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Mikiya Masuda

Publications and source records attributed to Mikiya Masuda.

At least 19 recordsLinked to original sources
arXiv2026

Six-dimensional GKM manifolds with four fixed points

In this paper, we study $6$-dimensional GKM manifolds with $4$ fixed points. We classify all possible GKM graphs, and for each type of graph we construct a manifold, proving the existence. We show that six types occur. (P1) complex projective space $\mathbb{C} P^3$ with standard complex structure (P2) blow up of $S^6$ at a fixed point, diffeomorphic to $\mathbb{C} P^3$ (P3) $\mathbb{C} P^3$ as the homogeneous space $\mathrm{Sp}(2)/(\mathrm{U}(1) \times \mathrm{Sp}(1))$ with non-standard almost complex structure (Q1) complex quadric $Q_3$ with standard complex structure (Q2) blow up of $S^6$ along isotropy $2$-sphere, diffeomorphic to $Q_3$ (S) $S^2 \times S^4$, obtained as equivariant gluing along orbits of two $S^6$'s

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arXiv2025

Notes on the cohomology of partial Hessenberg varieties

Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this paper, we study their cohomology by relating them to the cohomology of Hessenberg varieties in the full flag varieties.

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arXiv2025

Gamma vectors of partitioned permutohedra

We determine that $γ$-vectors of partitioned permutohedra, thereby generalizing a result of Foata and Schützenberger. Our result is closely related to a result of Athanasiadis on the representation of the symmetric group on the cohomology of the permutohedral variety. We explain how to derive Athanasiadis' result from ours and vice versa.

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arXiv2025

Symmetric matrices defined by plane vector sequences

Motivated by a work of Fu-So-Song, we associate a symmetric matrix $A$ to a plane vector sequence $v$ and give a formula to find the signature of $A$ in terms of the sequence $v$. When $A$ is nonsingular, we interpret the relation between $A$ and $A^{-1}$ from a topological viewpoint. Finally, we associate an omnioriented quasitoric orbifold $X$ of real dimension four to the sequence $v$ and show that $A^{-1}$ is the intersection matrix of the characteristic suborbifolds of $X$.

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arXiv2024

Automorphisms of GKM graphs and regular semisimple Hessenberg varieties

A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth subvariety of the full flag variety $\mathrm{Fl}(\mathbb{C}^n)$ associated with a regular semisimple matrix $S$ of order $n$ and a function $h$ from $\{1,2,\dots,n\}$ to itself satisfying a certain condition. We show that when $\mathrm{Hess}(S,h)$ is connected and not the entire space $\mathrm{Fl}(\mathbb{C}^n)$, the reductive part of the identity component $\mathrm{Aut}^0(\mathrm{Hess}(S,h))$ of the automorphism group $\mathrm{Aut}(\mathrm{Hess}(S,h))$ of $\mathrm{Hess}(S,h)$ is an algebraic torus of dimension $n-1$ and $\mathrm{Aut}(\mathrm{Hess}(S,h))/\mathrm{Aut}^0(\mathrm{Hess}(S,h))$ is isomorphic to a subgroup of $\mathfrak{S}_n$ or $\mathfrak{S}_n\rtimes \{\pm 1\}$, where $\mathfrak{S}_n$ is the symmetric group of degree $n$. As a byproduct of our argument, we show that $\mathrm{Aut}(X)/\mathrm{Aut}^0(X)$ is a finite group for any projective GKM manifold $X$.

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arXiv2024

Torus orbit closures in the flag variety

The study of torus orbit closures in the (complete) flag variety was initiated by Klyachko and Gelfand--Serganova in the mid-1980s, but it seems that not much has been done since then. In this chapter, we present some of the work by Klyachko and Gelfand--Serganova and our recent work on the topology, geometry, and combinatorics of torus orbit closures in the flag variety.

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arXiv2023

Toric Schubert varieties and directed Dynkin diagrams

A flag variety is a homogenous variety $G/B$ where $G$ is a simple algebraic group over the complex numbers and $B$ is a Boel subgroup of $G$. A Schubert variety $X_w$ is a subvariety of $G/B$ indexed by an element $w$ in the Weyl group of $G$. It is called toric if it is a toric variety with respect to the maximal torus of $G$ in $B$. In this paper, we associate an edge-labeled digraph $\mathcal{G}_w$ with a toric Schubert variety $X_w$ and classify toric Schubert varieties up to isomorphism. We also give a simple criterion of when a toric Schubert variety $X_w$ is (weak) Fano in terms of $\mathcal{G}_w$. Finally, we discuss whether toric Schubert varieties can be distinguished by their integral cohomology rings up to isomorphism and show that this is the case when $G$ is of simply-laced type.

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arXiv2023

Modular law through GKM theory

The solution of Shareshian-Wachs conjecture by Brosnan-Chow and Guay-Paquet tied the graded chromatic symmetric functions on indifference graphs (or unit interval graphs) and the cohomology of regular semisimple Hessenberg varieties with the dot action. A similar result holds between unicellular LLT polynomials and twins of regular semisimple Hessenberg varieties. A recent result by Abreu-Nigro enabled us to prove these results by showing the modular law for the geometrical objects, and this is indeed done by Precup-Sommers and Kiem-Lee. In this paper, we give elementary and simpler proofs to the modular law through GKM theory.

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arXiv2023

Regular semisimple Hessenberg varieties with cohomology rings generated in degree two

A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth subvariety of the flag variety determined by a square matrix $S$ with distinct eigenvalues and a Hessenberg function $h$. The cohomology ring $H^*(\mathrm{Hess}(S,h))$ is independent of the choice of $S$ and is not explicitly described except for a few cases. In this paper, we characterize the Hessenberg function $h$ such that $H^*(\mathrm{Hess}(S,h))$ is generated in degree two as a ring. It turns out that such $h$ is what is called a (double) lollipop.

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arXiv2022

Toric Richardson varieties of Catalan type and Wedderburn-Etherington numbers

We associate a complete non-singular fan with a polygon triangulation. Such a fan appears from a certain toric Richardson variety, called of Catalan type introduced in this paper. A toric Richardson variety of Catalan type is a Fano Bott manifold. We show that toric Richardson varieties of Catalan type are classified up to isomorphism in terms of unordered binary trees. In particular, the number of isomorphism classes of $n$-dimensional toric Richardson varieties of Catalan type is the $(n+1)$th Wedderburn--Etherington number.

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arXiv2022

Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties

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.

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arXiv2022

The second cohomology of regular semisimple Hessenberg varieties from GKM theory

We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group $\mathfrak{S}_n$ by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the isomorphism class of the second cohomology as an $\mathfrak{S}_n$-module. Our formula is not exactly the same as the known formula by Chow or Cho-Hong-Lee but they are equivalent. We also discuss its higher degree generalization.

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arXiv2022

How is a graph not like a manifold?

For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of $j$-independency of tangent weights at each fixed point implies $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r>j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of $(n-1)$-dimensional torus, under certain colorability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. Such observation underlines certain similarity between actions of complexity one and torus manifolds.

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arXiv2021

On the enumeration of Fano Bott manifolds

Fano Bott manifolds bijectively correspond to signed rooted forests with some equivalence relation. Using this bijective correspondence, we enumerate the isomorphism classes of Fano Bott manifolds and the diffeomorphism classes of indecomposable Fano Bott manifolds. We also observe that the signed rooted forests with the equivalence relation bijectively correspond to rooted triangular cacti.

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arXiv2021

Toric orbifolds associated with partitioned weight polytopes in classical types

Given a root system $Φ$ of type $A_n$, $B_n$, $C_n$, or $D_n$ in Euclidean space $E$, let $W$ be the associated Weyl group. For a point $p \in E$ not orthogonal to any of the roots in $Φ$, we consider the $W$-permutohedron $P_W$, which is the convex hull of the $W$-orbit of $p$. The representation of $W$ on the rational cohomology ring $H^\ast(X_Φ)$ of the toric variety $X_Φ$ associated to (the normal fan to) $P_W$ has been studied by various authors. Let $\{s_1,\ldots,s_n\}$ be a complete set of simple reflections in $W$. For $K \subseteq [n]$, let $W_K$ be the standard parabolic subgroup of $W$ generated by $\{s_k:k \in K\}$. We show that the fixed subring $H^\ast(X_Φ)^{W_K}$ is isomorphic to the cohomology ring of the toric variety $X_Φ(K)$ associated to a polytope obtained by intersecting $P_W$ with half-spaces bounded by reflecting hyperplanes for the given generators of $W_K$. By a result of Balibanu--Crooks, the cohomology rings $H^\ast(X_Φ(K))$ are isomorphic with cohomology rings of certain regular Hessenberg varieties.

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arXiv2020

Torus orbit closures in flag varieties and retractions on Weyl groups

A finite Coxeter group $W$ has a natural metric $d$ and if $\mathcal{M}$ is a subset of $W$, then for each $u\in W$, there is $q\in \mathcal{M}$ such that $d(u,q)=d(u,\mathcal{M})$. Such $q$ is not unique in general but if $\mathcal{M}$ is a Coxeter matroid, then it is unique, and we define a retraction $\mathcal{R}^m_{\mathcal{M}}\colon W\to \mathcal{M}\subset W$ so that $\mathcal{R}^m_{\mathcal{M}}(u)=q$. The $T$-fixed point set $Y^T$ of a $T$-orbit closure $Y$ in a flag variety $G/B$ is a Coxeter matroid, where $G$ is a semisimple algebraic group, $B$ is a Borel subgroup, and $T$ is a maximal torus of $G$ contained in $B$. We define a retraction $\mathcal{R}^g_{Y}\colon W\to Y^T\subset W$ geometrically, where $W$ is the Weyl group of $G$, and show that $\mathcal{R}^g_{Y}=\mathcal{R}^m_{Y^T}$. We introduce another retraction $\mathcal{R}^a_{\mathcal{M}}\colon W\to \mathcal{M}\subset W$ algebraically for an arbitrary subset $\mathcal{M}$ of $W$ when $W$ is a Weyl group of classical Lie type, and show that $\mathcal{R}^a_{\mathcal{M}}=\mathcal{R}^m_{\mathcal{M}}$ when $\mathcal{M}$ is a Coxeter matroid.

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arXiv2020

On Schubert varieties of complexity one

Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a maximal torus contained in $B$. Then $\mathbb{T}$ acts on $\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is $\mathbb{T}$-invariant. We say that a Schubert variety is of complexity $k$ if a maximal $\mathbb{T}$-orbit in $X_w$ has codimension $k$. In this paper, we discuss topology, geometry, and combinatorics related to Schubert varieties of complexity one.

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