arXiv subjects

Chul-hee Lee

Publications and source records attributed to Chul-hee Lee.

14 recordsLinked to original sources

arXiv2026

Lean-GAP: A Dataset of Formalized Graduate Algebra Problems

We present Lean-GAP (Lean-Graduate Agebra Problems), 430 formalized graduate-level algebra problems from the textbook Abstract Algebra by Dummit and Foote. We develop a scalable pipeline consisting of PDF-to-LaTeX preprocessing, autoformalization into Lean 4, and verification of informal-formal correspondence. While the preprocessing and autoformalization stages can be largely automated, we find that verification remains the most subtle and labor-intensive component, requiring careful human oversight. Our contributions include (i) the construction of a structured dataset of formalized exercises, (ii) a systematic methodology for formalizing textbook mathematics, and (iii) an analysis of recurring challenges in the formalization process. We also compare the performance of different autoformalization models and highlight key bottlenecks in translating informal statements into formal language.

cs.LO↗

arXiv2024

Verlinde rings and cluster algebras arising from quantum affine algebras

We formulate a positivity conjecture relating the Verlinde ring associated with an untwisted affine Lie algebra at a positive integer level and a subcategory of finite-dimensional representations over the corresponding quantum affine algebra with a cluster algebra structure. Specifically, we consider a ring homomorphism from the Grothendieck ring of this representation category to the Verlinde ring and conjecture that every object in the category has a positive image under this map. We prove this conjecture in certain cases where the underlying simple Lie algebra is simply-laced with level 2 or of type $A_1$ at an arbitrary level. The proof employs the close connection between this category and cluster algebras of finite cluster type. As further evidence for the conjecture, we show that for any level, all objects have positive quantum dimensions under the assumption that some Kirillov-Reshetikhin modules have positive quantum dimensions.

math.RT↗

arXiv2023

Spectrum of the Laplacian on the Fricke-Macbeath surface

The Fricke-Macbeath surface is the unique Hurwitz surface of genus 7 with 504 conformal automorphisms. In this paper, we prove that the first eigenvalue of the Laplacian on the Fricke-Macbeath surface has a sevenfold multiplicity and contained in the interval $[1.23, 1.26]$. Further, we numerically identify the 7-dimensional representation of its automorphism group corresponding to the eigenspace associated to the first eigenvalue. We also determine the Dirichlet domain centered at 0 for a Fuchsian group that uniformizes the Fricke-Macbeath surface, identifying the algebraic coordinates for its vertices.

math.RT↗

arXiv2020

An Elliptic Hypergeometric Function Approach to Branching Rules

We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures exhibiting a novel type of vanishing behaviour involving partitions with empty 2-cores.

math.CO↗

arXiv2020

Computation of Gross-Keating invariants

The Gross-Keating invariant of a half-integral matrix over a $p$-adic integer ring is a fundamental concept in the study of quadratic forms, and has important applications to Siegel modular forms and arithmetic geometry. We introduce the Mathematica package computeGK, a computer program for calculating the Gross-Keating invariant and the Siegel series of a half-integral matrix over $\mathbb{Z}_p$, as well as other related quantities. As a by-product, we obtain a table of the arithmetic intersection numbers related to the classical modular polynomials using the explicit formula of Gross and Keating.

math.NT↗

arXiv2019

Minor corrections to Nipp's tables of quaternary quadratic forms

We make some corrections to Nipp's tables of positive definite integral quaternary quadratic forms. They only affect $p$-adic densities and $p$-adic Jordan splittings in the Appendix of Nipp's book. There are no errors found in Nipp's list of quaternary forms for each genus.

math.NT↗

arXiv2019

On Polyhedral Formulas for Kirillov-Reshetikhin Modules

We propose a method to prove a polyhedral branching formula for Kirillov-Reshetikhin (KR) modules over an untwisted quantum affine algebra. When the underlying simple Lie algebra is of exceptional type, such a formula remains conjectural in many cases. Using a linear recurrence relation satisfied by the characters of KR modules, we convert the verification of a polyhedral formula into an identity between two rational functions of a single variable with only simple poles at known locations. It is then sufficient to compare the residues at those poles, which are explicitly computable quantities. By applying this strategy, we obtain new, computer-assisted and easily verifiable proofs of known polyhedral formulas in types $F_4$ and $G_2$ within a uniform framework.

math.RT↗

arXiv2018

Product formula for the limits of normalized characters of Kirillov-Reshetikhin modules

The normalized characters of Kirillov-Reshetikhin modules over a quantum affine algebra have a limit as a formal power series. Mukhin and Young found a conjectural product formula for this limit, which resembles the Weyl denominator formula. We prove this formula except for some cases in type $E_8$ by employing an algebraic relation among these limits, which is a variant of $Q\widetilde{Q}$-relations.

math.QA↗

arXiv2017

Linear recurrence relations in $Q$-systems via lattice points in polyhedra

We prove that the sequence of the characters of the Kirillov-Reshetikhin (KR) modules $W_{m}^{(a)}, m\in \mathbb{Z}_{m\geq 0}$ associated to a node $a$ of the Dynkin diagram of a complex simple Lie algebra $\mathfrak{g}$ satisfies a linear recurrence relation except for some cases in types $E_7$ and $E_8$. To this end we use the $Q$-system and the existing lattice point summation formula for the decomposition of KR modules, known as domino removal rules when $\mathfrak{g}$ is of classical type. As an application, we show how to reduce some unproven lattice point summation formulas in exceptional types to finite problems in linear algebra and also give a new proof of them in type $G_2$, which is the only completely proven case when KR modules have an irreducible summand with multiplicity greater than 1. We also apply the recurrence to prove that the function $\dim W_{m}^{(a)}$ is a quasipolynomial in $m$ and establish its properties. We conjecture that there exists a rational polytope such that its Ehrhart quasipolynomial in $m$ is $\dim W_{m}^{(a)}$ and the lattice points of its $m$-th dilate carry the same crystal structure as the crystal associated with $W_{m}^{(a)}$.

math.RT↗

arXiv2017

Positivity and periodicity of $Q$-systems in the WZW fusion ring

We study properties of solutions of $Q$-systems in the WZW fusion ring obtained by the Kirillov-Reshetikhin modules. We make a conjecture about their positivity and periodicity and give a proof of it in some cases. We also construct a positive solution of the level $k$ restricted $Q$-system of classical types in the fusion rings. As an application, we prove some conjectures of Kirillov and Kuniba-Nakanishi-Suzuki on the level $k$ restricted $Q$-systems.

math.QA↗

arXiv2015

Linear recurrence relations in $Q$-systems and difference $L$-operators

We study linear recurrence relations in the character solutions of $Q$-systems obtained from the Kirillov-Reshetikhin modules. We explain how known results on difference $L$-operators lead to a uniform construction of linear recurrences in many examples, and formulate certain conjectural properties predicted in general by this construcion.

math.QA↗

arXiv2013

A Proof of the KNS conjecture : $D_r$ case

We prove the Kuniba-Nakanishi-Suzuki (KNS) conjecture concerning the quantum dimension solution of the $Q$-system of type $D_r$ obtained by a certain specialization of classical characters of the Kirillov-Reshetikhin modules. To this end, we use various symmetries of quantum dimensions. As a result, we obtain an explicit formula for the positive solution of the level $k$ restricted $Q$-system of type $D_r$ which plays an important role in dilogarithm identities for conformal field theories.

math.QA↗

arXiv2013

Nahm's conjecture and Y-systems

Nahm's conjecture relates $q$-hypergeometric modular functions to torsion elements in the Bloch group. An interesting class of such functions can be (conjecturally) obtained from a pair $(X,X')$ of diagrams, each of which is either a Dynkin diagram of type $ADE$ or a diagram of type $T$. Using properties of Y-systems, we prove that for a matrix of the form $A=\mathcal{C}(X)\otimes \mathcal{C}(X')^{-1}$ where $\mathcal{C}(X)$ and $\mathcal{C}(X')$ are the corresponding Cartan matrices, every solution of the equation $\mathbf{x}=(1-\mathbf{x})^A$ gives rise to a torsion element of the Bloch group.

math.QA↗

arXiv2010

A Note on Nahm's Conjecture in Rank 2 Case

The aim of this paper is to get a complete list of positive definite symmetric matrices with integer entries $\a&b\b&d\$ such that all complex solutions to the system of equations $1-x_1=x_1^ax_2^b\ 1-x_2=x_1^bx_2^d$ are real. This result is related to Nahm's conjecture in rank 2 case.

math-ph↗