Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

Graaf, Willem A. de

doi:10.1112/s1461157000000607

Cited by 40 publications

(105 citation statements)

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“…In the remaining types and (ultimately for all truncated biparabolics, we found upper and lower bounds [10,Sects. 5,6] and [16, 6.7]) on ch Y (g x β ) which imply via [10, 7.1] that GK dim Y (g x β ) = rk g. Again, by the general theory developed already in [13,Sect. 4], it follows that Y (g x β ) is polynomial up to localization by x β .…”

Section: 5mentioning

confidence: 96%

“…Elashvili had asked if index g x = rk g. The Vinberg inequality implies that index g x rk g. The question of equality easily reduces via Jordan decomposition to the nilpotent case. Following this, Yakimova [32] has shown that equality holds for g classical and the remaining cases have been checked by de Graaf using computer aided calculations [6]. It is now even known that Y (g x ) is polynomial, for example, by Brown and Brundan [4] in type A and in types A and C by Panyushev, Premet and Yakimova [28].…”

Section: 2mentioning

confidence: 99%

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Compatible Adapted Pairs and a Common Slice Theorem for Some Centralizers

Joseph

2008

Transformation Groups

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Let g be a simple Lie algebra and x ∈ g nilpotent. We derive a criterion for when the G-orbits in g * and the G x -orbits in (g x ) * admit a common slice, applicable (in principle) when x is of Bala-Carter type. When g is of type A, or if x is the highest root vector with g not of type E 8 , we show that this criterion is satisfied. In these cases we also show that the Mishchenko-Fomenko shift of argument produces a maximal Poisson commutative polynomial subalgebra of S(g x ) which maps isomorphically by restriction of functions to an affine translate of a subspace of (g x ) * . It is conjectured that the above criterion is satisfied when the dimensions of certain weights spaces, which can be computed purely combinatorially, are decreasing.

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Section: 5mentioning

confidence: 96%

Section: 2mentioning

confidence: 99%

Compatible Adapted Pairs and a Common Slice Theorem for Some Centralizers

Joseph

2008

Transformation Groups

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“…Remark 2. In view of the now established truth of the Elashvili conjecture ( [39], [11]), (ii) also follows from [31,Cor. 2.3] (see also [19, 3.1, Remarks]), where in addition both sides equal rank g. However it can be useful to have an independent proof.…”

Section: Lemmamentioning

confidence: 99%

Polynomiality of invariants, unimodularity and adapted pairs

Joseph

Shafrir

2010

Transformation Groups

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Let a be a finite-dimensional Lie algebra and Y (a) the a invariant subalgebra of its symmetric algebra S(a) under adjoint action. Recently there has been considerable interest in studying situations when Y (a) may be polynomial on index a generators, for example if a is a biparabolic or a centralizer g x in a semisimple Lie algebra g.Here a sum rule of Ooms and Van den Bergh on the degrees d i of the generators is extended to the case when a is unimodular and the fundamental semi-invariant pa of a is an invariant. It is noted that these last two conditions always hold for a centralizer g x and moreover that one has GKdim Y (g x ) = index g x , even though S(g x ) may admit proper semi-invariants.Applications are given to the notion of an adapted pair (h, η) for a. Under the above conditions it is shown that the set of eigenvalues {m i } of − ad h on a η must equal {(d i − 1)}. This gives rise to a condition under which Y (a) cannot be polynomial on index a generators.Finally the theorem of Bolsinov is extended to the case when pa is nonscalar.

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“…In this sense our approach stresses the relationship between nilpotent orbits in g and those in the ambient gl n (k). We also note that in [2] De Graaf presents a probabilistic 'trial and error' algorithm for computing representatives of nilpotent orbits over any algebraically closed field. Whilst the parameters may be set so as to deliver arbitrarily large probability of success, the representative obtained is not canonical.…”

Section: Introductionmentioning

confidence: 99%

Computing nilpotent and unipotent canonical forms: a symmetric approach

Clarke¹

2011

Math. Proc. Camb. Phil. Soc.

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Abstract. Let k be an algebraically closed field of any characteristic except 2, and let G = GLn(k) be the general linear group, regarded as an algebraic group over k. Using an algebro-geometric argument and Dynkin-Kostant theory for G we begin by obtaining a canonical form for nilpotent Ad(G)-orbits in gln(k) which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi,j) → (xn+1−j,n+1−i)), with entries in {0, 1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups GUn(Fq) for all prime powers q.

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Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

Cited by 40 publications

References 4 publications

Compatible Adapted Pairs and a Common Slice Theorem for Some Centralizers

Compatible Adapted Pairs and a Common Slice Theorem for Some Centralizers

Polynomiality of invariants, unimodularity and adapted pairs

Computing nilpotent and unipotent canonical forms: a symmetric approach

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