Searcharxiv⌕ Search

arXiv subjects

Nadia Romero

Publications and source records attributed to Nadia Romero.

16 recordsLinked to original sources
arXiv2026

On the separability of some Green biset functors

We show that the Green biset functor $R_{\mathbb{C}}$ of complex characters over $\mathbb{Z}$, is not separable, i.e. it is not projective as a bimodule over itself. Also, we show that $RB_G$, the Burnside biset functor shifted by a finite group $G$, over a commutative ring $R$, is separable if and only if $|G|$ is invertible in $R$. Finally, to address the question of the relation between functors and their evaluations, we show that the Burnside $R$-algebra $RB(G)$ is separable if and only if $|G|$ is invertible in $R$.

math.GR↗
arXiv2025

Koszul resolution for linear monoidal functors

We introduce regular sequences and associated Koszul resolutions for monoids in the category of functors over an essentially small linear symmetric monoidal category. Next we define polynomials over such monoids. We compute the Hochschild cohomology functors and prove a relative analogue of Hilbert's syzygy theorem for polynomials over tensor idempotent commutative monoids.

math.CT↗
arXiv2023

Hochschild cohomology for functors on linear symmetric monoidal categories

Let $R$ be a commutative ring with unit. We develop a Hochschild cohomology theory in the category $\mathcal{F}$ of linear functors defined from an essentially small symmetric monoidal category enriched in $R$-Mod, to $R$-Mod. The category $\mathcal{F}$ is known to be symmetric monoidal too, so one can consider monoids in $\mathcal{F}$ and modules over these monoids, which allows for the possibility of a Hochschild cohomology theory. The emphasis of the article is in considering natural hom constructions appearing in this context. These homs, together with the abelian structure of $\mathcal{F}$ lead to nice definitions and provide effective tools to prove the main properties and results of the classical Hochschild cohomology theory.

math.RT↗
arXiv2022

On the essential algebra of the shifted Burnside biset functor

We describe the essential algebra, $\widehat{kB_T}(G)$, of the Burnside biset functor shifted by a group $T$, at a group $G$, in two cases. First, when $G$ and $T$ are both finite abelian groups and $k$ is a field of characteristic $0$. In this case, $\widehat{kB_T}(G)$ is isomorphic to a quotient of the shifted star algebra, which is defined in terms of the subgroups of $G\times G\times T$. The second case is when $G$ and $T$ are any finite groups satisfying $(|G|, |T|)=1$ and $k$ is a commutative unitary ring. In this case, $\widehat{kB_T}(G)$ is isomorphic to a semidirect product of $Out(G)$ and $kB^{Z(G)}(T)$, the monomial Burnside ring of $T$ with coefficients in $Z(G)$. The aim of the article is to consider the natural set of generators of $\widehat{kB_T}(G)$ coming from the transitive elements in $kB_T(G\times G)$ and explore some cases in which it is possible to give a basis for $\widehat{kB_T}(G)$ in this set.

math.RT↗
arXiv2022

Green fields

We introduce {\em Green fields}, as commutative Green biset functors with no non-trivial ideals. We state some of their properties and give examples of known Green biset functors which are Green fields. Among the properties, we prove some criterions ensuring that a Green field is semisimple. Finally, we describe a type of Green field for which its category of modules is equivalent to a category of vector spaces over a field.

math.CT↗
arXiv2022

Deflation and tensor induction on the Frobenius-Wielandt morphism

We explore conditions for the Frobenius-Wielandt morphism to commute with the operations of deflation and tensor induction on the Burnside ring. In doing this, we review the commutativity with induction. The techniques used for induction and tensor induction no longer work for deflation, so in this case we make use of tools coming from the theory of biset functors.

math.GR↗
arXiv2022

The center of a Green biset functor

For a Green biset functor $A$, we define the commutant and the center of $A$ and we study some of their properties and their relationship. This leads in particular to the main application of these constructions: the possibility of splitting the category of $A$-modules as a direct product of smaller abelian categories. We give explicit examples of such decompositions for some classical shifted representation functors. These constructions are inspired by similar ones for Mackey functors for a fixed finite group.

math.GR↗
arXiv2019

The additive completion of the biset category

Let $R$ be a commutative unital ring. We construct a category $\mathcal{C}_R$ of fractions $X/G$, where $G$ is a finite group and $X$ is a finite $G$-set, and with morphisms given by $R$-linear combinations of spans of bisets. This category is an additive, symmetric monoidal and self-dual category, with a Krull-Schmidt decomposition for objects. We show that $\mathcal{C}_R$ is equivalent to the additive completion of the biset category and that the category of biset functors over $R$ is equivalent to the category of $R$-linear functors from $\mathcal{C}_R$ to $R$-Mod. We also show that the restriction of one of these functors to a certain subcategory of $\mathcal{C}_R$ is a fused Mackey functor.

math.CT↗
arXiv2018

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P , the subgroup Cl\_1 (ZP) of SK\_1 (ZP), in terms of a genetic basis of P. We also introduce a deflation map Cl\_1 (ZP) $\rightarrow$ Cl\_1 (Z(P/N)) , for a normal subgroup N of P , and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of SK\_1 (ZP), when P is an elementary abelian p-group.

math.GR↗
arXiv2015

Computing Whitehead groups using genetic bases

We combine results about Whitehead groups of finite groups with results about genetic bases of finite $p$-groups to compute the Whitehead groups of some metacyclic $p$-groups. Let $C_{p^n}$ denote a cyclic group of order $p^n$ for $p$ an odd prime and $n$ a positive integer. We present a conjecture for the torsion part of the Whitehead group of $C_{p^n}\times C_{p^n}$ and we show that torsion part of the Whitehead group of the semidirect product of $C_{p^{n-1}}$ by $C_p$ is isomorphic to $C_p^{(n-2)(p-1)}$. The techniques we use can be applied to any abelian $p$-group and to many other $p$-groups for $p$ odd.

math.GR↗
arXiv2015

Cyclic cellularity and active sums

Let $G$ be a group and let $\mathcal{F}$ be a family of subgroups of $G$ closed under conjugation. For a positive integer $n$, let $C_n$ denote a cyclic group of order $n$. We show that if there exists an integer $n$ such that every group in $\mathcal{F}$ is $C_n$-cellular and has finite exponent diving $n$, then the active sum $S$ of $\mathcal{F}$ is $C_n$-cellular. We obtain a couple of interesting consequences of this result, using results about cellularity. Finally, we give different proofs of the facts that Coxeter groups are $C_2$-cellular and that many groups of the form $\mathrm{SL}(n,\,q)$ for $n\geq3$ are $C_3$-cellular.

math.GR↗
arXiv2014

Finite metacyclic groups as active sums of cyclic subgroups

The notion of active sum provides an analogue for groups of that of direct sum for abelian groups. One natural question then is which groups are the active sum of cyclic subgroups. Many groups have been found to give a positive answer to this question, while the case of finite metacyclic groups remained unknown. In this note we show that every finite metacyclic group can be recovered as the active sum of a discrete family of cyclic subgroups.

math.GR↗
arXiv2013

On fibred biset functors with fibres of order prime and four

This note has two purposes: First, to present a counterexample to a conjecture parametrizing the simple modules over Green biset functors, appearing in an author's previous article. This parametrization fails for the monomial Burnside ring over a cyclic group of order four. Second, to classify the simple modules for the monomial Burnside ring over a group of prime order, for which the above-mentioned parametrization holds.

math.GR↗
arXiv2012

Simple modules over Green biset functors

We present three examples of Green biset functors for which their simple modules can be parametrized. These are particular cases of a conjecture by Serge Bouc classifying the simple modules over a Green biset functor A, that generalizes the classification of simple biset functors. We also prove this conjecture under certain hypothesis for A.

math.RT↗
arXiv2011

On primordial groups for the Green ring

Consider the Mackey functor assigning to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p>0. Thevenaz foresaw in 1988 that the class of primordial groups for this functor is the family of k-Dress groups. In this paper we prove that this is true for the subfunctor defined by the Green ring of finitely generated kG-modules of trivial source.

math.RT↗