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Soojin Cho

Publications and source records attributed to Soojin Cho.

15 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.

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arXiv2026

Hall-Littlewood functions in noncommuting variables

In 2022 Aliniaeifard, Li, and van Willigenburg defined Schur functions in the algebra of symmetric functions in noncommuting variables (NCSym), answering an open question posed by Rosas and Sagan in 2004. These Schur functions are not monomial positive, since they are defined via a noncommutative analogue of the Jacobi-Trudi determinant. We introduce Hall-Littlewood functions ${\bf P}_{\pi}({\bf x};t)$ indexed by set partitions $\pi$ in noncommuting variables ${\bf x}=({\bf x}_1,{\bf x}_2,\ldots)$, and define Schur functions in noncommuting variables to be ${\bf s}_{\pi}({\bf x})={\bf P}_{\pi}({\bf x};0)$. We prove that the set of Hall-Littlewood functions $\{{\bf P}_{\pi}({\bf x};t)\}$ for all set partitions $\pi$ of $[n]$ forms a $\mathbb{Q}[t]$-basis of NCSym of homogeneous degree $n$, and that this basis is invariant under any permutation acting on set partitions. These Hall-Littlewood functions in NCSym map to classical Hall-Littlewood functions under commutation, up to a scalar factor. We also show that the Hall-Littlewood functions ${\bf P}_{\pi}({\bf x};t)$ naturally refine the lifted Hall-Littlewood functions in NCSym. Specifically, the Schur functions ${\bf s}_{\pi}({\bf x})$ are monomial positive and refine the lifted Schur function introduced by Rosas and Sagan. Moreover, we introduce a star product of two polynomials in NCSym and develop the star-multiplication rule for a lifted and a non-lifted Hall-Littlewood functions in NCSym. This rule is a noncommutative analogue of the product rule for two Hall-Littlewood functions and, in particular, of the Littlewood-Richardson rule. Finally, our approach extends to the algebra of quasisymmetric functions in noncommuting variables (NCQSym) indexed by set compositions.

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arXiv2023

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.

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arXiv2023

Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties

We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting of the classes that naturally arise from the Bialynicki-Birula decomposition of the Hessenberg varieties. We give an explicit combinatorial description of the support of each class, which enables us to compute the symmetric group actions on the classes in our bases. We then successfully apply the results to the permutohedral varieties to explicitly write down each class and to construct permutation submodules that constitute summands of a decomposition of cohomology space of each degree. This resolves the problem posed by Stembridge on the geometric construction of permutation module decomposition and also the conjecture posed by Chow on the construction of bases for the equivariant cohomology spaces of permutohedral varieties.

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arXiv2022

Permutation module decomposition of the second cohomology of a regular semisimple Hessenberg variety

Regular semisimple Hessenberg varieties admit actions of associated Weyl groups on their cohomology space of each degree. In this paper, we consider the module structure of the cohomology spaces of regular semisimple Hessenberg varieties of type $A$. We define a subset of the Bialynicki-Birula basis of the cohomology space so that they become a module generator set of the cohomology module of each degree. We then use those generators to construct permutation submodules of the degree two cohomology module and show that they form a permutation module decomposition. Our construction is consistent with a known combinatorial result by Chow on chromatic quasisymmetric functions.

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arXiv2022

Slide multiplicity free key polynomials

Schubert polynomials are refined by the key polynomials of Lascoux-Schützenberger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles. In this paper we determine which fundamental slide polynomial refinements of key polynomials, indexed by strong compositions, are multiplicity free. We also give a recursive algorithm to determine all terms in the fundamental slide polynomial refinement of a key polynomial indexed by a strong composition. From here, we apply our results to begin to classify which fundamental slide polynomial refinements, indexed by weak compositions, are multiplicity free. We completely resolve the cases when the weak composition has at most two nonzero parts or the sum has at most two nonzero terms.

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arXiv2019

An analogue of chromatic bases and $p$-positivity of skew Schur $Q$-functions

We investigate chromatic symmetric functions in the relation to the algebra $Γ$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $Γ$ in terms of chromatic symmetric functions. We also consider the $p$-positivity of skew Schur $Q$-functions and find a class of $p$-positive ribbon Schur $Q$-functions, making a conjecture that they are \emph{all}. We include many concrete computational results that support our conjecture.

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arXiv2019

Geometric representations of finite groups on real toric spaces

We develop a framework to construct geometric representations of finite groups $G$ through the correspondence between real toric spaces $X^{\mathbb R}$ and simplicial complexes with characteristic matrices. We give a combinatorial description of the $G$-module structure of the homology of $X^{\mathbb R}$. As applications, we make explicit computations of the Weyl group representations on the homology of real toric varieties associated to the Weyl chambers of type $A$ and $B$, which show an interesting connection to the topology of posets. We also realize a certain kind of Foulkes representation geometrically as the homology of real toric varieties.

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arXiv2017

On $e$-positivity and $e$-unimodality of chromatic quasisymmetric functions

The $e$-positivity conjecture and the $e$-unimodality conjecture of chromatic quasisymmetric functions are proved for some classes of natural unit interval orders. Recently, J. Shareshian and M. Wachs introduced chromatic quasisymmetric functions as a refinement of Stanley's chromatic symmetric functions and conjectured the $e$-positivity and the $e$-unimodality of these functions. The $e$-positivity of chromatic quasisymmetric functions implies the $e$-positivity of corresponding chromatic symmetric functions, and our work resolves Stanley's conjecture on chromatic symmetric functions of $(3+1)$-free posets for two classes of natural unit interval orders.

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arXiv2017

Chromatic classical symmetric functions

In this note we classify when a skew Schur function is a positive linear combination of power sum symmetric functions. We then use this to determine precisely when any scalar multiple of a skew Schur function is the chromatic symmetric function of some graph. From here we are able to prove that of the classical bases for symmetric functions only certain scalar multiples of the elementary symmetric functions can be realised as the chromatic symmetric function of some graph, namely a particular union of complete graphs.

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arXiv2016

Chromatic bases for symmetric functions

In this note we obtain numerous new bases for the algebra of symmetric functions whose generators are chromatic symmetric functions. More precisely, if $\{ G_ k \}_{k\geq 1}$ is a set of connected graphs such that $G_k$ has $k$ vertices for each $k$, then the set of all chromatic symmetric functions $\{ X_{G_ k} \}_{k\geq 1}$ generates the algebra of symmetric functions. We also obtain explicit expressions for the generators arising from complete graphs, star graphs, path graphs and cycle graphs.

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arXiv2015

A combinatorial proof of a symmetry of $(t,q)$-Eulerian numbers of type $B$ and type $D$

A symmetry of $(t,q)$-Eulerian numbers of type $B$ is combinatorially proved by defining an involution preserving many important statistics on the set of permutation tableaux of type $B$. This involution also proves a symmetry of the generating polynomial $\hat{D}_{n, k}(p,q,r)$ of number of crossings and alignments, and hence $q$-Eulerian numbers of type $A$ defined by L. Williams. By considering a restriction of our bijection, we were led to define a new statistic on the permutations of type $D$ and $(t,q)$-Eulerian numbers of type $D$, which is proved to have a nice symmetry as well. We conjecture that our new statistic is in the family of Eulerian statistics for the permutations of type $D$.

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arXiv2014

Permutation statistics and weak {B}ruhat order in permutation tableaux of type $B$

Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type $B$: Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. This leads us to relate the number of alignments and crossings with other statistics of signed permutations and also to characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$.

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arXiv2014

The Combinatorics of $\mathsf{A_2}$-webs

The nonelliptic $\mathsf{A_2}$-webs with $k$ "$+$"s on the top boundary and $3n-2k$ "$-$"s on the bottom boundary combinatorially model the space $\mathsf{Hom}_{\mathfrak{sl}_3}(\mathsf{V}^{\otimes (3n-2k)}, \mathsf{V}^{\otimes k})$ of $\mathfrak{sl}_3$-module maps on tensor powers of the natural $3$-dimensional $\mathfrak{sl}_3$-module $\mathsf{V}$, and they have connections with the combinatorics of Springer varieties. Petersen, Pylyavskyy, and Rhodes showed that the set of such $\mathsf{A_2}$-webs and the set of semistandard tableaux of shape $(3^n)$ and type $\{1^2,\dots,k^2,k+1,\dots, 3n-k\}$ have the same cardinalities. In this work, we use the $\sf{m}$-diagrams introduced by Tymoczko and the Robinson-Schensted correspondence to construct an explicit bijection, different from the one given by Russell, between these two sets. In establishing our result, we show that the pair of standard tableaux constructed using the notion of path depth is the same as the pair constructed from applying the Robinson-Schensted correspondence to a $3\,2\,1$-avoiding permutation. We also obtain a bijection between such pairs of standard tableaux and Westbury's $\mathsf{A_2}$ flow diagrams.

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