Equivariant coherent sheaves on the exotic nilpotent cone

Nandakumar, Vinoth

doi:10.1090/s1088-4165-2013-00444-2

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…Achar, Henderson and Sommers make an explicit connection between special pieces for N and those for the ordinary nilpotent cone in [AHS11]. The Lusztig-Vogan bijection can also be extended to the exotic nilpotent cone, as shown by the first author in [Nan13]. These results all demonstrate a strong connection between the exotic nilpotent cone and the ordinary nilpotent cone of type C. In some respects, the former has properties which are better than those of the latter; the present work is another example of this.…”

Section: Introductionmentioning

confidence: 99%

Irreducible components of exotic Springer fibres

Nandakumar

Rosso

Saunders

2018

J. London Math. Soc.

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Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces. Contents

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Section: Introductionmentioning

confidence: 99%

Irreducible components of exotic Springer fibres

Nandakumar

Rosso

Saunders

2018

J. London Math. Soc.

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“…Many of its properties relating to e.g. intersection cohomology of orbit closures (see [AH08] and [SS14]), theory of special pieces (see [AHS11]), and the Lusztig-Vogan bijection (see [Nan13]) have been explored in follow-up work to [Kat09].…”

Section: Introductionmentioning

confidence: 99%

Exotic Springer fibers for orbits corresponding to one-row bipartitions

Saunders,

Wilbert

2018

Preprint

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We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the two-boundary Temperley-Lieb algebra. This provides a first step in generalizing the geometric versions of Khovanov's arc algebra to the exotic setting.

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“…Let N (gl 2n ) be the nilpotent cone for GL 2n and let N (S) = N (gl 2n ) ∩ S, where S is the Sp 2n -complement to sp 2n in gl 2n viewed as an Sp 2n -module. Kato's exotic nilpotent cone for Sp 2n is the variety N = C 2n × N (S) which is the Hilbert nullcone of the Sp 2n -module C 2n ⊕ S. In [Kat09], Kato constructs an exotic Springer correspondence, and showed that the Sp 2n -orbits on N are in bijection with the bipartitions of n, which also parametrise the irreducible representations of the Weyl group of type C. In subsequent work, many other Springer theoretic results have been extended to the exotic settingintersection cohomology of orbit closures, (see Achar and Henderson, [AH08], and Shoji-Sorlin, [SS14]), theory of special pieces (see Achar-Henderson-Sommers, [AHS11]), and the Lusztig-Vogan bijection (see [Nan13]). In many respects, the exotic nilpotent cone behaves more nicely than the ordinary nilpotent cone of type C, and our present paper is another illustration of this.…”

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confidence: 99%

Irreducible components of exotic Springer fibres II : The exotic Robinson–Schensted algorithm

2021

Self Cite

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Kato's exotic nilpotent cone was introduced as a substitute for the ordinary nilpotent cone of type C with nicer properties. The geometric Robinson-Schensted correspondence is obtained by parametrizing the irreducible components of the Steinberg variety (the conormal variety for the action of a semisimple group on two copies of its flag variety) in two different ways. In type A the correspondence coincides with the classical Robinson-Schensted algorithm for the symmetric group. Here we give an explicit combinatorial description of the geometric bijection that we obtained in our previous paper by replacing the ordinary type C nilpotent cone with the exotic nilpotent cone in the setting of the geometric Robinson-Schensted correspondence. This "exotic Robinson-Schensted algorithm" is a new algorithm which is interesting from a combinatorial perspective, and not a naive extension of the type A Robinson-Schensted bijection.

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Equivariant coherent sheaves on the exotic nilpotent cone

Cited by 4 publications

References 22 publications

Irreducible components of exotic Springer fibres

Irreducible components of exotic Springer fibres

Exotic Springer fibers for orbits corresponding to one-row bipartitions

Irreducible components of exotic Springer fibres II : The exotic Robinson–Schensted algorithm

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