Characteristic free description of semi-invariants of 2×22\times 2 matrices

A minimal homogeneous generating system of the algebra of semi-invariants of tuples of two-by-two matrices over an infinite field of characteristic two or over the ring of integers is given. In an alternative interpretation this yields a minimal system of homogeneous generators for the vector invariants of the special orthogonal group of degree four over a field of characteristic two or over the ring of integers. An irredundant separating system of semi-invariants of tuples of two-by-two matrices is also determined, it turns out to be independent of the characteristic.Comment: A crucial reference to a paper of A. Lopatin was adde

On singularities of quiver moduli

Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth.Comment: a known side result removed, a reference added, minor changes in expositio

Degree bound for separating invariants of abelian groups

It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is strictly smaller than the universal degree bound for generators of polynomial invariants, unless the goup is cyclic or is the direct product of $r$ even order cyclic groups where the number of two-element direct factors is not less than the integer part of the half of $r$. A characterization of separating sets of monomials is given in terms of zero-sum sequences over abelian groups

Helly dimension of algebraic groups

It is shown that for a linear algebraic group G over a field of characteristic zero, there is a natural number \kappa(G) such that if a system of Zariski closed cosets in G has empty intersection, then there is a subsystem consisting of at most \kappa(G) cosets with empty intersection. This is applied to the study of algebraic group actions on product varieties.Comment: 18 page

Noether bound for invariants in relatively free algebras

Let $\mathfrak{R}$ be a weakly noetherian variety of unitary associative algebras (over a field $K$ of characteristic 0), i.e., every finitely generated algebra from $\mathfrak{R}$ satisfies the ascending chain condition for two-sided ideals. For a finite group $G$ and a $d$-dimensional $G$-module $V$ denote by $F({\mathfrak R},V)$ the relatively free algebra in $\mathfrak{R}$ of rank $d$ freely generated by the vector space $V$. It is proved that the subalgebra $F({\mathfrak R},V)^G$ of $G$-invariants is generated by elements of degree at most $b(\mathfrak{R},G)$ for some explicitly given number $b(\mathfrak{R},G)$ depending only on the variety $\mathfrak{R}$ and the group $G$ (but not on $V$). This generalizes the classical result of Emmy Noether stating that the algebra of commutative polynomial invariants $K[V]^G$ is generated by invariants of degree at most $\vert G\vert$