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Pham Huu Tiep

Publications and source records attributed to Pham Huu Tiep.

At least 19 recordsLinked to original sources
arXiv2026

Degrees of $p$-rational characters and normality of Sylow $p$-subgroups

Several refinements of (the normality part of) the celebrated Itô--Michler theorem were obtained during the last two decades, in which the condition of having $p'$-degree, for a fixed prime $p$, is imposed only on some subsets of complex irreducible characters of a finite group $G$. We prove further extensions of these results, where this condition is now imposed on the irreducible characters which lie above the principal character of a Sylow $p$-subgroup and are either $p$-rational, or strongly real when $p=2$.

math.GR↗
arXiv2026

Zeros of characters and orders of elements in finite groups

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked ``prime by prime". For these groups, we show that a strong form of Wilde's conjecture holds in many important cases, and for primes $p>5$ we are able to show the required statement for most classes of nearly simple groups. The few remaining cases, however, seem to require information on extensions of irreducible characters that are not available at the present time.

math.RT↗
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.

math.RT↗
arXiv2025

Irreducible restrictions of spin representations of symmetric and alternating groups

Let $\mathbb{F}$ be an algebraically closed field and $G$ be an almost quasi-simple group. An important problem in representation theory is to classify the subgroups $H<G$ and $\mathbb{F} G$-modules $L$ such that the restriction $L\downarrow_H$ is irreducible. This problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where $G$ is the Schur's double cover of alternating or symmetric group.

math.RT↗
arXiv2025

Principal blocks, irreducible restriction, fields and degrees

Several recent problems in the representation theory of finite groups require determining whether certain characters of almost simple groups belong to the principal block. Since the values of these characters are not yet known, we employ alternative group-theoretical techniques to address the "going down" case. This approach enables us to reduce the block version of well-known results by the third and fourth authors to a question about almost simple groups. Moreover, this suggests a Galois analogue of the height-zero-equal-degree conjecture of Malle and Navarro, which we formulate. However, the "going up" case of irreducible extensions of principal block characters remains unresolved.

math.RT↗
arXiv2025

A reduction theorem for the Feit conjecture

We prove that if all the simple groups involved in a finite group $G$ satisfy the `inductive Feit condition', then Walter Feit's conjecture from 1980 holds for $G$. In particular, this would solve Brauer's Problem 41 from 1963 in the affirmative. This inductive Feit condition implies that some features of all the irreducible characters of finite groups can be found locally.

math.RT↗
arXiv2025

Character estimates for finite classical groups and the asymptotic Thompson Conjecture

If $G$ is a finite classical group, linear or unitary in any characteristic, and orthogonal in odd characteristic, we give an approximate formula for $χ(g)$ in which the error term is much smaller than the estimate, when $g\in G$ is an element with large centralizer and $χ\in \mathrm{Irr}(G)$ is an irreducible character of low degree. As an application, we prove Thompson's conjecture for all sufficiently large finite simple groups: each such group contains a conjugacy class whose square is the whole group.

math.GR↗
arXiv2024

Non-congruence presentations of finite simple groups

We prove two results on some special generators of finite simple groups and use them to prove that every non-abelian finite simple group $S$ admits a non-congruence presentation (as conjectured in [CLT24]), and that if $S$ has a non-trivial Schur multiplier, then it admits a smooth cover (as conjectured in [CFLZ]).

math.GR↗
arXiv2024

Tensor product Markov chains and Weil representations

We obtain sharp bounds on the convergence rate of Markov chains on irreducible representations of finite general linear, unitary, and symplectic groups (in both odd and even characteristic) given by tensoring with Weil representations.

math.RT↗
arXiv2024

Representation growth of Fuchsian groups and modular forms

Let $Γ$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $Γ$. Then $|\mathrm{Hom}(Γ,\mathrm{GL}_n(q))|\sim c_{q,n} q^{(1-χ(Γ))n^2}$, where $c_{q,n}$ is periodic in $n$. As a function of $q$, $c_{q,n}$ can be expressed as a Puiseux series in $1/q$ whose coefficients are periodic in $n$ and $q$. Moreover, this series is essentially the $q$-expansion of a meromorphic modular form of half-integral weight.

math.GR↗
arXiv2024

Brauer's Height Zero Conjecture

We complete the proof of Brauer's Height Zero Conjecture from 1955 by establishing the open implication for all odd primes.

math.RT↗
arXiv2024

An extension of Gow's theorem

We extend Gow's theorem on products of semisimple regular conjugacy classes to finite groups whose generalized Fitting subgroup is Z(G)S where S is a quasisimple group of Lie type in characteristic p and Z(G) has order prime to p.

math.GR↗
arXiv2024

Uniform Character Bounds for Finite Classical Groups

For every finite quasisimple group of Lie type $G$, every irreducible character $χ$ of $G$, and every element $g$ of $G$, we give an exponential upper bound for the character ratio $|χ(g)|/χ(1)$ with exponent linear in $\log_{|G|} |g^G|$, or, equivalently, in the ratio of the support of $g$ to the rank of $G$. We give several applications, including a proof of Thompson's conjecture for all sufficiently large simple symplectic groups, orthogonal groups in characteristic $2$, and some other infinite families of orthogonal and unitary groups

math.RT↗
arXiv2023

Finite simple characteristic quotients of the free group of rank 2

In this paper we describe how to explicitly construct infinitely many finite simple groups as characteristic quotients of the rank 2 free group $F_2$. This shows that a "baby" version of the Wiegold conjecture fails for $F_2$, and provides counterexamples to two conjectures in the theory of noncongruence subgroups of $\text{SL}_2(\mathbb{Z})$. Our main result explicitly produces, for every prime power $q\ge 7$, the groups $\text{SL}_3(\mathbb{F}_q)$ and $\text{SU}_3(\mathbb{F}_q)$ as characteristic quotients of $F_2$. Our strategy is to study specializations of the Burau representation for the braid group $B_4$, exploiting an exceptional relationship between $F_2$ and $B_4$ first observed by Dyer, Formanek, and Grossman. Weisfeiler's strong approximation theorem guarantees that our specializations are surjective for infinitely many primes, but it is not effective. To make our result effective, we give another proof of surjectivity via a careful analysis of the maximal subgroup structures of $\text{SL}_3(\mathbb{F}_q)$ and $\text{SU}_3(\mathbb{F}_q)$. We also show that our examples of $\text{PSL}_3(\mathbb{F}_q)$ and $\text{PSU}_3(\mathbb{F}_q)$ are minimal in the sense that no group of the form $\text{PSL}_2(\mathbb{F}_q)$ is a characteristic quotient of $F_2$.

math.GR↗
arXiv2023

Squares of Conjugacy Classes in Alternating Groups

We extend to alternating groups $A_n$ several results about symmetric groups asserting that under various conditions on a conjugacy class, or more generally, a normal subset, $C$ of $S_n$, we have $C^2 \supseteq A_n\setminus\{1\}$

math.GR↗
arXiv2023

The Field of Values of the Height Zero Characters

We determine what are the fields of values of the irreducible $p$-height zero characters of all finite groups for $p=2$; we conjecture what they should be for odd primes, and reduce this statement to a problem on blocks of quasi-simple groups.

math.GR↗