Harm Derksen
Unipotent group actions on affine varieties.
Tue, 02/23/2010 - 16:01 — SomNambulAuthors: Harm Derksen, Stefan Maubach, Arno van den Essen, David R. Finston
Subjects: Algebraic Geometry
AbstractAlgebraic actions of unipotent groups $U$ actions on affine $k-$varieties $X$
($k$ an algebraically closed field of characteristic 0) for which the algebraic
quotient $X//U$ has small dimension are considered$.$ In case $X$ is factorial,
$O(X)^{\ast}=k^{\ast},$ and $X//U$ is one-dimensional, it is shown that
$O(X)^{U}$=$k[f]$, and if some point in $X$ has trivial isotropy, then $X$ is
$U$ equivariantly isomorphic to $U\times A^{1}(k).$ The main results are given
distinct geometric and algebraic proofs.General Presentations of Algebras.
Thu, 11/26/2009 - 16:01 — SomNambulAuthors: Harm Derksen, Jiarui Fei
Subjects: Rings and Algebras
AbstractFor any finite dimensional basic associative algebra, we study
subrepresentations and the canonical decomposition of a general presentation.
As a special case, we consider rigid presentations. We construct a simplicial
complex governing the canonical decompositions of rigid presentations. We show
how to complete a rigid presentation and study the number of nonisomorphic
direct summands and different complements.Valuative invariants for polymatroids.
Fri, 08/21/2009 - 19:51 — SomNambulAuthors: Harm Derksen, Alex Fink
Subjects: Combinatorics
AbstractMany important invariants for matroids and polymatroids, such as the Tutte
polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant
$\mathcal G$ introduced by the first author, are valuative. In this paper we
construct the $\Z$-modules of all $\Z$-valued valuative functions for labeled
matroids and polymatroids on a fixed ground set, and their unlabeled
counterparts, the $\Z$-modules of valuative invariants. We give explicit bases
for these modules and for their dual modules generated by indicator functions
of polytopes, and explicit formulas for their ranks.Valuative invariants for polymatroids.
Fri, 08/21/2009 - 19:51 — SomNambulAuthors: Harm Derksen, Alex Fink
Subjects: Combinatorics
AbstractMany important invariants for matroids and polymatroids, such as the Tutte
polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant
$\mathcal G$ introduced by the first author, are valuative. In this paper we
construct the $\Z$-modules of all $\Z$-valued valuative functions for labeled
matroids and polymatroids on a fixed ground set, and their unlabeled
counterparts, the $\Z$-modules of valuative invariants. We give explicit bases
for these modules and for their dual modules generated by indicator functions
of polytopes, and explicit formulas for their ranks.
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