arXiv subjects

Seung Jin Lee

Publications and source records attributed to Seung Jin Lee.

At least 19 recordsLinked to original sources

arXiv2026

Hall--Littlewood expansions of chromatic quasisymmetric functions using linked rook placements

In this work, we obtain a Hall--Littlewood expansion of the chromatic quasisymmetric function arising from a natural unit interval order and describe the coefficients in terms of linked rook placements. Applying the Carlsson--Mellit relation between chromatic quasisymmetric functions and unicellular LLT polynomials, we also obtain a combinatorial description for the coefficients of the unicellular LLT polynomials expanded in terms of the modified transformed Hall--Littlewood polynomials.

math.CO↗

arXiv2026

Ubiquitous Antiparallel Domains in 2D Hexagonal Boron Nitride Uncovered by Interferometric Nonlinear Optical Imaging

Hexagonal boron nitride (hBN) supports a wide range of two-dimensional (2D) technologies, yet assessing its crystalline quality over large areas remains a fundamental challenge. Both antiparallel domains, an intrinsic outcome of epitaxy on high-symmetry substrates, and associated structural defects have long evaded optical detection. Here, we show that interferometric second-harmonic generation (SHG) imaging provides a powerful, nondestructive probe of lattice orientation and structural integrity in chemical vapor deposition-grown hBN. This approach reveals the ubiquitous formation of antiparallel domains and quantifies their impact on crystalline order. SHG intensity also emerges as a direct optical metric of domain disorder, spanning three orders of magnitude across films produced by ten different growth routes. Correlation with Raman spectroscopy establishes a unified framework for evaluating crystalline quality. Beyond hBN, this method offers a high-throughput route to wide-area structural imaging in various non-centrosymmetric materials, advancing their deployment in electronics, photonics, and quantum technologies.

cond-mat.mtrl-sci↗

arXiv2026

Toward Butler's conjecture

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

math.CO↗

arXiv2025

LLM Agents at the Roundtable: A Multi-Perspective and Dialectical Reasoning Framework for Essay Scoring

The emergence of large language models (LLMs) has brought a new paradigm to automated essay scoring (AES), a long-standing and practical application of natural language processing in education. However, achieving human-level multi-perspective understanding and judgment remains a challenge. In this work, we propose Roundtable Essay Scoring (RES), a multi-agent evaluation framework designed to perform precise and human-aligned scoring under a zero-shot setting. RES constructs evaluator agents based on LLMs, each tailored to a specific prompt and topic context. Each agent independently generates a trait-based rubric and conducts a multi-perspective evaluation. Then, by simulating a roundtable-style discussion, RES consolidates individual evaluations through a dialectical reasoning process to produce a final holistic score that more closely aligns with human evaluation. By enabling collaboration and consensus among agents with diverse evaluation perspectives, RES outperforms prior zero-shot AES approaches. Experiments on the ASAP dataset using ChatGPT and Claude show that RES achieves up to a 34.86% improvement in average QWK over straightforward prompting (Vanilla) methods.

cs.CL↗

arXiv2025

Lusztig $q$-weight multiplicities and Kirillov-Reshetikhin crystals

Lusztig $q$-weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this work, we investigate these multiplicities through the framework of Kirillov-Reshetikhin crystals. Specifically, for type $C$ with dominant weights and type $B$ with dominant spin weights, we present a combinatorial formula for Lusztig $q$-weight multiplicities in terms of energy functions of Kirillov-Reshetikhin crystals, generalizing the charge statistic on semistandard Young tableaux for type $A$. Additionally, we introduce level-restricted $q$-weight multiplicities for nonexceptional types, and prove positivity by providing their combinatorial formulas.

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

Combinatorial description of Lusztig $q$-weight multiplicity

We conjecture a precise relationship between Lusztig $q$-weight multiplicities for type $C$ and Kirillov-Reshetikhin crystals. We also define $\mathfrak{gl}_n$-version of $q$-weight multiplicity for type $C$ and conjecture the positivity.

math.RT↗

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

Explicit formulas for e-positivity of chromatic quasisymmetric functions

In 1993, Stanley and Stembridge conjectured that a chromatic symmetric function of any $(3+1)$-free poset is $e$-positive. Guay-Paquet reduced the conjecture to $(3+1)$- and $(2+2)$-free posets which are also called natural unit interval orders. Shareshian and Wachs defined chromatic quasisymmetric functions, generalizing chromatic symmetric functions, and conjectured that a chromatic quasisymmetric function of any natural unit interval order is $e$-positive and $e$-unimodal. For a given natural interval order, there is a corresponding partition $λ$ and we denote the chromatic quasisymmetric function by $X_λ$. The first author introduced local linear relations for chromatic quasisymmetric functions. In this paper, we prove a powerful generalization of the above-mentioned local linear relations, called a rectangular lemma, which also generalizes the result of Huh,Nam and Yoo. Such a lemma can be applied to describe explicit formulas for $e$-positivity of a chromatic symmetric function $X_λ$ where $λ$ is contained in a rectangle. We also suggest some conjectural formulas for $e$-positivity when $λ$ is not contained in a rectangle by applying the rectangular lemma.

math.CO↗

arXiv2021

Back stable K-theory Schubert calculus

We study the back stable $K$-theory Schubert calculus of the infinite flag variety. We define back stable (double) Grothendieck polynomials and double $K$-Stanley functions and establish coproduct expansion formulae. Applying work of Weigandt, we extend our previous results on bumpless pipedreams from cohomology to $K$-theory. We study finiteness and positivity properties of the ring of back stable Grothendieck polynomials, and divided difference operators in $K$-homology.

math.CO↗

arXiv2021

Back stable Schubert calculus

We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of double and triple Stanley symmetric functions; 3) a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman-Greene insertion algorithm; 5) the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case; 6) equivariant Pieri rules for the homology of the infinite Grassmannian; 7) homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.

math.CO↗

arXiv2020

On the coproduct in affine Schubert calculus

The cohomology of the affine flag variety of a complex reductive group is a comodule over the cohomology of the affine Grassmannian. We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and K-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author.

math.CO↗

arXiv2019

Crystal structure on King tableaux and semistandard oscillating tableaux

In 1976, King defined certain tableaux model, called King tableaux in this paper, counting weight multiplicities of irreducible representation of the symplectic group $Sp(2m)$ for a given dominant weight. Since Kashiwara defined crystals, it is an open problem to provide a crystal structure on King tableaux. In this paper, we present crystal structures on King tableaux and semistandard oscillating tableaux. The semistandard oscillating tableaux naturally appear as $Q$-tableaux in the symplectic version of RSK algorithms. As an application, we discuss Littlewood-Richardson coefficients for $Sp(2m)$ in terms of semistandard oscillating tableaux.

math.CO↗

arXiv2018

Linear relations on LLT polynomials and their k-Schur positivity for k=2

LLT polynomials are $q$-analogues of product of Schur functions that are known to be Schur-positive by Grojnowski and Haiman. However, there is no known combinatorial formula for the coefficients in the Schur expansion. Finding such a formula also provides Schur positivity of Macdonald polynomials. On the other hand, Haiman and Hugland conjectured that LLT polynomials for skew partitions lying on $k$ adjacent diagonals are $k$-Schur positive, which is much stronger than Schur positivity. In this paper, we prove the conjecture for $k=2$ by analyzing unicellular LLT polynomials. We first present a linearity theorem for unicellular LLT polynomials for $k=2$. By analyzing linear relations between LLT polynomials with known results on LLT polynomials for rectangles, we provide the $2$-Schur positivity of the unicellular LLT polynomials as well as LLT polynomials appearing in Haiman-Hugland conjecture for $k=2$.

math.CO↗

arXiv2018

Combinatorial description of the cohomology of the affine flag variety

We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutative algebra generated by these operators is isomorphic to the cohomology of the affine flag variety. We show that the cohomology of the affine flag variety is product of the cohomology of an affine Grassmannian and a flag variety, which are generated by MN elements and Dunkl elements respectively. The Schubert classes in cohomology of the affine Grassmannian (resp. the flag variety) can be identified with affine Schur functions (resp. Schubert polynomials) in a quotient of the polynomial ring. Affine Schubert polynomials, polynomial representatives of the Schubert class in the cohomology of the affine flag variety, can be defined in the product of two quotient rings using the Bernstein-Gelfand-Gelfand operators interpreted as divided difference operators acting on the affine Fomin-Kirillov algebra. As for other applications, we obtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. We also define $k$-strong-ribbon tableaux from Murnaghan-Nakayama elements to provide a new formula of $k$-Schur functions. This formula gives the character table of the representation of the symmetric group whose Frobenius characteristic image is the $k$-Schur function.

math.CO↗

arXiv2017

Positivity of Cylindric skew Schur functions

Cylindric skew Schur functions, a generalization of skew Schur functions, are closely related to the famous problem finding a combinatorial formula for the 3-point Gromov-Witten invariants of Grassmannian. In this paper, we prove cylindric Schur positivity of the cylindric skew Schur functions, conjectured by McNamara. We also show that all coefficients appearing in the expansion are the same as $3$-point Gromov-Witten invariants. We start discussing properties of affine Stanley symmetric functions for general affine permutations and $321$-avoiding affine permutations, and explain how these functions are related to cylindric skew Schur functions. We also provide an effective algorithm to compute the expansion of the cylindric skew Schur functions in terms of the cylindric Schur functions, and the expansion of affine Stanley symmetric functions in terms of affine Schur functions.

math.CO↗

arXiv2016

Chern class of Schubert cells in the flag manifold and related algebras

We discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies the nonnegativity of the Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of the Bott-Samelson varieties. We also discuss the refined positivity conjectures of the Chern-Schwartz-MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in Fomin-Kirillov algebra.

math.CO↗

arXiv2015

Pieri rule for the affine flag variety

We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar's work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule.

math.CO↗