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Gabriel Navarro

Publications and source records attributed to Gabriel Navarro.

At least 19 recordsLinked to original sources
arXiv2026

On The Eaton-Moret\'o Conjecture for Principal Blocks of Finite Groups

Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $\chi \in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<\chi(1)_p\le m(P)$, giving one inequality of the Eaton-Moret\'o conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moret\'o conjecture for principal blocks.

math.RT↗
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

An order-oriented approach to scoring hesitant fuzzy elements

Traditional scoring approaches on hesitant fuzzy sets often lack a formal base in order theory. This paper proposes a unified framework, where each score is explicitly defined with respect to a given order. This order-oriented perspective enables more flexible and coherent scoring mechanisms. We examine several classical orders on hesitant fuzzy elements, that is, nonempty subsets in [0,1], and show that, contrary to prior claims, they do not induce lattice structures. In contrast, we prove that the scores defined with respect to the symmetric order satisfy key normative criteria for scoring functions, including strong monotonicity with respect to unions and the G\"ardenfors condition. Following this analysis, we introduce a class of functions, called dominance functions, for ranking hesitant fuzzy elements. They aim to compare hesitant fuzzy elements relative to control sets incorporating minimum acceptability thresholds. Two concrete examples of dominance functions for finite sets are provided: the discrete dominance function and the relative dominance function. We show that these can be employed to construct fuzzy preference relations on typical hesitant fuzzy sets and support group decision-making.

cs.AI↗
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

Adjoint and duality for rank-metric codes in a skew polynomial framework

Skew polynomial rings provide a fundamental example of noncommutative principal ideal domains. Special quotients of these rings yield matrix algebras that play a central role in the theory of rank-metric codes. Recent breakthroughs have shown that specific subsets of these quotients produce the largest known families of maximum rank distance (MRD) codes. In this work, we present a systematic study of transposition and duality operations within quotients of skew polynomial rings. We develop explicit skew-polynomial descriptions of the transpose and dual code constructions, enabling us to determine the adjoint and dual codes associated with the MRD code families recently introduced by Sheekey et al. Building on these results, we compute the nuclear parameters of these codes, and prove that, for a new infinite set of parameters, many of these MRD codes are inequivalent to previously known constructions in the literature.

cs.IT↗
arXiv2025

A counterexample to a conjecture of A. R. Miller

Let $\chi$ be an irreducible character of a finite group $G$. A. R. Miller conjectured that the proportion of elements $g\in G$ such that $\chi(g)$ is zero or a root of unity is at least 1/2. We construct a character of a perfect group of order 69120 such that this proportion is 511/1152.

math.RT↗
arXiv2025

Feit's conjecture, the canonical Brauer induction formula, and Adams operations

This paper is motivated by a strong version of Feit's conjecture, first formulated by the authors in joint work with A. Kleshchev and P. H. Tiep in 2025, concerning the conductor $c(\chi)$ of an irreducible character $\chi$ of a finite group $G$. We connect the conjecture with the following construction: For any positive integer $n$ dividing the exponent of $G$ and for any character $\chi$ of $G$, we introduce an integer-valued invariant $S(G,\chi,n)$ which can be defined as the sum of certain coefficients of the canonical Brauer induction formula of $\chi$, or alternatively as the multiplicity of the trivial character in a specified integral linear combination of Adams operations of $\chi$. We show two facts about this invariant. The first seems of independent interest (apart from Feit's conjecture): $S(G,\chi,n)$ is always non-negative, and it is positive if and only if a representation affording $\chi$ involves an eigenvalue of order $n$. Secondly, the strong version of Feit's conjecture holds for an irreducible character $\chi$ if and only if $S(G,\chi, c(\chi))>0$.

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

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↗
arXiv2025

Brain-Inspired Quantum Neural Architectures for Pattern Recognition: Integrating QSNN and QLSTM

Recent advances in the fields of deep learning and quantum computing have paved the way for innovative developments in artificial intelligence. In this manuscript, we leverage these cutting-edge technologies to introduce a novel model that emulates the intricate functioning of the human brain, designed specifically for the detection of anomalies such as fraud in credit card transactions. Leveraging the synergies of Quantum Spiking Neural Networks (QSNN) and Quantum Long Short-Term Memory (QLSTM) architectures, our approach is developed in two distinct stages, closely mirroring the information processing mechanisms found in the brain's sensory and memory systems. In the initial stage, similar to the brain's hypothalamus, we extract low-level information from the data, emulating sensory data processing patterns. In the subsequent stage, resembling the hippocampus, we process this information at a higher level, capturing and memorizing correlated patterns. We will compare this model with other quantum models such as Quantum Neural Networks among others and their corresponding classical models.

cs.ET↗
arXiv2025

On the number and sizes of double cosets of Sylow subgroups of the symmetric group

Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_n$. We investigate the number and sizes of the $P_n\setminus S_n\ /\ P_n$ double cosets, showing that most double cosets have maximal size when $p$ is odd, or equivalently, that $P_n\cap P_n^x=1$ for most $x\in S_n$ when $n$ is large. We also find that all possible sizes of such double cosets occur, modulo a list of small exceptions.

math.GR↗
arXiv2024

Characters, Hall subgroups, and Normal Complements

If $H$ is a Hall subgroup of a finite group $G$, it was proven in 1989 using the classification of finite simple groups that all the irreducible complex characters of $H$ extend to $G$ if and only if there is $N\trianglelefteq G$ such that $HN=G$ and $H\cap N=1$. In this note we offer a modular version of this result under more general hypothesis.

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↗