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Martha Precup - One of the best experts on this subject based on the ideXlab platform.
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A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
Mathematische Zeitschrift, 2020Co-Authors: Megumi Harada, Martha Precup, Tatsuya Horiguchi, Satoshi Murai, Julianna TymoczkoAbstract:Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $$GL(n,{\mathbb {C}})/B$$ G L ( n , C ) / B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $$GL(n-1,{\mathbb {C}})/B$$ G L ( n - 1 , C ) / B , showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of “Hessenberg Schubert polynomials” in the context of regular nilpotent Hessenberg varieties, which generalize the classical Schubert polynomials. We also outline several open questions pertaining to them.
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A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
arXiv: Algebraic Geometry, 2019Co-Authors: Megumi Harada, Martha Precup, Tatsuya Horiguchi, Satoshi Murai, Julianna TymoczkoAbstract:Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincare polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of "Hessenberg Schubert polynomials" in the context of regular nilpotent Hessenberg varieties, and outline several open questions pertaining to them.
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Hessenberg varieties associated to ad-nilpotent ideals
arXiv: Combinatorics, 2019Co-Authors: Martha PrecupAbstract:We consider Hessenberg varieties in the flag variety of $GL_n(\mathbb{C})$ with the property that the corresponding Hessenberg function defines an ad-nilpotent ideal. Each such Hessenberg variety is contained in a Springer fiber. We extend a theorem of Tymoczko to this setting, showing that these varieties have an affine paving obtained by intersecting with Schubert cells. Our method of proof constructs an an affine paving for each Springer fiber that restricts to an affine paving of the Hessenberg variety. We use the combinatorial properties of this paving to prove that Hessenberg varieties of this kind are connected.
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The singular locus of semisimple Hessenberg varieties
Journal of Algebra, 2019Co-Authors: Erik Insko, Martha PrecupAbstract:Abstract Although regular semisimple Hessenberg varieties are smooth and irreducible, semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of semisimple Hessenberg varieties corresponding to the standard Hessenberg space in all Lie types. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of this nature do not hold for all semisimple Hessenberg varieties.
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THE BETTI NUMBERS OF REGULAR Hessenberg VARIETIES ARE PALINDROMIC
Transformation Groups, 2017Co-Authors: Martha PrecupAbstract:Recently Brosnan and Chow have proven a conjecture of Shareshian and Wachs describing a representation of the symmetric group on the cohomology of regular semisimple Hessenberg varieties for GL n (ℂ). A key component of their argument is that the Betti numbers of regular Hessenberg varieties for GL n (ℂ) are palindromic. In this paper, we extend this result to all complex reductive algebraic groups, proving that the Betti numbers of regular Hessenberg varieties are palindromic.
Julianna Tymoczko - One of the best experts on this subject based on the ideXlab platform.
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A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
Mathematische Zeitschrift, 2020Co-Authors: Megumi Harada, Martha Precup, Tatsuya Horiguchi, Satoshi Murai, Julianna TymoczkoAbstract:Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $$GL(n,{\mathbb {C}})/B$$ G L ( n , C ) / B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $$GL(n-1,{\mathbb {C}})/B$$ G L ( n - 1 , C ) / B , showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of “Hessenberg Schubert polynomials” in the context of regular nilpotent Hessenberg varieties, which generalize the classical Schubert polynomials. We also outline several open questions pertaining to them.
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A formula for the cohomology and K-class of a regular Hessenberg variety
Journal of Pure and Applied Algebra, 2020Co-Authors: Erik Insko, Julianna Tymoczko, Alexander WooAbstract:Abstract Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on h than here.
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A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
arXiv: Algebraic Geometry, 2019Co-Authors: Megumi Harada, Martha Precup, Tatsuya Horiguchi, Satoshi Murai, Julianna TymoczkoAbstract:Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincare polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of "Hessenberg Schubert polynomials" in the context of regular nilpotent Hessenberg varieties, and outline several open questions pertaining to them.
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Affine Grassmannians and Hessenberg Schubert Cells
Association for Women in Mathematics Series, 2019Co-Authors: Linda Chen, Julianna TymoczkoAbstract:We give an overview of the linear algebra, geometry, and combinatorics of affine Grassmannians along the lines of Fulton’s Young Tableaux for classical Grassmannians. We discuss geometric and linear algebraic aspects of the decomposition of the affine Grassmannian into affine Schubert cells in terms of coset representatives and linear models. We describe (Grassmannian) Hessenberg Schubert cells and show that every affine Schubert cell can be realized as a Hessenberg Schubert cell in a complete flag variety and as a Grassmannian Hessenberg Schubert cell in a finite Grassmannian.
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Hessenberg varieties of parabolic type
arXiv: Algebraic Geometry, 2017Co-Authors: Martha Precup, Julianna TymoczkoAbstract:This paper proves a combinatorial relationship between two well-studied subvarieties of the flag variety: certain Hessenberg varieties, which are a family of subvarieties of the flag variety that includes Springer fibers, and Schubert varieties, which induce a well-known basis for the cohomology of the flag variety. The main result shows that the Betti numbers of parabolic Hessenberg varieties decompose into a combination of those of Springer fibers and Schubert varieties associated to the parabolic. As a corollary we show that the Betti numbers of some parabolic Hessenberg varieties in Lie type A are equal to those of a specific union of Schubert varieties. The corollary uses (and generalizes) recent work of the same authors that proves the analogous result for certain Springer fibers in Lie type A.
Megumi Harada - One of the best experts on this subject based on the ideXlab platform.
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A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
Mathematische Zeitschrift, 2020Co-Authors: Megumi Harada, Martha Precup, Tatsuya Horiguchi, Satoshi Murai, Julianna TymoczkoAbstract:Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $$GL(n,{\mathbb {C}})/B$$ G L ( n , C ) / B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $$GL(n-1,{\mathbb {C}})/B$$ G L ( n - 1 , C ) / B , showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of “Hessenberg Schubert polynomials” in the context of regular nilpotent Hessenberg varieties, which generalize the classical Schubert polynomials. We also outline several open questions pertaining to them.
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A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
arXiv: Algebraic Geometry, 2019Co-Authors: Megumi Harada, Martha Precup, Tatsuya Horiguchi, Satoshi Murai, Julianna TymoczkoAbstract:Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincare polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of "Hessenberg Schubert polynomials" in the context of regular nilpotent Hessenberg varieties, and outline several open questions pertaining to them.
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Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies
Selecta Mathematica, 2018Co-Authors: Hiraku Abe, Lauren Dedieu, Federico Galetto, Megumi HaradaAbstract:In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plucker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety.
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geometry of Hessenberg varieties with applications to newton okounkov bodies
Selecta Mathematica-new Series, 2018Co-Authors: Hiraku Abe, Lauren Dedieu, Federico Galetto, Megumi HaradaAbstract:In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plucker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety.
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The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture
arXiv: Combinatorics, 2017Co-Authors: Megumi Harada, Martha PrecupAbstract:We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the $S_n$-representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case, and generalizes results obtained by Shareshian-Wachs and Teff. Our proof uses previous work of Stanley, Gasharov, Shareshian-Wachs, and Brosnan-Chow, as well as results of the second author on the geometry and combinatorics of Hessenberg varieties. As part of our arguments, we obtain inductive formulas for the Poincare polynomials of regular abelian Hessenberg varieties.
Erik Insko - One of the best experts on this subject based on the ideXlab platform.
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A formula for the cohomology and K-class of a regular Hessenberg variety
Journal of Pure and Applied Algebra, 2020Co-Authors: Erik Insko, Julianna Tymoczko, Alexander WooAbstract:Abstract Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on h than here.
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The singular locus of semisimple Hessenberg varieties
Journal of Algebra, 2019Co-Authors: Erik Insko, Martha PrecupAbstract:Abstract Although regular semisimple Hessenberg varieties are smooth and irreducible, semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of semisimple Hessenberg varieties corresponding to the standard Hessenberg space in all Lie types. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of this nature do not hold for all semisimple Hessenberg varieties.
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The singular locus of semisimple Hessenberg varieties
arXiv: Algebraic Geometry, 2017Co-Authors: Erik Insko, Martha PrecupAbstract:Although regular semisimple Hessenberg varieties are smooth and irreducible, semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of semisimple Hessenberg varieties corresponding to the standard Hessenberg space. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of this nature do not hold for all semisimple Hessenberg varieties.
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Equivariant cohomology and local invariants of Hessenberg varieties
1Co-Authors: Erik InskoAbstract:Nilpotent Hessenberg varieties are a family of subvarieties of the flag variety, which include the Springer varieties, the Peterson variety, and the whole flag variety. In this thesis I give a geometric proof that the cohomology of the flag variety surjects onto the cohomology of the Peterson variety; I provide a combinatorial criterion for determing the singular loci of a large family of regular nilpotent Hessenberg varieties; and I describe the equivariant cohomology of any regular nilpotent Hessenberg variety whose cohomology is generated by its degree two classes.
Hiraku Abe - One of the best experts on this subject based on the ideXlab platform.
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Fano and weak Fano Hessenberg varieties.
arXiv: Algebraic Geometry, 2020Co-Authors: Hiraku Abe, Naoki Fujita, Haozhi ZengAbstract:Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anti-canonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.
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Geometry of regular Hessenberg varieties
Transformation Groups, 2020Co-Authors: Hiraku Abe, Naoki Fujita, Haozhi ZengAbstract:Let $$ \mathfrak{g} $$ be a complex semisimple Lie algebra. For a regular element x in $$ \mathfrak{g} $$ and a Hessenberg space H ⊆ $$ \mathfrak{g} $$, we consider a regular Hessenberg variety X(x, H) in the ag variety associated with $$ \mathfrak{g} $$. We take a Hessenberg space so that X(x, H) is irreducible, and show that the higher cohomology groups of the structure sheaf of X(x, H) vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.
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A survey of recent developments on Hessenberg varieties
arXiv: Algebraic Geometry, 2019Co-Authors: Hiraku Abe, Tatsuya HoriguchiAbstract:This article surveys recent developments on Hessenberg varieties, emphasizing some of the rich connections of their cohomology and combinatorics. In particular, we will see how hyperplane arrangements, representations of symmetric groups, and Stanley's chromatic symmetric functions are related to the cohomology rings of Hessenberg varieties. We also include several other topics on Hessenberg varieties to cover recent developments.
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Hessenberg varieties, Slodowy slices, and integrable systems
Mathematische Zeitschrift, 2019Co-Authors: Hiraku Abe, Peter CrooksAbstract: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, \(X(H_0)\), 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 \(X(H_0)\). We also show \(X(H_0)\) to have an open dense symplectic leaf isomorphic to \(G/Z \times S_{\text {reg}}\), where Z is the centre of G and \(S_{\text {reg}}\) is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on \(G\times S_{\text {reg}}\), as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties.
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Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies
Selecta Mathematica, 2018Co-Authors: Hiraku Abe, Lauren Dedieu, Federico Galetto, Megumi HaradaAbstract:In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plucker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety.