Related papers: Orbits of vectors in some representations. II

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such…

We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root…

The study of rational point sets on circles over the Euclidean plane is discussed in a more general framework, i.e. we generalize the notion rational and consider these circular point sets over arbitrary fields. We also determine the…

Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$, and $G$ is not metacyclic. Then $G$ always has a regular orbit on $V$ except for a few "small" cases. We completely…

For a representation of a finite group $G$ on a complex vector space $V$ we determine when a holomorphic $\binom{p}{q}$-tensor field on the principle stratum of the orbit space $V/G$ can be lifted to a holomorphic $G$-invariant tensor field…

Let $G$ be a finite group, $\pi(G)$ be the set of prime divisors dividing the order of $G$ and $\pi_e(G)$ (spectrum) denote the set of element orders of $G$. We define $w_o(G)$ = $|\pi(G)|$ the width of order of $G$ and $w_s(G)$ =…

In this note we obtain some distortion results for spirallike functions with respect to a boundary point. In particular, we find the maximal domain covered by all spirallike functions of order $\beta$.

The orbit space of a distributive binary $G$-space is studied. A number of its properties in the case of a compact binarily acting group $G$ are established.

Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the…

The semiclassical orbitals of a relativistic electron on a rotating sphere threaded by an intense magnetic dipole field are examined. Several physically distinct regimes emerge, depending on the relative sizes of the mass, total energy,…

We prove that a finite coprime linear group G in characteristic p>=(|G|-1)/2 has a regular orbit. This bound on p is best possible. We also give an application to blocks with abelian defect groups.

Let $\mathcal P_{\Phi}$ be the root polytope of a finite irreducible crystallographic root system $\Phi$, i.e., the convex hull of all roots in $\Phi$. The polar of $\mathcal P_{\Phi}$, denoted $\mathcal P_{\Phi}^*$, coincides with the…

Any gentle algebra $A$ with one maximal path corresponds to a unique quasi-diagram $\alpha$. We introduce the regularity for $\alpha$, and show that $A$ has finite global dimension if and only if $\alpha$ is regular. We characterize regular…

Let $L$ be a field of positive characteristic $p$ with a fixed algebraic closure $\overline{L}$, and let $\alpha_1,\alpha_2,\beta\in L$. For an integer $d\ge 2$, we consider the family of polynomials $f_{\lambda}(z) := z^d+\lambda$,…

Let $F$ be a non-Archimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. The orbit of an irreducible representation $\rho$ of $\mathrm{GL}_n(\mathcal{O}_F)$ is a conjugacy class in $\mathfrak{gl}_n(\mathcal{O}_F)$…

A group $G$ is called root graded if it has a family of subgroups $G_\alpha$ indexed by roots from a root system $\Phi$ satisfying natural conditions similar to Chevalley groups over commutative unital rings. For any such group there is a…

Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $\phi$ of $W$ over the finite field $F_2$ of two elements. The action of $\phi(W)$ on $F_2^n$ by left multiplication is…

Let $R$ be a finite local ring of odd characteristic and $\beta$ a non-degenerate symmetric bilinear form on $R^2$. In this short note, we determine the largest possible cardinality of pairwise orthogonal sets of unimodular vectors in…

Using a group-theoretic approach, a method for determining the equivalence classes (also called orbits) of the set of rules of one-dimensional cellular automata induced by the symmetry operations of reflection and permutation and their…

Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study…