Polarization of Separating Invariants

Draisma, Jan; Kemper, Gregor; Wehlau, David L.

doi:10.4153/cjm-2009-027-x

Cited by 23 publications

(39 citation statements)

References 5 publications

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“…Weyl's theorem no longer holds in positive characteristic, though a weaker statement is still true [12]. However, an analogue of Weyl's theorem, for separating invariants, is true in arbitrary characteristic [6]-and, again, implies the last statement of Proposition 1.…”

Section: Proposition 1 If P ≥ Q Then C(m P ) ≥ C(m Q ) If In Additimentioning

confidence: 90%

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

Burgin

Draisma

2006

Math. Z.

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We describe the null-cone of the representation of G on M p , where either G = SL(W) × SL(V) and M = Hom(V, W) (linear maps), or G = SL(V) and M is one of the representations S 2 (V * ) (symmetric bilinear forms), 2 (V * ) (skew bilinear forms), or V * ⊗ V * (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and M p is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on M p . Results of Kraft and Wallach predict that their number stabilises at a certain value of p, and we determine this value. We also answer the question of when the null-cone in M p is defined by the polarisations of the invariants on M; typically, this is only the case if either dim V or p is small. A fundamental tool in our proofs is the Hilbert-Mumford criterion for nilpotency (also known as unstability).

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Section: Proposition 1 If P ≥ Q Then C(m P ) ≥ C(m Q ) If In Additimentioning

confidence: 90%

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

Burgin

Draisma

2006

Math. Z.

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“…Consequently, it is noteworthy that for any positive characteristic and finite group H, the algebra of vector invariants H k[M n,d ] is contained in the p-root closure of the polarizations of the invariants of a single vector. This result follows immediately from Theorem 2.15 in [14] (whose proof is valid in any characteristic) and Theorem 6 above or from Theorem 2.4 in [9] and Theorems 2(c) and 6 above. In stating these results, ])).…”

Section: Saturated Subgroupsmentioning

confidence: 59%

“…This theorem is not true when char k = p > 0; a simple example when H is a torus is given below in Section 2; the case where H is a finite group whose order is divisible by p, was considered by Richman [18]; Domokos, et al, have given examples involving the classical groups [2], [3], [4], [5]. Recently, Draisma, Kemper and Wehlau have shown that at least a set of separating invariants can be obtained by polarization [9].…”

Section: Introductionmentioning

confidence: 99%

Vector Invariants in Arbitrary Characteristic

Grosshans

2007

Transformation Groups

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Let k be an algebraically closed field of characteristic p 0. Let H be a subgroup of GLn(k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d vectors, d n, are obtained from those of n vectors by polarization. This result is not true when char k = p > 0 even in the case where H is a torus. However, we show that the algebra of invariants is always the p-root closure of the algebra of polarized invariants. We also give conditions for the two algebras to be equal, relating equality to good filtrations and saturated subgroups. As applications, we discuss the cases where H is finite or a classical group.

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“…On the other hand, they are "really" about separating invariants, in the sense that they have no analogues for generating invariants, even in characteristic zero. From the point of view of polarizations, our results can be viewed as an addendum to [6] making possible some computational simplifications in the construction of a separating set in k[W ⊕ V m ] G for large m. Namely, one takes a separating set S 0 in k[W ⊕ V n ] G with n = dim(V ). Then it is sufficient to take those polarizations of the elements of S 0 that depend only on 2n (resp., n + 1 for reductive G) type V variables to get a (multihomogeneous) separating set in k[W ⊕ V m ] G .…”

Section: Discussionmentioning

confidence: 99%

Typical separating invariants

Domokos

2007

Transformation Groups

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It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: If two points in the direct sum of the G-modules W and m copies of V can be separated by polynomial invariants, then they can be separated by invariants depending only on 2 dim(V ) variables of type V ; when G is reductive, invariants depending only on dim(V ) + 1 variables suffice. A similar result is valid for rational invariants. Explicit bounds on the number of type V variables in a complete system of typical separating invariants are given for the binary polyhedral groups, and this is applied to the invariant theory of binary forms.

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Polarization of Separating Invariants

Cited by 23 publications

References 5 publications

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

Vector Invariants in Arbitrary Characteristic

Typical separating invariants

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