Instability in Invariant Theory

Kempf, George R.

doi:10.2307/1971168

Cited by 319 publications

(378 citation statements)

References 9 publications

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“…Our basic reference for this material will be Humphreys' book [12] (see also [33]). In particular, no mention will be made of rationality problems (which will ultimately be studied using the techniques of [13]). …”

Section: N»0mentioning

confidence: 99%

Classification of Semisimple Algebraic Monoids

Renner¹

1985

Transactions of the American Mathematical Society

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Abstract.Let A' be a semisimple algebraic monoid with unit group G. Associated with E is its polyhedral root system (X, 0, C), where X = X(T) is the character group of the maximal torus T c G, $ c X(T) is the set of roots, and C = X(T) is the character monoid of T c E (Zariski closure).The correspondence £-»(A\ O, C) is a complete and discriminating invariant of the semisimple monoid £, extending the well-known classification of semisimple groups. In establishing this result, monoids with preassigned root data are first constructed from linear representations of G. That done, we then show that any other semisimple monoid must be isomorphic to one of those constructed. To do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory. This, ultimately, yields the desired conclusions.Consider the classification problem for semisimple, algebraic monoids over the algebraically closed field k.What sort of problem is this? First the definition: A semisimple, algebraic monoid is an irreducible, affine, algebraic variety E, defined over k together with an associative morphism m: E X E -* E and a two-sided unit 1 e E for m. We assume further that E has a 0, the unit group G (which is always linear, algebraic and dense in E), is reductive (e.g. G\n(k)), dim ZG = 1 and is is a normal variety.The problem then, presents us with two familiar objects. Let Tc G be a maximal torus. Then Z = T c E (Zariski closure) is an affine, torus embedding and G c E is a reductive, algebraic group. Torus embeddings have been introduced by Demazure in [6] in his study of Cremona groups, and are classified numerically using rational, polyhedral cones [14].On the other hand, reductive groups have been studied, at least in principle, since the nineteenth century; their classification in the modern sense being achieved largely by Chevalley [4]. That numerical classification uses the now familiar root systems [11, Chapter 3] of Killing that were introduced by him [15] in his penetrating formulation of the classification (E. Cartan's!) of semisimple Lie algebras.Thus, in the classification of semisimple monoids we are compelled to consider the root system (X, $) = (X(T), ®(T)) of G, and the polyhedral cone C = X(Z) c X(T) of T c Z. The two objects are canonically related by the Weyl group action on X, which leaves C stable.

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Section: N»0mentioning

confidence: 99%

Classification of Semisimple Algebraic Monoids

Renner¹

1985

Transactions of the American Mathematical Society

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“…By above, every such curve degenerates isotrivially to a double hyperplane section in P 5 of a projective cone over a genus one curve of degree 5 in P 4 . The semistability of this non-reduced curve follows from Kempf's results [14].…”

Section: Line Bundles On the Moduli Stack Of Gorenstein Curvesmentioning

confidence: 87%

“…Moreover, the ideal of the Veronese surface is generated by the quadrics containing C. Proof . This also follows immediately from [14,Corollary 5.3], as the Veronese surface is simply P 2 embedded in P 5 via the complete linear system |O P 2 (2)|. 4 An answer to the riddle…”

Section: Degenerations To Rational Normal Surface Scrollsmentioning

confidence: 91%

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Stability of 2nd Hilbert Points of Canonical Curves

Fedorchuk

Jensen

2012

International Mathematics Research Notices

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“…For x A X let G Á x denote the G-orbit of x in X and C G ðxÞ the stabilizer of x in G. Let x A X and let C be the unique closed orbit in the closure of G Á x; we refer the reader to [15, (1.3)] for a proof that there is a unique closed G-orbit in G Á x. The Kempf-Rousseau theory tells us that there exists a nonempty subset WðxÞ of X X ðGÞ consisting of so called optimal cocharacters l such that lim t!0 lðtÞ Á x exists and belongs to C; we refer the reader to [9] or [16] for information on the Kempf-Rousseau theory and the definition of optimal cocharacters. Moreover, there exists a parabolic subgroup PðxÞ of G so that PðxÞ ¼ P l for every l A WðxÞ, and we have that WðxÞ is a single PðxÞ-orbit.…”

Section: Commuting Varietiesmentioning

confidence: 99%

On conjugacy of unipotent elements in finite groups of Lie type

Goodwin¹,

Röhrle²

2009

Journal of Group Theory

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Abstract. Let G be a connected reductive algebraic group defined over F q , where q is a power of a prime p that is good for G. Let F be the Frobenius morphism associated with the F q -structure on G and set G ¼ G F , the fixed point subgroup of F . Let P be an F -stable parabolic subgroup of G and let U be the unipotent radical of P; set P ¼ P F and U ¼ U F . Let G uni be the set of unipotent elements in G. In this note we show that the number of conjugacy classes of U in G uni is given by a polynomial in q with integer coe‰cients.

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Instability in Invariant Theory

Cited by 319 publications

References 9 publications

Classification of Semisimple Algebraic Monoids

Classification of Semisimple Algebraic Monoids

Stability of 2nd Hilbert Points of Canonical Curves

On conjugacy of unipotent elements in finite groups of Lie type

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