10 research outputs found
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
The Noether number of the non-abelian group of order 3p
It is proven that for any representation over a field of characteristic 0 of
the non-abelian semidirect product of a cyclic group of prime order p and the
group of order 3 the corresponding algebra of polynomial invariants is
generated by elements of degree at most p+2. We also determine the exact degree
bound for any separating system of the polynomial invariants of any
representation of this group in characteristic not dividing 3p.Comment: 12 page
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 even
order cyclic groups where the number of two-element direct factors is not less
than the integer part of the half of . A characterization of separating sets
of monomials is given in terms of zero-sum sequences over abelian groups
Separating invariants for arbitrary linear actions of the additive group
We consider an arbitrary representation of the additive group G_a
over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants
Separating invariants for arbitrary linear actions of the additive group
We consider an arbitrary representation of the additive group G_a
over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants
Explicit separating invariants for cyclic p-groups
Cataloged from PDF version of article.We consider a finite-dimensional indecomposable modular representation of a cyclic p-group and we give a recursive description of an associated separating set: We show that a separating set for a representation can be obtained by adding, to a separating set for any subrepresentation, some explicitly defined invariant polynomials. Meanwhile, an explicit generating set for the invariant ring is known only in a handful of cases for these representations. © 2010 Elsevier Inc. All rights reserved
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 typically strictly smaller than the universal degree bound for generators of polynomial invariants. More precisely, these degree bounds can be equal only if the group is cyclic or is the direct sum of r even order cyclic groups where the number of two-element direct summands 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. © 2017 American Mathematical Society
Mapping toric varieties into low dimensional spaces
A smooth d-dimensional projective variety X can always be embedded into 2d + 1-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any d-dimensional projective variety can be mapped injectively to 2d + 1-dimensional projective space. A natural question then arises: what is the minimal m such that a projective variety can be mapped injectively to m-dimensional projective space? In this paper we investigate this question for normal toric varieties, with our most complete results being for Segre-Veronese varieties
Degree bounds for fields of rational invariants of and other finite groups
Degree bounds for algebra generators of invariant rings are a topic of
longstanding interest in invariant theory. We study the analogous question for
field generators for the field of rational invariants of a representation of a
finite group, focusing on abelian groups and especially the case of
. The inquiry is motivated by an application to signal
processing. We give new lower and upper bounds depending on the number of
distinct nontrivial characters in the representation. We obtain additional
detailed information in the case of two distinct nontrivial characters. We
conjecture a sharper upper bound in the case, and pose
questions for further investigation.Comment: 39 pages, 1 tabl