Manifolds of Isospectral Matrices and Hessenberg Varieties

Ayzenberg, Anton; Buchstaber, Victor Matveevich

doi:10.1093/imrn/rnz388

Cited by 13 publications

(7 citation statements)

References 19 publications

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“…The quotient ring (1.2), which is also a graded S n -module, is a Poincaré duality algebra. We observe that it agrees with the cohomology ring of the twin Y (h) of X(h) introduced by Ayzenberg-Buchstaber [4] 1 , even as graded S n -modules with a naturally defined S n -action on H * (Y (h)). Consequently, we have…”

Section: Introductionsupporting

confidence: 82%

“…, x n ) is the cohomology ring of some T -manifold. In this section we observe that Y(h) is indeed the cohomology ring of the twin of X(h) introduced by Ayzenberg-Buchstaber [4].…”

Section: Twin Of Regular Semisimple Hessenberg Varietymentioning

confidence: 90%

“…Remark. As pointed out in [4], the twin Y (h) may be regarded as the quotient T \Z(h) of Z(h) by the left T -action on U(n). Indeed, the diffeomorphism ι above induces a diffeomorphism from…”

Section: Twin Of X(h)mentioning

confidence: 99%

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Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties

Masuda¹,

Satō²

2022

Preprint

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The solution of Shareshian-Wachs conjecture by Brosnan-Chow linked together the cohomology of regular semisimple Hessenberg varieties and graded chromatic symmetric functions on unit interval graphs. On the other hand, it is known that unicellular LLT polynomials have similar properties to graded chromatic symmetric functions. In this paper, we link together the unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties introduced by Ayzenberg-Buchstaber. We prove their palindromicity from topological viewpoint. We also show that modules of a symmetric group generated by faces of a permutohedron are related to a shifted unicellular LLT polynomial and observe the e-positivity of shifted unicellular LLT polynomials, which is established by Alexandersson-Sulzgruber in general, for path graphs and complete graphs through the cohomology of the twins. |h| i=0 ch(H 2i (X(h)) ⊗ U

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Section: Introductionsupporting

confidence: 82%

“…, x n ) is the cohomology ring of some T -manifold. In this section we observe that Y(h) is indeed the cohomology ring of the twin of X(h) introduced by Ayzenberg-Buchstaber [4].…”

Section: Twin Of Regular Semisimple Hessenberg Varietymentioning

confidence: 90%

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Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties

Masuda¹,

Satō²

2022

Preprint

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“…To broaden parts of the discussion started above, one notes that the Peterson variety is an example of a Hessenberg variety. Introduced by the works [25,26], Hessenberg varieties are closed subvarieties of G/B with natural manifestations in topology [2,6,12,58,66,67], algebraic geometry [3,4,40,41], combinatorics [11,34,38,61], and representation theory [5,7,14,18,36]. It turns out that each Lagrangian leaf in the Toda lattice is naturally compactified by an appropriate Hessenberg variety (as we explain momentarily), generalizing the above-discussed relationship between the Peterson variety and a specific leaf.…”

mentioning

confidence: 91%

Hessenberg varieties, Slodowy slices, and integrable systems

Abe

Crooks

2019

Math. Z.

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This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties.We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, X(H0), is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko-Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to X(H0). We also show X(H0) to have an open dense symplectic leaf isomorphic to G/Z × Sreg, where Z is the centre of G and Sreg is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on G × Sreg, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties. 10 3.2. The B-orbit stratification of V Toda 11 3.3. The Toda lattice 13 4. Symplectic geometry on G/Z × Sreg 14 4.

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“…Remark A.2. In the type A case LLT H ≃ H * (X H ) where X H is the smooth manifold of Hermitian matrices having a particular staircase form (determined by H) and a given fixed simple spectrum (determined by s) [AB20].…”

mentioning

confidence: 99%

Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

Precup¹,

Sommers²

2022

Preprint

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We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This leads to several implications for the dual picture, where we consider maps that generalize the Grothendieck-Springer resolution of the whole Lie algebra. In particular we are able to prove a conjecture of Brosnan.As we vary the maps, the cohomology of the corresponding nilpotent Hessenberg varieties often satisfy a relation we call the geometric modular law, which also has origins in the work on De Concini, Lusztig, and Procesi. We connect this relation in type A with a combinatorial modular law defined by Guay-Paquet that is satisfied by certain symmetric functions and deduce some consequences of that connection.

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Manifolds of Isospectral Matrices and Hessenberg Varieties

Cited by 13 publications

References 19 publications

Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties

Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties

Hessenberg varieties, Slodowy slices, and integrable systems

Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

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