Kohls, M., Kraft, H.: Degree bounds for separating invariants
Preprint Juni 2010
If 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.
Kraft, H., Lötscher, R., Schwarz, G.W.: Compression of finite group actions
and covariant dimension, II
J. Algebra vol. 322 (2009) 94-107
This is a continuation of a previous paper with the same title (see below).
In that paper we investigated covariant dimension and were able to determine it in many cases.
Our techniques largely depended upon finding homogeneous faithful covariants. After publication
of that paper, the junior author of this article pointed out several gaps in our proofs.
Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants,
which have enabled us to extend and complete the results, simplify the proofs and fill the
gaps.
Kraft,
H., Wallach, N.: Polarization and nullcone of representations of reductive groups
Progress in Math. Vol. 278 (2009), p.
The 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 <= m copies of V if and only if every linear subspace L of the nullcone of V
of dimension <= m is annhilated by a one-parameter subgroup (shortly a 1-PSG).
This means that there is a group homomorphism h: C* -> G such that lim_{t->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,C) 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.)
Kraft, H., Schwarz, G.W.: Compression of finite group actions
and covariant dimension
J. Algebra vol. 313 (2007) 268-291
Let 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.
Draisma, J., Kraft, H., Kuttler, J.: Nilpotent subspaces of
maximal dimension in semisimple Lie algebras
Compositio Math. vol. 142 (2006), 464-476
We 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.
Kraft,
H., Wallach, N.: On the nullcone of representations of reductive groups
Pacific J. Math. vol. 224 (2006), 119-140
We 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.
Kraft,
H.: A result of Hermite and equations of degree 5 and 6
J. Algebra vol. 297 (2006), 234-253
A 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.
Kraft, H.: Free C+ actions on affine threefolds
In: Affine Algebraic Geometry (Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu, eds.),
Contemporary Mathematics vol. 369, AMS, (2005)
We 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.
Kraft,
H., Wallach, N.R.: Polarizations and nullcone of representations of reductive groups
Preprint October 2004
The paper starts with the following simple observation. Let
V be a representation of a reductive group G, and let f1,f2,...,fn be homogeneous invariant
functions.
Then the polarizations of f1,f2,...,fn define the nullcone
of k <= m copies of V if and only if every linear subspace of the
nullcone of V of dimension <= m is annhilated by a 1-PSG. This is then applied to
many examples. A surprising result is about the group SL_2 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 where 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" will be
published separately in Pacific J. Math.)
Kraft,
H., Wallach, N.R.: On the separation property of orbits in
representation spaces
J. Algebra 258 (2002), 228-254
A subset X of a vector space V is said to have the "Separation
Property" if it separates linear forms in the following sense:
Given a pair (a,b) of linearly independent linear forms on V there
is a point x on X such that a(x) = 0 and b(x) is non-zero. A more
geometric way to express this is the following: Every linear
subspace H of V of codimension 1 is linearly spanned by its
intersection with X.
The separation property was first asked for conjugacy classes in
simple Lie algebras, in connection with some classification
problems. We give a general answer for orbits in representation
spaces of algebraic groups and discuss in detail some special
cases. We also introduce a strong and a weak
separation property which come up very naturally in our setting.
It turns out that these separation properies have a number of very
nice features. For example, we discovered the surprising fact that
in an irreducible representation V of a connected semisimple group
every linear hyperplane meets every orbit, and we show that a
generic orbit in V always has the separation property.
Kraft,
H., Schwarz, G.W.: Rational covariants of reductive groups and
homaloidal polynomials
Math. Research Letters 8 (2002) 641-650
Let 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.
Kraft,
H., Small, L.W., Wallach, N.R.: Properties and examples of
FCR-algebras
Manuscripta math. 104 (2001) 443-450
An 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.
Kraft,
H., Procesi, C.: Classical Invariant Theory, a
Primer
Lecture Notes, Version 2000
Abstract: The lecture notes give an elementary introduction
to classical invariant theory. We start with the First Fundamental
Theorem for GL(n) and explain its relation with endomorphisms of
tensors. The representation theory of GL(n) in characteristic zero
is first apporached via highest weights and then via character
theory (Schur polynomials). Next we can give some applications to
classical invariant theory and differential operators. We then
explain Weyl's theorems and the theory of Capelli. The last
chapter gives a proof of the Fundamental Theorems for the
orthogonal and symplectic groups. The text also contains many
exercises.
Kraft,
H., Weyman, J.: Degree bounds for invariants and covariants of
binary forms
Preprint 1999
Abstract: In this paper we solve an old problem of
Classical Invariant Theory of binary forms. We give a modern
approach and rigorous proof for the degree estimates of a
generating system of invariants and covariants of binary form due
to Camille Jordan. Moreover, we show that this approach can be
efficiently used to calculate the generators in low
degrees.
Kraft,
H., Small, L.W., Wallach, N.R.: Hereditary properties of direct
summands of algebras
Math. Research Letters 6 (1999), 371-376
Abstract: We 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.
Derksen,
H., Kraft, H.: Constructive invariant
theory
In: "Séminaires et Congrès",
Société Mathématique de France, Editors Alev,
J. et al. (1997) pp. 221-244
Abstract: Invariant 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.
Howe,
R., Kraft, H.: Principal covariants, multiplicity-free actions,
and the K-types of holomorphic series
In: "Geometry and Representation Theory of Real and p-adic Lie
Groups", Editors J. Tirao, D. Vogen, J. Wolf, Progress in Math.
vol 158, pp. 147-161, Birkhäuser Verlag 1998
Abstract: We 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
$(Sp_{2n}, O_m)$.
Kraft,
H.: Challenging problems in affine n-space
Séminaire Bourbaki, Juin 1995, 47ème
année, 1994/95, Exp. no. 802, 5 Astérisque 237
(1996), pp. 295-317, Paris
Abstract: Complex 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.
Kraft,
H., Kutzschebauch, F.: Equivariant affine line bundles and
linearization
Math. Research Letters 3 (1996) 619-628
Abstract: We 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)
Jonqui`ere 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.
Kraft,
H., Schwarz, G.: Finite automorphisms of affine
n-space
In: Proceedings of the Curacao Conference on "Automorphisms of
Affine Space," 1995
Kraft,
H.: On a question of E. Stein
In: Proceedings of the Curacao Conference on "Automorphisms of
Affine Space," 1995
Kraft,
H., Small, L.W.: Invariant algebras and completely reducible
representations
Math. Research Letters 1 (1994) 297-307
Abstract: We 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.
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