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Nguyen N. Hung

Publications and source records attributed to Nguyen N. Hung.

13 recordsLinked to original sources
arXiv2026

On the $p$-rationality of Deligne--Lusztig characters

Among finite simple groups, character values of alternating and sporadic groups have relatively low irrationality at any prime $p$, whereas those of simple groups of Lie type can have arbitrarily high $p$-irrationality. We provide concrete evidence supporting this phenomenon. In particular, we show that if $χ:=R_{\mathbf{T}}^{\mathbf{G}}(θ)$ is a Deligne--Lusztig character of a finite reductive group $\mathbf{G}^F$, with $θ$ an irreducible character of a maximal torus $\mathbf{T}^F$, and if $χ$ has degree prime to $p$, then the so-called $p$-rationality level of $χ$ coincides precisely with that of $θ$. We present further evidence suggesting that Lusztig induction preserves $p$-rationality for characters of $p'$-degree.

math.RT↗
arXiv2026

Problems on the conductor of finite group characters

This paper reviews recent results and open problems on the conductor of finite group characters, highlighting their connections to one another and to broader topics in the representation theory of finite groups.

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arXiv2026

Character values and conductors of low-rank groups of Lie type

Let $χ$ be a complex irreducible character of a finite group $G$. The conductor of $χ$, denoted $c(χ)$, is the smallest positive integer $n$ such that $χ(x)\in \mathbb{Q}(\exp({2πi/n}))$ for all $x\in G$. We show that for certain rank $1$ finite groups of Lie type, the conductor $c(χ)$ is realized at a single group element; that is, there exists $g\in G$ such that $c(χ)=c(χ(g))$. In some quasisimple cases, we further prove that the field of values \(\mathbb{Q}(χ)\) is generated by a single value. This phenomenon, which is related to a well-known conjecture of W.~Feit, was recently observed by Boltje \emph{et al.} in their reduction of the conjecture to finite simple groups. Our approach uses techniques from algebraic number theory together with the known character tables of these groups.

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arXiv2026

Sum of the squares of the $p'$-character degrees

We study the sum of the squares of the irreducible character degrees not divisible by some prime $p$, and its relationship with the the corresponding quantity in a $p$-Sylow normalizer. This leads to study a recent conjecture by E. Giannelli, which we prove for $p=2$ and in some other cases.

math.GR↗
arXiv2026

Primes and The Field of Values of Characters

Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In this work, we extend this conjecture to characters whose degrees are divisible by arbitrary powers of $p$, and we provide some evidence supporting its validity.

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arXiv2026

Hall $π$-subgroups and characters of $π'$-degree

We study the relationship between the existence of Hall $π$-subgroups and that of irreducible characters of $π'$-degree with prescribed fields of values in finite groups. This work extends a result of Navarro and Tiep from a single odd prime to multiple odd primes.

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arXiv2025

Wreath products and the non-coprime $k(GV)$ problem

Let $G = X \wr H$ be the wreath product of a nontrivial finite group $X$ with $k$ conjugacy classes and a transitive permutation group $H$ of degree $n$ acting on the set of $n$ direct factors of $X^n$. If $H$ is semiprimitive, then $k(G) \leq k^n$ for every sufficiently large $n$ or $k$. This result solves a case of the non-coprime $k(GV)$ problem and provides an affirmative answer to a question of Garzoni and Gill for semiprimitive permutation groups. The proof does not require the classification of finite simple groups.

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arXiv2024

The $p$-rationality of height-zero characters

We propose and present evidence for a conjectural global-local phenomenon concerning the $p$-rationality of $p$-height-zero characters. Specifically, if $χ$ is a height-zero character of a finite group $G$ and $D$ is a defect group of the $p$-block of $G$ containing $χ$, then the $p$-rationality of $χ$ can be captured inside the normalizer $N_G(D)$.

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arXiv2024

The codegree isomorphism problem for finite simple groups II

Let $H$ be a nonabelian finite simple group. Huppert's conjecture asserts that if $G$ is a finite group with the same set of complex character degrees as $H$, then $G\cong H\times A$ for some abelian group $A$. Over the past two decades, several specific cases of this conjecture have been addressed. Recently, attention has shifted to the analogous conjecture for character codegrees: if $G$ has the same set of character codegrees as $H$, then $G\cong H$. Unfortunately, both problems have primarily been examined on a case-by-case basis. In this paper and the companion [HM22], we present a more unified approach to the codegree conjecture and confirm it for several families of simple groups.

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arXiv2023

Common zeros of irreducible characters

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $S_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group $G$, with non-linear irreducible characters of $G$ as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by 3. Lastly, the result for $S_n$ is applied to prove the non-equivalence of the metrics on permutations induced from faithful irreducible characters of the group.

math.GR↗
arXiv2023

The codegree isomorphism problem for finite simple groups

We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups and only depends on the classification by means of Artin-Tits' simple order theorem.

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arXiv2023

The continuity of $p$-rationality and a lower bound for $p'$-degree characters of finite groups

Let $p$ be a prime and $G$ a finite group. We propose a strong bound for the number of $p'$-degree irreducible characters of $G$ in terms of the commutator factor group of a Sylow $p$-subgroup of $G$. The bound arises from a recent conjecture of Navarro and Tiep [NT21] on fields of character values and a phenomenon called the continuity of $p$-rationality level of $p'$-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture [Nav04]. We achieve both the bound and the continuity property for $p=2$.

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