Parametrization, structure and Bruhat order of certain spherical quotients

Chaput, Pierre–Emmanuel; Fresse, Lucas; Gobet, Thomas

doi:10.1090/ert/584

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…Besides the presence of a Schubert variety is detected as a consequence. For us, the latter is a base point for a more general construction, using results on parabolic induction (see [CFG21] by P-E. Chaput, L. Fresse, T. Gobet) and symmetric subgroups (see the work of R-W. Richardson and T-A. Springer [RS90]), which can treat all the types A, B, C 11 , D together (see our Lemma 2.1) and can exactly recover, for type A and some precautions, the previous construction (see (38) of Appendix B).…”

mentioning

confidence: 99%

Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D

Jacques¹

2022

Preprint

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Let G be a reductive algebraic group in classical types A, B, D and e be an element of its Lie algebra with Z its centraliser in G for the adjoint action. We suppose that e identifies with an nilpotent matrix of order two, which guarantees the number of Z-orbits in the flag variety of G is finite. For types B, D in characteristic two, we also suppose the image of e is totally isotropic. We show that any closure Y of such orbit is normal. We also prove that Y is Cohen-Macaulay with rational singularities provided that the base field is of characteristic zero, and that Cohen-Macaulayness remains in any characteristic for type A. We exhibit a birational, rational morphism onto Y involving Schubert varieties. Our work generalizes a result by N. Perrin and E. Smirnov on Springer fibers ([PS12]).

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confidence: 99%

Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D

Jacques¹

2022

Preprint

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“…Other parametrizations of the B-orbits in N 2 have been given by Chaput, Fresse and Gobet in the recent preprint [7]. In the case of SL n (k), there is provided yet another description of the partial order among the B-orbits.…”

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Nilpotent orbits of height 2 and involutions in the affine Weyl group

Gandini¹,

Frajria²,

Papi³

2019

Preprint

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Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B ⊂ G be a Borel subgroup. Then B acts with finitely many orbits on the variety N 2 ⊂ g of the nilpotent elements whose height is at most 2. We provide a parametrization of the B-orbits in N 2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g. Introduction.Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let N ⊂ g be the nilpotent cone. It is well known that G acts with finitely many orbits on N : for instance, when G is a classical group, the nilpotent G-orbits are parametrized in terms of partitions, and the partial order defined by the inclusions of their closures is nicely expressed in terms of the dominance order of partitions.Given e ∈ N a natural index of nilpotency is the height, defined as ht(e) = max{n ∈ N | ad(e) n = 0}.

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Fully commutative elements and spherical nilpotent orbits

Gandini

2022

Journal of Algebra

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Parametrization, structure and Bruhat order of certain spherical quotients

Cited by 3 publications

References 18 publications

Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D

Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D

Nilpotent orbits of height 2 and involutions in the affine Weyl group

Fully commutative elements and spherical nilpotent orbits

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