edoc: No conditions. Results ordered -Date Deposited. 2024-10-08T10:31:53ZEPrintshttps://edoc.unibas.ch/images/uni-logo.jpghttps://edoc.unibas.ch/2020-09-21T14:44:47Z2020-10-12T10:08:08Zhttps://edoc.unibas.ch/id/eprint/64412This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/644122020-09-21T14:44:47ZCompression of finite group actions and covariant dimensionLet G be a finite group and f: V -< W an equivariant morphism of finite dimensional G-modules, classically called a "covariant". We say that f is faithful if G acts faithfully on the image f(V). The covariant dimension of G is the minimum of the dimension of f(V) taken over all faithful covariants f. The essential dimension of G is defined in the same way, but allows for rational equivariant morphisms. The essential dimension and covariant dimension of G are related to cohomological invariants, generic polynomials and other topics, see the work of Buehler-Reichstein [BuR97]. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify the groups of covariant dimension less or equal to 2. It turns out that they are the finite subgroups of GL(2,C). A byproduct of our investigations is the existence of a purely transcendental field of definition of degree n-3 for a generic field extension of degree n < 5. Hanspeter KraftGerald W. Schwarz2018-08-24T06:48:48Z2018-08-24T06:48:48Zhttps://edoc.unibas.ch/id/eprint/64709This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/647092018-08-24T06:48:48ZAutomorphism groups of affine varieties and a characterization of affine n-spaceWe show that the automorphism group of affine -space determines up to isomorphism: If is a connected affine variety such that as ind-groups, then as varieties. We also show that every torus appears as for a suitable irreducible affine variety , but that cannot be isomorphic to a semisimple group. In fact, if is finite-dimensional and if , then the connected component is a torus. Concerning the structure of we prove that any homomorphism of ind-groups either factors through where is the Jacobian determinant, or it is a closed immersion. For we show that every nontrivial homomorphism is a closed immersion. Finally, we prove that every nontrivial homomorphism is an automorphism, and that is given by conjugation with an element from . Hanspeter Kraft2018-08-24T06:41:37Z2018-08-24T06:41:37Zhttps://edoc.unibas.ch/id/eprint/64687This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/646872018-08-24T06:41:37ZAutomorphisms of the Lie algebra of vector fields on affine n-spaceWe show that every Lie algebra automorphism of the vector fields Vec( A n ) of affine n -space A n , of the vector fields Vec c ( A n ) with constant divergence, and of the vector fields Vec 0 ( A n ) with divergence zero is induced by an automorphism of A n . This generalizes results of the second author obtained in dimension 2, see [Reg13]. The case of Vec( A n ) goes back to Kulikov [Kul92]. Hanspter KraftAndriy Regeta2018-08-02T09:12:21Z2019-02-06T15:10:03Zhttps://edoc.unibas.ch/id/eprint/59709This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/597092018-08-02T09:12:21ZIs the affine space determined by its automorphism group?Hanspeter KraftAndriy RegetaImmanuel van Santen2016-10-20T07:00:49Z2018-02-22T15:53:14Zhttps://edoc.unibas.ch/id/eprint/43325This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/433252016-10-20T07:00:49ZRepresentations with a Reduced Null ConeLet G be a complex reductive group and V a G- module. Let π: V → V/G be the quotient morphism and set N(V) = π−1(π(0)). We consider the following question. Is the null cone N (V) reduced, i.e., is the ideal of N (V) generated by G- invariant polynomials? We have complete results when G is SL2, SL3 or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible. Hanspeter KraftGerald W. Schwarz2016-10-18T14:58:39Z2016-10-18T14:58:39Zhttps://edoc.unibas.ch/id/eprint/43326This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/433262016-10-18T14:58:39ZInvariants and separating morphisms for algebraic group actionsThe first part of this paper is a refinement of Winkelmann’s work on invariant rings and quotients of algebraic groups actions an affine varieties where we take a more geometric point of view. We show that the (schematic) quotient X//G given by the possibly not finitely generated ring of invariants is “almost” an algebraic variety, and that the quotient morphism π : X → X//G has a number of nice properties. One of the main difficulties comes from the fact that the quotient morphism is not surjective. These general results are then refined for actions of the additive group Ga where we can say much more. We get a rather explicit description of the so-called plinth variety and of the separation variety which measures how much orbits are separated by invariants. The most complete results are obtained for representations. We also give a complete and detailed analysis of Roberts’ famous example of a 7-dimensional representation of Ga with a non-finitely generated ring of invariants. Emilie DufresneHanspeter Kraft2016-10-18T14:49:29Z2016-10-18T14:49:29Zhttps://edoc.unibas.ch/id/eprint/43324This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/433242016-10-18T14:49:29ZFamilies of Group Actions, Generic Isotriviality, and LinearizationWe study families of reductive group actions on A2 parametrized by curves and show that every faithful action of a non-finite reductive group on A3 is linearizable, i.e. G-isomorphic to a representation of G. The difficulties arise for non-connected groups G. We prove a Generic Equivalence Theorem which says that two affine mor- phisms p: S → Y and q: T → Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant ́etale base change φ : U → Y . A special case is the following result. Call a morphism φ: X → Y a fibration with fiber F if φ is flat and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an ́etale dominant morphism μ: U → Y such that the pull-back is a trivial fiber bundle: U ×Y X ≃ U × F . As an application we give short proofs of the following two (known) results: (a) Every affine A1-fibration over a normal variety is locally trivial in the Zariski-topology; (b) Every affine A2-fibration over a smooth curve is locally trivial in the Zariski-topology. Hanspeter KraftPeter Russell2016-10-18T13:54:59Z2016-10-18T13:54:59Zhttps://edoc.unibas.ch/id/eprint/43323This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/433232016-10-18T13:54:59ZAutomorphisms of the affine Cremona groupWe show that every automorphism of the group Gn := Aut(An) of polynomial automorphisms of complex affine n-space An = Cn is inner up to field automorphisms when restricted to the subgroup TGn of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension n = 2 where all automorphisms are tame: TG2 = G2. The methods are different, based on arguments from algebraic group actions. Hanspeter KraftImmanuel Stampfli2014-11-07T08:29:07Z2015-12-31T10:56:33Zhttps://edoc.unibas.ch/id/eprint/34691This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/346912014-11-07T08:29:07ZVarieties characterized by their endomorphismsWe show that two varietes X and Y with isomorphic endomorphism semigroups are isomorphic up to field automorphism if one of them is affine and contains a copy of the affine line. A holomorphic version of this result is due to the first author. Rafael B. AndristHanspeter Kraft2012-06-08T06:52:11Z2017-10-11T06:25:32Zhttps://edoc.unibas.ch/id/eprint/18867This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/188672012-06-08T06:52:11ZFinite automorphisms of affine n-spaceHanspeter KraftGerald W. Schwarz2012-06-08T06:52:11Z2017-09-19T07:50:27Zhttps://edoc.unibas.ch/id/eprint/18868This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/188682012-06-08T06:52:11ZOn a question of Yosef SteinHanspeter Kraft2012-06-08T06:48:38Z2017-09-20T10:47:05Zhttps://edoc.unibas.ch/id/eprint/18548This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/185482012-06-08T06:48:38ZConstructive invariant theoryInvariant theory was a major subject of research in the 19th century. One of the highlights was Gordan's famous theorem from 1868 showing that the invariants and covariants of binary forms have a finite basis. His method was constructive and led to explicit degree bounds for a system of generators (Jordan 1876/79). In 1890, Hilbert presented a very general finiteness result using completely different methods such as his famous 'Basissatz'. He was heavily attacked because his proof didn't give any tools to construct a system of generators. In his second paper from 1893 he again introduced new techniques in order to make his approach more constructive. This paper contains the 'Nullstellensatz', 'Noether's Normalization Lemma', and the 'Hilbert-Mumford Criterion'. We shortly overview this development, discuss in detail the degree bounds given by Popov, Wehlau and Hiss and describe some exciting new development relating these bounds with the (geometric) degree of projective varieties and with the Eisenbud-Goto conjecture. The challenge is still the fact that the degree bounds for binary forms given by Jordan are much better than those obtained from the work of Popov and Hiss. PS. Very recently, Harm Derksen was able to give polynomial bounds for the generators of the invariant ring for any representation of a reductive group. Hanspeter KraftHarm Derksen2012-06-08T06:48:27Z2015-12-31T10:49:03Zhttps://edoc.unibas.ch/id/eprint/18538This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/185382012-06-08T06:48:27ZFree C+ actions on affine threefoldsWe study algebraic actions of the additive group C+ on an affine threefold X and prove a smoothness property for the quotient morphism X -< X//C+. Then, following Shulim Kaliman, we give a proof of the conjecture that every free C+ action on affine 3-space C^3 is a translation. H. Kraft2012-06-08T06:48:11Z2017-11-06T11:26:07Zhttps://edoc.unibas.ch/id/eprint/18524This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/185242012-06-08T06:48:11ZOn the nullcone of representations of reductive groupsWe study the geometry of the nullcone N(V^k) for several copies of a representation V of a reductive group G and its behavior for different k. We show that for large k there is a certain 'stability' with respect to the irreducible components. In the case of the so-called theta-representations, this can be made more precise by using the combinatorics of the weight system as a subset of the root system. All this finally allows to calculate explicitly and in detail a number of important examples, e.g. the cases of 3- and 4-qubits which play a fundamental role in quantum computing. Hanspeter KraftNolan Wallach2012-06-08T06:48:09Z2015-12-31T10:49:03Zhttps://edoc.unibas.ch/id/eprint/18520This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/185202012-06-08T06:48:09ZEquivariant affine line bundles and linearizationWe show that every algebraic action of a linearly reductive group on affine n-space C^n which is given by Jonqui`ere automorphisms is linearizable. Similarly, every holomorphic action of a compact group K by (holomorphic) Jonquière automorphisms is linearizable. Moreover, any holomorphic action of K on C^2 by overshears is linearizable, too. These results are based on the fact that equivariant algebraic or holomorphic affine line bundles over C^n are trivial. Hanspeter KraftFrank Kutzschebauch2012-06-08T06:48:09Z2015-12-31T10:49:03Zhttps://edoc.unibas.ch/id/eprint/18521This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/185212012-06-08T06:48:09ZDegree bounds for separating invariantsIf V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f from S such that f(v) is different from f(v'). It is known that there always exist finite separating sets. Moreover, if the group G is finite, then the invariant functions of degree ≤ |G| form a separating set. We show that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist. If G is finite, we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We show that for a subgroup H of G we have b(H) ≤ b(G) ≤ [G:H] b(H), and that b(G) ≤ b(G/H) b(H) in case H is normal. Moreover, we calculate b(G) for some specific finite groups. Hanspeter KraftMartin Kohls2012-06-08T06:48:08Z2015-12-31T10:49:03Zhttps://edoc.unibas.ch/id/eprint/18518This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/185182012-06-08T06:48:08ZProperties and examples of FCR-algebrasAn algebra A over a field k is FCR if every finite dimensional representation of A is completely reducible and the intersection of the kernels of these representations is zero. We give a useful characterization of FCR-algebras and apply this to C*-algebras and to localizations. Moreover, we show that 'small' products and sums of FCR-algebras are again FCR. Hanspeter KraftLance W. SmallNolan R. Wallach2012-06-08T06:48:04Z2017-10-26T06:50:25Zhttps://edoc.unibas.ch/id/eprint/18511This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/185112012-06-08T06:48:04ZNilpotent subspaces of maximal dimension in semisimple Lie algebrasWe show that a linear subspace of a reductive Lie algebra g that consists of nilpotent elements has dimension at most equal to the number of positive roots, and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of g. This generalizes a classical theorem of Gerstenhaber which states this fact for the algebra of n x n matrices. Jan DraismaHanspeter KraftJochen Kuttler2012-06-08T06:47:46Z2017-09-28T05:58:25Zhttps://edoc.unibas.ch/id/eprint/18476This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/184762012-06-08T06:47:46ZA result of Hermite and equations of degree 5 and 6A classical result from 1861 due to Hermite says that every separable equation of degree 5 can be transformed into an equation of the form x^5 + b x^3 + c x + d = 0. Later this was generalized to equations of degree 6 by Joubert. We show that both results can be understood as an explicit analysis of certain covariants of the symmetric groups S_5 and S_6. In case of degree 5, the classical invariant theory of binary forms of degree 5 come into play whereas in degree 6 the existence of an outer automorphism of S_6 plays an essential role. Hanspeter Kraft2012-06-08T06:47:42Z2017-09-21T06:46:54Zhttps://edoc.unibas.ch/id/eprint/18469This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/184692012-06-08T06:47:42ZChallenging problems in affine n-spaceComplex affine n-space C^n, the basic object of algebraic geometry, offers a number of exciting and striking problems. The most famous one, the Jacobian Conjecture is the still unsolved. Others are the Cancellation Problem (Does Y times C^k simeq C^{n+k} imply that Y simeq C^n?), the Linearization Problem (Is every automorphism of C^n of finite order conjugate to a linear automorphism?), or the Embedding Problem (Are there other embeddings of C^{n-1} into C^n than the standard ones?). It turns out that these questions and several others are intimately related and have very interesting connections with problems arising from algebraic group actions and orbit spaces. We give a survey on these problems and discuss some recent progress and examples. Hanspeter Kraft2012-06-08T06:47:20Z2017-09-21T06:56:26Zhttps://edoc.unibas.ch/id/eprint/18449This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/184492012-06-08T06:47:20ZPrincipal covariants, multiplicity-free actions, and the K-types of holomorphic seriesWe prove a result on the structure of the $K$-types for holomorphic discrete series of $Sp(2n,R)$. The proof applies the theory of multiplicity-free actions to the realization of holomorphic discrete series by means of the dual pair $(Sp_{2n}, O_m)$. Roger HoweHanspeter Kraft2012-06-08T06:47:14Z2015-12-31T10:49:01Zhttps://edoc.unibas.ch/id/eprint/18445This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/184452012-06-08T06:47:14ZHereditary properties of direct summands of algebrasWe consider subrings S of rings R such that R = S oplus V with V either two sided invariant under multiplication by S or invariant under the commutator with S. We show that some important properties of R are inherited by S under such conditions. One is the FCR-property which says that every finite dimensional representation is completely reducible. Another application gives a characterization (in characteristic zero) of reductive subgroups of reductive groups. Hanspeter KraftLance W. SmallNolan R. Wallach2012-06-08T06:47:14Z2015-12-31T10:49:01Zhttps://edoc.unibas.ch/id/eprint/18446This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/184462012-06-08T06:47:14ZRational covariants of reductive groups and homaloidal polynomialsLet G be a complex reductive group, V a G-module and f a nonconstant homogenous invariant polynomial on V. We investigate relations between the following properties:- The differential df: V -< V* is dominant;- The invariant f is homaloidal, i.e., df induces a birational map P(V) -< P(V*);- V is a stable representation, i.e., the generic G-orbit in V is closed.If f generates the invariants, we show that the properties are equivalent, generalizing results of Sato-Kimura on prehomogeneous vector spaces. Hanspeter KraftGerald W. Schwarz2012-06-08T06:47:10Z2015-12-31T10:49:00Zhttps://edoc.unibas.ch/id/eprint/18439This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/184392012-06-08T06:47:10ZInvariant algebras and completely reducible representationsWe give a general construction of affine noetherian algebras with the property that every finite dimensional representation is completely reducible. Starting from enveloping algebras of semi simple Lie algebras in characteristic zero we obtain explicit examples and describe some of their properties. Hanspeter KraftLance W. Small2012-03-22T14:15:23Z2012-03-22T14:30:35Zhttps://edoc.unibas.ch/id/eprint/15533This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/155332012-03-22T14:15:23ZSymmetry and spaces : in honor of Gerry Schwarz2012-03-22T13:59:57Z2018-06-21T14:22:06Zhttps://edoc.unibas.ch/id/eprint/11723This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/117232012-03-22T13:59:57ZPolarizations and Nullcone of Representations of Reductive GroupsThe paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f_1,f_2,...,f_n be homogeneous invariant functions. Then the polarizations of f_1,f_2,...,f_n define the nullcone of k 0} h(t) x = 0 for all x in L. This is then applied to many examples. A surprising result is about the group SL(2,C) where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of $SL_2$ on the n-fold tensor product C^2 otimes C^2 otimes ... otimes C^2. Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n <= 3. (An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title ``On the nullcone of representations of reductive groups'' is published in Pacific J. Math. {bf 224} (2006), 119--140.) Hanspeter KraftNolan Wallach2012-03-22T13:57:13Z2020-10-12T14:46:32Zhttps://edoc.unibas.ch/id/eprint/10947This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/109472012-03-22T13:57:13ZCompression of finite group actions and covariant dimension, IIHanspeter KraftRoland LötscherGerald W. Schwarz