutation group acting on $\Omega$. A base for $G$ is a subset $B \subseteq \Omega$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the…

We determine the minimal number of separating invariants for the invariant ring of a matrix group $G < \mathrm{GL}_n(\mathbb{F}_q)$ over the finite field $\mathbb{F}_q$. We show that this minimal number can be obtained with invariants of…

We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new…

Answering a question of I. M. Isaacs, we show that the largest degree of irreducible complex representations of any finite non-abelian simple group can be bounded in terms of the smaller degrees. We also study the asymptotic behavior of…

This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying…

In a recent paper we found conditions for a nilpotent Lie group $N$ to have a filtration by normal subgroups whose successive quotients have square integrable representations, and such that these square integrable representations fit…

In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes),…

A right-angled Coxeter group is a group with a given set of generators of order two, subject only to the relations that certain pairs of the generators commute. Various papers have shown how homological properties of the Coxeter group are…

The Medvedev degree of a subshift is a dynamical invariant of computable origin that can be used to compare the complexity of subshifts that contain only uncomputable configurations. We develop theory to describe how these degrees can be…

We compute for reflection groups of type $A,B,D,F_4,H_3$ and for dihedral groups a statistic counting the maximal cardinality of a set of elements in the group whose generalized inversions yield the full set of inversions and which are…