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Seungjai Lee

Publications and source records attributed to Seungjai Lee.

13 recordsLinked to original sources
arXiv2026

Zeta functions of solvable Lie algebras over finite fields -- with calculations in detail

Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also discuss the implications of our data, in particular in relation to the general theory of Lie algebras over finite fields and zeta functions of Lie algebras over commutative rings.

math.RA↗
arXiv2025

Cotype zeta functions enumerating subalgebras of $R$-algebras

We introduce and study subalgebra cotype zeta functions, multivariate zeta functions enumerating fixed-index subalgebras of $R$-algebras of a given cotype. This generalizes and unifies previous works on subalgebra zeta functions and cotype zeta functions of $R$-algebras. We prove the local functional equations for the generic Euler factors of these zeta functions, and give an explicit formula for the subalgebra cotype zeta function of a general $\mathbb{Z}$-Lie algebra $L$ of rank 3. We also give an asymptotic formula for the number of subalgebras $Λ$ of $L$ of index at most $X$ for which $L/Λ$ has rank at most, answering a question of Chinta, Kaplan, and Koplewitz. In particular, we show that unlike $\mathbb{Z}^3$, $\mathbb{Z}$-Lie algebras of rank 3 with additional multiplication structure exhibit different distribution of cocyclic subalgebras.

math.RA↗
arXiv2025

Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)

In physics, complex systems are often simplified into minimal, solvable models that retain only the core principles. In machine learning, layerwise linear models (e.g., linear neural networks) act as simplified representations of neural network dynamics. These models follow the dynamical feedback principle, which describes how layers mutually govern and amplify each other's evolution. This principle extends beyond the simplified models, successfully explaining a wide range of dynamical phenomena in deep neural networks, including neural collapse, emergence, lazy and rich regimes, and grokking. In this position paper, we call for the use of layerwise linear models retaining the core principles of neural dynamical phenomena to accelerate the science of deep learning.

stat.ML↗
arXiv2025

Zeta functions of quadratic lattices of a hyperbolic plane

In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane -- that is, a nondegenerate isotropic quadratic space of dimension $2$. We derive explicit formulas for the associated zeta functions and obtain a combinatorial way to compute them. Their analytic properties lead to the intriguing consequence that a large proportion of proper classes are one-lattice classes.

math.NT↗
arXiv2024

Zeta functions enumerating subforms of quadratic forms

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas linking this Dirichlet series with Dirichlet series counting ideal classes of the imaginary quadratic field associated with the quadratic form. Utilizing the result, we provide explicit formulas of the Dirichlet series for several lattices, including square lattice and hexagonal lattice. Moreover, we investigate some analytic properties of this Dirichlet series.

math.NT↗
arXiv2022

Uniformity in Higher class Free Lie algebras

Let $\mathfrak{f}_{c,2}$ denote a free class-$c$ Lie rings on $2$ generators. We investigate the zeta functions enumerating graded ideals in $\mathfrak{f}_{c,2}(\mathbb{F}_p)$ for $c\leq6$, prove that they are uniformly given by polynomials in $p$ for $c\leq5$ and not uniformly given by a polynomial in $p$ for $c=6$. We also show that the zeta functions enumerating one-step graded ideals $\mathfrak{f}_{c,2}(\mathbb{F}_p)$ is always given by a polynomial in $p$ for all $c$.

math.RA↗
arXiv2022

Zeta functions of $\mathbb{F}_p$-Lie algebras and finite $p$-groups

We study zeta functions enumerating subalgebras or ideals of Lie algebras over finite field of prime order $\mathbb{F}_p$. We first develop a general blueprint method for computing zeta functions of $\mathbb{F}_p$-Lie algebras, and demonstrate its practical applications in detail to obtain explicit formulas for various interesting new examples that are not covered in any known literature yet. For nilpotent cases this also provides zeta functions counting subgroups and normal subgroups of finite $p$-groups of exponent $p$ for almost all primes via the Lazard correspondence. We investigate their connections to the study of finite $p$-groups, and discuss what can be deduced from these finite Dirichlet polynomials.

math.RA↗
arXiv2021

Zeta functions of integral nilpotent quiver representations

We introduce and study multivariate zeta functions enumerating subrepresentations of integral quiver representations. For nilpotent such representations defined over number fields, we exhibit a homogeneity condition that we prove to be sufficient for local functional equations of the generic Euler factors of these zeta functions. This generalizes and unifies previous work on submodule zeta functions including, specifically, ideal zeta functions of nilpotent (Lie) rings and their graded analogues.

math.RA↗
arXiv2020

Normal zeta functions of small $\mathfrak{T}_2$-groups and their behaviour on residue classes

Let $G$ be a finitely generate nilpotent class-2 torsion-free group. We study how the zeta function enumerating normal subgroups of G varies on residue classes. In particular, we show that for small such $G$ of Hirsch length less than or equal to 7, the normal zeta functions are generically always rational functions on residue classes. We then show that there are examples of groups with Hirsch length 8 whose normal zeta function is not a rational function on residue classes. We observe the connection to Higman's PORC conjecture.

math.GR↗
arXiv2016

Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free $d$-generator Lie rings $\mathfrak{f}_{c,d}$ of nilpotency class $c$ for all $c\leq2$ and for $(c,d)\in\{(3,3),(3,2),(4,2)\}$. We apply our computations to obtain information about $\mathfrak{p}$-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.

math.RA↗