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Carolina Vallejo

Publications and source records attributed to Carolina Vallejo.

15 recordsLinked to original sources
arXiv2026

The McKay conjecture with coprime group automorphisms and the Okuyama-Wajima argument

Let $A$ and $G$ be finite groups. Suppose that $A$ acts coprimely on $G$ stabilizing $N\triangleleft G$. Let $θ\in \rm{Irr}(N)$ be $A$-invariant. We prove that the number of $A$-invariant irreducible characters of $G$ that lie over $θ$ can be counted in terms of the $(A, θ)$-good conjugacy classes of $G_θ/N$, where $G_θ$ is the inertia subgroup of $θ$ in $G$. This result generalizes a classic result of Gallagher and can be used to prove the following: if $P$ is an $A$-invariant Sylow $p$-subgroup of $G$ and $G$ is $p$-solvable, then there exists an $A$-equivariant (McKay) bijection between the irreducible characters of degree prime to $p$ of $G$ and those of $\textbf{N}_G(P)$. While this is a consequence of a recent result of D. Rossi, our approach here is independent of Rossi's and follows the original idea of the proof of the McKay conjecture for $p$-solvable groups. In particular, we rely on the so-called Okuyama-Wajima argument to deal with characters above Glauberman correspondents.

math.RT↗
arXiv2026

Cyclotomic character fields and sets of primes

Let $π=\{ 2, q \}$ where $q$ is an odd prime. Let $G$ be a finite group of order divisible by a prime $p \in π$. We show that the principal $p$-block of $G$ contains a nontrivial irreducible character of degree not divisible by $2$ nor $q$ and with field of values contained in the $q$th cyclotomic extension. This statement simultaneously provides a principal block version of results of Navarro--Tiep and Giannelli--Hung--Schaeffer Fry--Vallejo.

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arXiv2024

The continuity of $p$-rationality of characters and the principal block

We study rationality properties of irreducible characters of finite groups. We show that the continuity of $2$-rationality is a phenomenon that can be detected in the principal $2$-block, thus refining a recent result of N. N. Hung. We also propose a conjectural group theoretical criterion for the continuity gap at level $1$ for all primes

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arXiv2024

On almost $p$-rational characters in principal blocks

Let p be a prime. In this paper we provide a lower bound for the number of almost p-rational characters of degree coprime to p in the principal p-block of a finite group of order divisible by p. We further describe the p-local structure of the groups for which the above-mentioned bound is sharp.

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arXiv2024

Characters and Sylow $3$-subgroup abelianization

We characterize when a finite group G possesses a Sylow 3-subgroup P with abelianization of order 9 in terms of the number of height zero characters lying in the principal 3-block of G, settling a conjecture put forward by Navarro, Sambale, and Tiep in 2018. Along the way, we show that a recent result by Laradji on the number of character of height zero in a block that lie above a given character of some normal subgroup holds, without any hypothesis on the group for blocks of maximal defect.

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↗
arXiv2021

Principal blocks with 5 irreducible characters

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

math.GR↗
arXiv2021

Sylow branching coefficients and a conjecture of Malle and Navarro

We prove that a finite group $G$ has a normal Sylow $p$-subgroup $P$ if, and only if, every irreducible character of $G$ appearing in the permutation character $({\bf 1}_P)^G$ with multiplicity coprime to $p$ has degree coprime to $p$. This confirms a prediction by Malle and Navarro from 2012. Our proof of the above result depends on a reduction to simple groups and ultimately on a combinatorial analysis of the properties of Sylow branching coefficients for symmetric groups.

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arXiv2020

Galois action on the principal block and cyclic Sylow subgroups

We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.

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arXiv2019

Character correspondeces in solvable groups with a self-normalizing Sylow subgroup

In 1973, I. M. Isaacs described a correspondence between characters of degree not divisible by a fixed prime $p$ of a finite solvable group $G$ and those of the normalizer of Sylow $p$-subgroup of $G$, whenever the index of the normalizer in $G$ is odd. This correspondence is natural in the sense that an algorithm is provided to compute it, and the result of the application of the algorithm does not depend on choices made. Later on, for $p$-solvable groups with self-normalizing Sylow $p$-subgroup, G. Navarro showed that every irreducible character of degree not divisible by $p$ has a unique linear constituent when restricted to a Sylow $p$-subgroup. Furthermore, the process of choosing the unique linear constituent of the restriction defines a bijection. Navarro's bijection is obviously natural in the sense described above. We show that these two correspondences are the same under the intersection of the hypotheses.

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arXiv2019

A reduction theorem for the Galois-McKay conjecture

We generalize the theory of ordering character triples, developed by Navarro and Späth, by taking into account the action of Galois automorphisms on characters. This new technique, together with previous results of Ladisch and Turull, allows us to reduce the Galois--McKay conjecture to a question about simple groups.

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arXiv2019

Characters of $π'$-degree

Let $G$ be a finite group and let $π$ be a set of primes. Write $\mathrm{Irr}_{π'}(G)$ for the set of irreducible characters of degree not divisible by any prime in $π$. We show that if $π$ contains at most two prime numbers and the only element in $\mathrm{Irr}_{π'}(G)$ is the principal character, then $G=1$.

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arXiv2018

Brauer correspondent blocks with one simple module

One of the main problems in representation theory is to understand the exact relationship between Brauer corresponding blocks of finite groups. The case where the local correspondent has a unique simple module seems key. We characterize this situation for the principal p-blocks where p is odd.

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arXiv2017

Even degree characters in principal blocks

We characterise finite groups such that for an odd prime $p$ all the irreducible characters in its principal $p$-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by $p$ unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\neq 7$ or if $M_{22}$ is not a composition factor of a group $G$, then the condition above is equivalent to $G/O_{p'}(G)$ having odd order.

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