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Jaeseong Oh

Publications and source records attributed to Jaeseong Oh.

At least 19 recordsLinked to original sources
arXiv2026

Haglund--Haiman--Loehr formula via Carlsson--Mellit Algebra

We prove a Haglund--Haiman--Loehr type combinatorial formula for the torus fixed point classes $I_{\mu,w}$ in the equivariant $K$-theory $K_{\mathbb{C}^{*}\times\mathbb{C}^{*}}(\operatorname{PFH}_{n,n-k})$ of parabolic flag Hilbert schemes. Under the identification of Bechtloff Weising and Orr, our result gives an HHL type formula for modified partially symmetric Macdonald functions in terms of weighted partial Dyck paths. When $w=\emptyset$, it specializes to the classical HHL formula. The proof relies essentially on the Carlsson--Gorsky--Mellit action of the Carlsson--Mellit algebra $\mathbb{A}_{q,t}$.

math.CO↗
arXiv2026

Shuffle theorem for torus link homology

We prove that the symmetric function $e_{(1^k)}[-MX^{m,n}] \cdot 1$, arising from the elliptic Hall algebra, equals the generating function for $k$-tuples of cyclic $(m,n)$-parking functions. This result resolves a conjecture of Gorsky--Mazin--Vazirani and Wilson, establishing that the elliptic Hall algebra governs the Khovanov--Rozansky homology of torus links $T(km,kn)$. Consequently, this provides an affirmative answer to a question of Galashin and Lam in the torus link case. As a key step in the proof, we develop a rational analogue of the Shareshian--Wachs involution originally introduced to prove the symmetry property of the chromatic quasisymmetric functions.

math.CO↗
arXiv2025

Hadamard Products of dual Jacobi-Trudi matrices

We study positivity properties of Hadamard products of Jacobi-Trudi matrices. Maló proved that the Hadamard (entrywise) product of two totally positive upper-triangular Toeplitz matrices whose Toeplitz sequences are the coefficient sequences of real-rooted polynomials with nonpositive zeros is again totally positive. Sokal conjectured that this result can be strengthened to total monomial positivity for the Hadamard product of Jacobi-Trudi matrices. In this paper we show that Temperley-Lieb immanants are Schur positive for Hadamard products of Jacobi-Trudi matrices given by ribbon-like skew shapes. In particular, we affirm Sokal's conjecture for minors given by ribbon-like skew shapes. Moreover, we provide a manifestly positive Schur expansion for Temperley-Lieb immanants evaluated on the Hadamard product of Jacobi-Trudi matrices indexed by ribbons. In addition, for the ribbon case, we construct a corresponding representation, offering a representation-theoretic proof of the Schur positivity.

math.CO↗
arXiv2025

Spectral analysis of $q$-deformed unitary ensembles with the Al-Salam--Carlitz weight

We study $q$-deformed random unitary ensembles associated with the weight function of the Al-Salam--Carlitz orthogonal polynomials, indexed by a parameter $a < 0$. In the special case $a = -1$, the model reduces to the $q$-deformed Gaussian unitary ensemble. Employing the Flajolet--Viennot theory together with the combinatorics of matchings, we derive an explicit positive-sum expression for the spectral moments. In the double-scaling regime $q = e^{-λ/N}$, where $N$ denotes the ensemble size and $λ> 0$ is fixed, we derive the first two terms in the large-$N$ expansion of the spectral moments. As a consequence, we obtain a closed-form expression for the limiting spectral density. Notably, this density exhibits two successive phase transitions as $λ$ increases, characterised by a reduction in the number of soft edges from two, to one, and eventually to none. Furthermore, we show that the limiting density coincides with the limiting zero distribution of the Al-Salam--Carlitz orthogonal polynomials under the same scaling.

math-ph↗
arXiv2025

Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials

An edge-weighted graph $G$, possibly with loops, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory, including the number of independent sets and proper vertex-colourings, as well as their relaxations in statistical physics. We obtain homomorphism inequalities for various graphs $H$ and antiferromagnetic graphs~$G$ of the form \[ \lvert\operatorname{Hom}(H,G)\rvert^2 \leq \lvert\operatorname{Hom}(H\times K_2,G)\rvert, \] where $H\times K_2$ denotes the tensor product of $H$ and $K_2$. Firstly, we show that the inequality holds for any $H$ obtained by blowing up vertices of a bipartite graph into complete graphs and any antiferromagnetic $G$. In particular, one can take $H=K_{d+1}$, which already implies a new result for the Sah--Sawhney--Stoner--Zhao conjecture on the maximum number of $d$-regular graphs in antiferromagnetic graphs. Secondly, the inequality also holds for $G=K_q$ and those $H$ obtained by blowing up vertices of a bipartite graph into complete multipartite graphs, paths or even cycles. Both results can be seen as the first progress towards Zhao's conjecture on $q$-colourings, which states that the inequality holds for any $H$ and $G=K_q$, after his own work. Our method leverages on the emerging theory of Lorentzian polynomials due to Brändén and Huh and log-concavity of the list colourings of bipartite graphs, which may be of independent interest.

math.CO↗
arXiv2025

Total positivity of Hadamard product of dual Jacobi--Trudi matrices

In 1992, Wagner proved that the Hadamard product of two totally positive lower triangular Toeplitz matrices is totally positive. In this work, we strengthen this result by establishing total monomial positivity for the Hadamard product of Jacobi--Trudi matrices. In particular, we resolve a conjecture of Sokal concerning the Hadamard square of Jacobi--Trudi matrices. Moreover, we provide a manifestly positive Schur expansion for the Hadamard square of Jacobi--Trudi matrices indexed by ribbons. In addition, we construct a corresponding representation, offering a representation-theoretic proof of the Schur positivity.

math.CO↗
arXiv2025

$α$-chromatic symmetric functions

In this paper, we introduce the \emph{$α$-chromatic symmetric functions} $χ^{(α)}_π[X;q]$, extending Shareshian and Wachs' chromatic symmetric functions with an additional real parameter $α$. We present positive combinatorial formulas with explicit interpretations. Notably, we show an explicit monomial expansion in terms of the $α$-binomial basis and an expansion into certain chromatic symmetric functions in terms of the $α$-falling factorial basis. Among various connections with other subjects, we highlight a significant link to $q$-rook theory, including a new solution to the $q$-hit problem posed by Garsia and Remmel in their 1986 paper introducing $q$-rook polynomials.

math.CO↗
arXiv2025

Zigzags, contingency tables, and quotient rings

Let $\mathbf{x}_{k \times p}$ be a $k \times p$ matrix of variables and let $\mathbb{F}[\mathbf{x}_{k \times p}]$ be the polynomial ring in these variables. Given two weak compositions $α,β\models_0 n$ of lengths $\ell(α) = k$ and $\ell(β) = p$, we study the ideal $I_{α,β} \subseteq \mathbb{F}[\mathbf{x}_{k \times \ell}]$ generated by row sums, column sums, monomials in row $i$ of degree $> α_i$, and monomials in column $j$ of degree $> β_j$. We prove results connecting algebraic properties of the quotient ring $R_{α,β} := \mathbb{F}[\mathbf{x}_{k \times \ell}]/I_{α,β}$ with the set $C_{α,β}$ of $α,β$-contingency tables. The standard monomial basis of $R_{α,β}$ with respect to a diagonal term order is encoded by the matrix-ball avatar of the RSK correspondence. We describe the Hilbert series of $R_{α,β}$ in terms of a zigzag statistic on contingency tables. The ring $R_{α,β}$ carries a graded action of the product $\mathrm{Stab}(α) \times \mathrm{Stab}(β)$ of symmetry groups of the sequences $α= (α_1,\dots,α_k)$ and $β= (β_1,\dots,β_p)$; we describe how to calculate the isomorphism type of this graded action. Our analysis regards the set $C_{α,β}$ as a locus in the affine space $\mathrm{Mat}_{k \times p}(\mathbb{F})$ and applies orbit harmonics to this locus.

math.CO↗
arXiv2025

Refinement of Hikita's $e$-positivity theorem via Abreu--Nigro's $g$-functions and restricted modular law

We study the symmetric functions \( g_{\mm,k}(x;q) \), introduced by Abreu and Nigro for a Hessenberg function \( \mm \) and a positive integer \( k \), which refine the chromatic symmetric function. Building on Hikita's recent breakthrough on the Stanley--Stembridge conjecture, we prove the \( e \)-positivity of \( g_{\mm,k}(x;1) \), refining Hikita's result. We also provide a Schur expansion of the sum \( \sum_{k=1}^n e_k(x) g_{\mm,n-k}(x;q) \) in terms of \( P \)-tableaux with 1 in the upper-left corner. We introduce a restricted version of the modular law as our main tool. Then, we show that any function satisfying the restricted modular law is determined by its values on disjoint unions of path graphs.

math.CO↗
arXiv2025

Shuffle formula in science fiction for Macdonald polynomials

We initiate the study of the Macdonald intersection polynomials $\operatorname{I}_{μ^{(1)},\dots,μ^{(k)}}[X;q,t]$, which are indexed by $k$-tuples of partitions $μ^{(1)},\dots,μ^{(k)}$. These polynomials are conjectured to be equal to the bigraded Frobenius characteristic of the intersection of Garsia-Haiman modules, as proposed by the science fiction conjecture of Bergeron and Garsia. In this work, we establish the vanishing identity and the shape independence of the Macdonald intersection polynomials. Additionally, we unveil a remarkable connection between $\operatorname{I}_{μ^{(1)},\dots,μ^{(k)}}$ and the character $\nabla e_{k-1}$ of diagonal coinvariant algebra by employing the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler. Furthermore, we establish a connection between $\operatorname{I}_{μ^{(1)},\dots,μ^{(k)}}$ and the shuffle formula $D_{k-1}[X;q,t]$, utilizing novel combinatorial tools such as the column exchange rule, a new fermionic formula for the shuffle formula, and the lightning bolt formula for Macdonald intersection polynomials. Notably, our findings provide a new proof for the shuffle theorem.

math.CO↗
arXiv2024

Extending the science fiction and the Loehr--Warrington formula

We introduce the Macdonald piece polynomial $\operatorname{I}_{μ,λ,k}[X;q,t]$, which is a vast generalization of the Macdonald intersection polynomial in the science fiction conjecture by Bergeron and Garsia. We demonstrate a remarkable connection between $\operatorname{I}_{μ,λ,k}$, $\nabla s_λ$, and the Loehr--Warrington formula $\operatorname{LW}_λ$, thereby obtaining the Loehr--Warrington conjecture as a corollary. To connect $\operatorname{I}_{μ,λ,k}$ and $\nabla s_λ$, we employ the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler, and to connect $\operatorname{I}_{μ,λ,k}$ and $\operatorname{LW}_λ$, we use our new findings on the combinatorics of $P$-tableaux together with the column exchange rule. We also present an extension of the science fiction conjecture and the Macdonald positivity by exploiting $\operatorname{I}_{μ,λ,k}$.

math.CO↗
arXiv2024

$q$-deformed Gaussian unitary ensemble: spectral moments and genus-type expansions

The eigenvalue probability density function of the Gaussian unitary ensemble permits a $q$-extension related to the discrete $q$-Hermite weight and corresponding $q$-orthogonal polynomials. A combinatorial counting method is used to specify a positive sum formula for the spectral moments of this model. The leading two terms of the scaled $1/N^2$ genus-type expansion of the moments are evaluated explicitly in terms of the incomplete beta function. Knowledge of these functional forms allows for the smoothed leading eigenvalue density and its first correction to be determined analytically.

math.PR↗
arXiv2023

Haglund's conjecture for multi-$t$ Macdonald polynomials

We provide new approaches to prove identities for the modified Macdonald polynomials via their LLT expansions. As an application, we prove a conjecture of Haglund concerning the multi-$t$-Macdonald polynomials of two rows.

math.CO↗
arXiv2022

Toward Butler's conjecture

For a partition $\nu$, let $\lambda,\mu\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Butler conjectured that the divided difference $\operatorname{I}_{\lambda,\mu}[X;q,t]=(T_\lambda\widetilde{H}_\mu[X;q,t]-T_\mu\widetilde{H}_\lambda[X;q,t])/(T_\lambda-T_\mu)$ of modified Macdonald polynomials of two partitions $\lambda$ and $\mu$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for $\operatorname{I}_{\lambda,\mu}[X;q,t]$, which is a positive monomial expansion. We also prove Butler's conjecture for some special cases.

math.CO↗
arXiv2021

Dualities and reciprocities on graphs on surfaces

We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.

math.CO↗
arXiv2021

Macdonald polynomials and cyclic sieving

The Garsia--Haiman module is a bigraded $\mathfrak{S}_n$-module whose Frobenius image is a Macdonald polynomial. The method of orbit harmonics promotes an $\mathfrak{S}_n$-set $X$ to a graded polynomial ring. The orbit harmonics can be applied to prove cyclic sieving phenomena which is a notion that encapsulates the fixed-point structure of finite cyclic group action on a finite set. By applying this idea to the Garsia--Haiman module, we provide cyclic sieving results regarding the enumeration of matrices that are invariant under certain cyclic row and column rotation and translation of entries.

math.CO↗
arXiv2020

Cyclic sieving and orbit harmonics

Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point locus in $\mathbb{C}^n$. The cyclic sieving phenomenon is a notion in enumerative combinatorics which encapsulates the fixed-point structure of the action of a finite cyclic group $C$ on a finite set $X$ in terms of root-of-unity evaluations of an auxiliary polynomial $X(q)$. We apply orbit harmonics to prove cyclic sieving results.

math.CO↗