Searcharxiv⌕ Search

arXiv subjects

Seonjeong Park

Publications and source records attributed to Seonjeong Park.

At least 19 recordsLinked to original sources
arXiv2026

Permutation module decomposition of the cohomology of Hessenberg varieties associated with lollipop graphs

We study the cohomology of regular semisimple Hessenberg varieties associated with lollipop graphs as a module under the dot action. Using the natural basis introduced by Cho, Hong, and Lee, which we call the CHL basis, we establish structural properties of the dot action, including a result for classes satisfying \(i\)-decomposability. We also obtain an explicit elementary symmetric function expansion of the chromatic quasisymmetric functions of lollipop graphs in terms of \(h\)-admissible permutations and their associated partitions. Combining these geometric and combinatorial results, we construct a permutation module decomposition of the cohomology of the corresponding Hessenberg varieties, thereby proving a conjecture of Cho, Hong, and Lee for lollipop graphs.

math.CO↗
arXiv2026

Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties

A \emph{Hessenberg Schubert variety} is an irreducible component of the intersection of a Schubert variety and a Hessenberg variety, defined as the closure of a Schubert cell intersected with the Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type $A$ in this paper. We consider the smoothness of the intersection of a Schubert variety and a Hessenberg variety to ensure the smoothness of the corresponding Hessenberg Schubert variety. Specifically, we analyze the structure of the GKM graphs of the intersection of a Schubert variety and a Hessenberg variety. Our results show that the regularity of these GKM graphs is completely characterized in terms of pattern avoidance, which is a necessary and sufficient condition for the intersection to be smooth. This shows that our pattern avoidance provides a sufficient condition for the smoothness of a Hessenberg Schubert variety.

math.CO↗
arXiv2025

Intersections of Schubert varieties and smooth $T$-stable subvarieties of flag varieties

A smooth projective variety with an action of a torus admits a cell decomposition, called the Bialynicki-Birula decomposition. Singularities of the closures of these cells are not well-known. One of the examples of such closures is a Schubert variety in a flag variety G/B, and there are several criteria for the smoothness of Schubert varieties. In this paper, we focus on the closures of Bialynicki-Birula cells in regular semisimple Hessenberg varieties Hess(s,h), called Hessenberg Schubert varieties. We first consider the intersection of the Schubert varieties with Hess(s,h) and investigate the irreducibility and the smoothness of this intersection, from which we get a sufficient condition for a Hessenberg Schubert variety to be smooth.

math.AG↗
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.

math.AG↗
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.

math.AG↗
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.

math.AG↗
arXiv2022

Toric varieties of Schröder type

A dissection of a polygon is obtained by drawing diagonals such that no two diagonals intersect in their interiors. In this paper, we define a toric variety of Schröder type as a smooth toric variety associated with a polygon dissection. Toric varieties of Schröder type are Fano generalized Bott manifolds, and they are isomorphic if and only if the associated Schröder trees are the same as unordered rooted trees. We describe the cohomology ring of a toric variety of Schröder type using the associated Schröder tree and discuss the cohomological rigidity problem.

math.AG↗
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.

math.AG↗
arXiv2021

Conic decomposition of a toric variety and its application to cohomology

We introduce the notion of a \emph{conic sequence} of a convex polytope. It is a way of building up a polytope starting from a vertex and attaching faces one by one with certain regulations. We apply this to a toric variety to obtain an iterated cofibration structure on it. This allows us to prove several vanishing results in the rational cohomology of a toric variety and to calculate Poincaré polynomials for a large class of singular toric varieties.

math.AT↗
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.

math.CO↗
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.

math.AT↗
arXiv2020

Unique toric structure on a Fano Bott manifold

We prove that if there exists a $c_1$-preserving graded ring isomorphism between integral cohomology rings of two Fano Bott manifolds, then they are isomorphic as toric varieties. As a consequence, we give an affirmative answer to McDuff's question on the uniqueness of a toric structure on a Fano Bott manifold.

math.SG↗
arXiv2020

Poincare polynomials of generic torus orbit closures in Schubert varieties

The closure of a generic torus orbit in the flag variety $G/B$ of type $A$ is known to be a permutohedral variety and its Poincare polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincare polynomial of a generic torus orbit closure in a Schubert variety in $G/B$. When the generic torus orbit closure in a Schubert variety is smooth, its Poincare polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

math.AT↗
arXiv2019

Toric Bruhat interval polytopes

For two elements $v$ and $w$ of the symmetric group $\mathfrak{S}_n$ with $v\leq w$ in Bruhat order, the Bruhat interval polytope $Q_{v,w}$ is the convex hull of the points $(z(1),\ldots,z(n))\in \mathbb{R}^n$ with $v\leq z\leq w$. It is known that the Bruhat interval polytope $Q_{v,w}$ is the moment map image of the Richardson variety $X^{v^{-1}}_{w^{-1}}$. We say that $Q_{v,w}$ is \emph{toric} if the corresponding Richardson variety $X_{w^{-1}}^{v^{-1}}$ is a toric variety. We show that when $Q_{v,w}$ is toric, its combinatorial type is determined by the poset structure of the Bruhat interval $[v,w]$ while this is not true unless $Q_{v,w}$ is toric. We are concerned with the problem of when $Q_{v,w}$ is (combinatorially equivalent to) a cube because $Q_{v,w}$ is a cube if and only if $X_{w^{-1}}^{v^{-1}}$ is a smooth toric variety. We show that a Bruhat interval polytope $Q_{v,w}$ is a cube if and only if $Q_{v,w}$ is toric and the Bruhat interval $[v,w]$ is a Boolean algebra. We also give several sufficient conditions on $v$ and $w$ for $Q_{v,w}$ to be a cube.

math.CO↗
arXiv2018

The volume polynomial of regular semisimple Hessenberg varieties and the Gelfand-Zetlin polytope

Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm{Flag}(\mathbb{C}^n)$ arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of Abe-Horiguchi-Masuda-Murai-Sato and Abe-DeDieu-Galetto-Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand-Zetlin polytope $\mathrm{GZ}(λ)$ for $λ=(λ_1,λ_2,\ldots,λ_n)$. The main results of this manuscript use and generalize tools developed by Anderson-Tymoczko, Kiritchenko-Smirnov-Timorin, and Postnikov, in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand-Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the $α_i := λ_i-λ_{i+1}$. In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial $(n-1)$-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in $\mathrm{Flag}(\mathbb{C}^n)$ as a sum of the cohomology classes of a certain set of Richardson varieties.

math.AG↗
arXiv2018

Graph invariants and Betti numbers of real toric manifolds

For a graph $G$, a graph cubeahedron $\square_G$ and a graph associahedron $\triangle_G$ are simple convex polytopes which admit (real) toric manifolds. In this paper, we introduce a graph invariant, called the $b$-number, and we show that the $b$-numbers compute the Betti numbers of the real toric manifold $X^\mathbb{R}(\square_G)$ corresponding to a graph cubeahedron. The $b$-number is a counterpart of the notion of $a$-number, introduced by S. Choi and the second named author, which computes the Betti numbers of the real toric manifold $X^\mathbb{R}(\triangle_G)$ corresponding to a graph associahedron. We also study various relationships between $a$-numbers and $b$-numbers from a toric topological view. Interestingly, for a forest $G$ and its line graph $L(G)$, the real toric manifolds $X^\mathbb{R}(\triangle_G)$ and $X^\mathbb{R}(\square_{L(G)})$ have the same Betti numbers.

math.CO↗
arXiv2018

On shellability for a poset of even subgraphs of a graph

Given a simple graph $G$, a poset of its even subgraphs was firstly considered by S. Choi and H. Park to study the topology of a real toric manifold associated with $G$. S. Choi and the authors extended this to a graph allowing multiple edges, motivated by the work on the pseudograph associahedron of Carr, Devadoss and Forcey. In this paper, we completely characterize the graphs (allowing multiple edges) whose posets of even subgraphs are always shellable. By the result, we also compute the Betti numbers of a real toric manifold corresponding to a path with two multiple edges.

math.CO↗