43,243 research outputs found
Non-linear Group Actions with Polynomial Invariant Rings and a Structure Theorem for Modular Galois Extensions
Let be a finite -group and a field of characteristic . We
show that has a \emph{non-linear} faithful action on a polynomial ring
of dimension such that the invariant ring is also
polynomial. This contrasts with the case of \emph{linear and graded} group
actions with polynomial rings of invariants, where the classical theorem of
Chevalley-Shephard-Todd and Serre requires to be generated by
pseudo-reflections.
Our result is part of a general theory of "trace surjective -algebras",
which, in the case of -groups, coincide with the Galois ring-extensions in
the sense of \cite{chr}. We consider the \emph{dehomogenized symmetric algebra}
, a polynomial ring with non-linear -action, containing as a
retract and we show that is a polynomial ring. Thus turns out to be
\emph{universal} in the sense that every trace surjective -algebra can be
constructed from by "forming quotients and extending invariants". As a
consequence we obtain a general structure theorem for Galois-extensions with
given -group as Galois group and any prescribed commutative -algebra
as invariant ring. This is a generalization of the Artin-Schreier-Witt theory
of modular Galois field extensions of degree .Comment: 20 page
The action of the primitive Steenrod-Milnor operations on the modular invariants
We compute the action of the primitive Steenrod-Milnor operations on
generators of algebras of invariants of subgroups of general linear group
GL_n=GL(n,F_p) in the polynomial algebra with p an odd prime number.Comment: This is the version published by Geometry & Topology Monographs on 14
November 200
Constants of Weitzenb\"ock derivations and invariants of unipotent transformations acting on relatively free algebras
In commutative algebra, a Weitzenb\"ock derivation is a nonzero triangular
linear derivation of the polynomial algebra in several
variables over a field of characteristic 0. The classical theorem of
Weitzenb\"ock states that the algebra of constants is finitely generated. (This
algebra coincides with the algebra of invariants of a single unipotent
transformation.) In this paper we study the problem of finite generation of the
algebras of constants of triangular linear derivations of finitely generated
(not necessarily commutative or associative) algebras over assuming that
the algebras are free in some sense (in most of the cases relatively free
algebras in varieties of associative or Lie algebras). In this case the algebra
of constants also coincides with the algebra of invariants of some unipotent
transformation. \par The main results are the following: 1. We show that the
subalgebra of constants of a factor algebra can be lifted to the subalgebra of
constants. 2. For all varieties of associative algebras which are not nilpotent
in Lie sense the subalgebras of constants of the relatively free algebras of
rank are not finitely generated. 3. We describe the generators of the
subalgebra of constants for all factor algebras modulo a
-invariant ideal . 4. Applying known results from commutative
algebra, we construct classes of automorphisms of the algebra generated by two
generic matrices. We obtain also some partial results on relatively
free Lie algebras.Comment: 31 page
Cayley-Hamilton theorem for 2 × 2 matrices over the Grassmann algebra
AbstractIt is shown that the characteristic polynomial of matrices over a Lie nilpotent ring introduced recently by Szigeti is invariant with respect to the conjugation action of the general linear group. Explicit generators of the corresponding algebra of invariants in the case of 2 × 2 matrices over an algebra over a field of characteristic zero satisfying the identity [[x, y], z] = 0 are described. In this case the coefficients of the characteristic polynomial are expressed by traces of powers of the matrix, yielding a compact form of the Cayley-Hamilton equation of 2 × 2 matrices over the Grassmann algebra
- …