Related papers: Decomposition of reductive regular prehomogeneous …

Let $K$ be an algebraically closed field of characteristic zero, and let $G$ be a connected reductive algebraic group over $K$. We address the problem of classifying triples $(G,H,V)$, where $H$ is a proper connected subgroup of $G$, and…

We obtain a decomposition formula of a representation of Sp(p,q) and SO^\ast(2n) unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules.…

Let G be a complex reductive group and V a G-module. Let \pi: V \to V//G be the quotient morphism and set N(V) = \pi^{-1}(\pi(0)). We consider the following question. Is the null cone N(V) reduced, i.e., is the ideal of N(V) generated by…

Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary…

The notion of a \emph{$G$-completely reducible} subgroup is important in the study of algebraic groups and their subgroup structure. It generalizes the usual idea of complete reducibility from representation theory: a subgroup $H$ of a…

We expand the existing arsenal of methods for exploring the irreducible components of the varieties $Rep(A,\bold d)$ which parametrize the representations with dimension vector $\bold d$ of a finite dimensional algebra $A$. To do so, we…

Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of…

Let $p>0$ be a prime, $k$ a field of characteristic $p$ and $G$ and elementary abelian $p$-group of order $q = p^n$. Let $W$ be an indecomposable $kG$-module of dimension 2 and define $V_i=S^{i-1}(W^*)$ for each $i=1 \ldots q$. We show that…

We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra \g and the resonance varieties R(V,K)\subset V^* attached to irreducible \g-modules V and submodules K\subset V\wedge V. In the…

Fix any algebraic extension $\mathbb K$ of the field $\mathbb Q$ of rationals. In this article we study exponential sets $V\subset \mathbb R^n$. Such sets are described by the vanishing of so called exponential polynomials, i.e.,…

We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is…

The coadjoint representation of a connected algebraic group $Q$ with Lie algebra $\mathfrak q$ is a thrilling and fascinating object. Symmetric invariants of $\mathfrak q$ (= $\mathfrak q$-invariants in the symmetric algebra $S(\mathfrak…

Let G be a simple, simply connected and connected algebraic group over an algebraically closed field of characteristic p>0, and let V be a rational G-module such that dim V <= p. According to a result of Jantzen, V is completely reducible,…

Among connected linear algebraic groups, quasi-reductive groups generalize pseudo-reductive groups, which in turn form a useful relaxation of the notion of reductivity. We study quasi-reductive groups over non-archimedean local fields,…

This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $GL_d(\mathbb{F}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module…

Let $G$ be a simple algebraic group over an algebraically closed field. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if, whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi…

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible $A_1$ subgroups of exceptional algebraic groups $G$. Consequences are given…

Let $G$ be a connected reductive group. We find a necessary and sufficient condition for a quasiaffine homogeneous space of $G$ to be embeddable into an irreducible $G$-module. In addition, for an affine homogeneous space we find a…

Let A be a tame quasi-tilted algebra and d the dimension vector of an indecomposable A-module. In the paper we prove that each irreducible component of the variety of A-modules of dimension vector d is regular in codimension one.

We classify irreducible representations of connected compact Lie groups whose orbit space is isometric to the orbit space of a representation of a finite extension of (positive dimensional) toric group. They turn out to be exactly the…