688 research outputs found
Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer
We extend in several directions invariant theory results of Chevalley,
Shephard and Todd, Mitchell and Springer. Their results compare the group
algebra for a finite reflection group with its coinvariant algebra, and compare
a group representation with its module of relative coinvariants. Our extensions
apply to arbitrary finite groups in any characteristic.Comment: The applications and Examples in section 4 have been extende
Rationality of Hilbert series in noncommutative invariant theory
It is a fundamental result in commutative algebra and invariant theory that a
finitely generated graded module over a commutative finitely generated graded
algebra has rational Hilbert series, and consequently the Hilbert series of the
algebra of polynomial invariants of a group of linear transformations is
rational, whenever this algebra is finitely generated. This basic principle is
applied here to prove rationality of Hilbert series of algebras of invariants
that are neither commutative nor finitely generated. Our main focus is on
linear groups acting on certain factor algebras of the tensor algebra that
arise naturally in the theory of polynomial identities.Comment: Examples both from commutative and noncommutative invariant theory
are included, a problem is formulated and references are added. Comments for
v3: references added, minor revisio
The determination of integral closures and geometric applications
We express explicitly the integral closures of some ring extensions; this is
done for all Bring-Jerrard extensions of any degree as well as for all general
extensions of degree < 6; so far such an explicit expression is known only for
degree < 4 extensions. As a geometric application we present explicitly the
structure sheaf of every Bring-Jerrard covering space in terms of coefficients
of the equation defining the covering; in particular, we show that a degree-3
morphism f : Y --> X is quasi-etale if and only if the first Chern class of the
sheaf f_*(O_Y) is trivial (details in Theorem 5.3). We also try to get a
geometric Galoisness criterion for an arbitrary degree-n finite morphism; this
is successfully done when n = 3 and less satifactorily done when n = 5.Comment: Advances in Mathematics, to appear (no changes, just add this info
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