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Serge Bouc

Publications and source records attributed to Serge Bouc.

At least 19 recordsLinked to original sources
arXiv2026

The $D^{\Delta}$-pair biset category

Let $p$ be a prime number. A $D^{\Delta}$-pair is a pair consisting of a finite $p$-group and a $p'$-automorphism of the group. In this paper, we introduce diagonal $D^\Delta$-pair bisets and a category whose objects are $D^\Delta$-pairs and whose morphism groups are Grothendieck groups of diagonal $D^\Delta$-pair bisets. Our main result shows that, over suitable coefficient rings, the category of diagonal $p$-permutation functors is equivalent to the category of linear functors on a natural quotient of this new category. In this way, diagonal $p$-permutation functors can be studied through a category built only from finite $p$-groups and their automorphisms of $p'$-order.

math.RT↗
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

Functorial equivalence classes of $2$-blocks of tame representation type

For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal $p$-permutation functor over an algebraically closed field $\mathbb{F}$ of characteristic $0$ into a direct sum of simple functors. As a consequence we see that two blocks with dihedral, semidihedral, or generalized quaternion defect groups are functorially equivalent over $\mathbb{F}$ if and only if their fusion systems are isomorphic. It is an open question if two blocks (with arbitrary defect groups) that are functorially equivalent over $\mathbb{F}$ must have isomorphic fusion systems. The converse is wrong in general.

math.RT↗
arXiv2025

On Alperin's conjecture and functorial equivalence of blocks

Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider Alperin's weight conjecture (over $k$) from the point of view of (stable) functorial equivalence of blocks over $\mathbb{F}$. We formulate a functorial version of Alperin's blockwise weight conjecture, and show that it is equivalent to the original one. We also show that this conjecture holds stably, i.e., in the category of stable diagonal $p$-permutation functors over $\mathbb{F}$.

math.RT↗
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↗
arXiv2024

The diagonal $p$-permutation functor $kR_k$

Let $k$ be an algebraically closed field of positive characteristic $p$. We describe the full lattice of subfunctors of the diagonal $p$-permutation functor $kR_k$ obtained by $k$-linear extension from the functor $R_k$ of linear representations over $k$. This leads to the description of the ``composition factors'' $S_P$ of $kR_k$, which are parametrized by finite $p$-groups (up to isomorphism), and of the evaluations of these particular simple diagonal $p$-permutation functors over $k$.

math.GR↗
arXiv2024

Diagonal $p$-permutation functors in characteristic $p$

Let $p$ be a prime number. We consider diagonal $p$-permutation functors over a (commutative, unital) ring $\mathsf{R}$ in which all prime numbers different from $p$ are invertible. We first determine the finite groups $G$ for which the associated essential algebra $\mathcal{E}_\mathsf{R}(G)$ is non zero: These are groups of the form $G=L\rtimes \langle u\rangle$, where $(L,u)$ is a $D^\Delta$-pair. When $\mathsf{R}$ is an algebraically closed field $\mathbb{F}$ of characteristic 0 or $p$, this yields a parametrization of the simple diagonal $p$-permutation functors over $\mathbb{F}$ by triples $(L,u,W)$, where $(L,u)$ is a $D^\Delta$-pair, and $W$ is a simple $\mathbb{F}\mathrm{Out}(L,u)$-module. Finally, we describe the evaluations of the simple functor $\mathsf{S}_{L,u,W}$ parametrized by the triple $(L,u,W)$. We show in particular that if $G$ is a finite group and $\mathbb{F}$ has characteristic $p$, the dimension of $\mathsf{S}_{L,1,\mathbb{F}}(G)$ is equal to the number of conjugacy classes of $p$-regular elements of $G$ with defect isomorphic to $L$.

math.GR↗
arXiv2024

A general Greenlees-May splitting principle

In equivariant topology, Greenlees and May used Mackey functors to show that, rationally, the stable homotopy category of $G$-spectra over a finite group $G$ splits as a product of simpler module categories. We extend the algebraic part (also independently proved by Thévenaz and Webb) of this classical result to Mackey modules over an arbitrary Green functor, and use the case of the complex representation ring Green functor to obtain an algebraic model of the rational equivariant Kasparov category of $G$-cell algebras.

math.KT↗
arXiv2023

Stable functorial equivalence of blocks

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring and let $\mathcal{F}$ be an algebraically closed field of characteristic $0$. We introduce the category $\overline{\mathcal{F}_{Rpp_k}}$ of stable diagonal $p$-permutation functors over $R$. We prove that the category $\overline{\mathcal{F}_{\mathbb{F}pp_k}}$ is semisimple and give a parametrization of its simple objects in terms of the simple diagonal $p$-permutation functors. We also introduce the notion of a stable functorial equivalence over $R$ between blocks of finite groups. We prove that if $G$ is a finite group and if $b$ is a block idempotent of $kG$ with an abelian defect group $D$ and Frobenius inertial quotient $E$, then there exists a stable functorial equivalence over $\mathbb{F}$ between the pairs $(G,b)$ and $(D\rtimes E,1)$.

math.GR↗
arXiv2022

Diagonal $p$-permutation functors, semisimplicity, and functorial equivalence of blocks

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring, and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $R$-linear category $\mathcal{F}^Δ_{Rpp_k}$ of diagonal $p$-permutation functors over $R$. We first show that the category $\mathcal{F}^Δ_{\mathbb{F}pp_k}$ is semisimple, and we give a parametrization of its simple objects, together with a description of their evaluations. Next, to any pair $(G,b)$ of a finite group $G$ and a block idempotent $b$ of $kG$, we associate a diagonal $p$-permutation functor $RT^Δ_{G,b}$ in $\mathcal{F}^Δ_{Rpp_k}$. We find the decomposition of the functor $\mathbb{F}T^Δ_{G,b}$ as a direct sum of simple functors in $\mathcal{F}^Δ_{\mathbb{F}pp_k}$. This leads to a characterization of nilpotent blocks in terms of their associated functors in $\mathcal{F}^Δ_{\mathbb{F}pp_k}$. Finally, for such pairs $(G,b)$ of a finite group and a block idempotent, we introduce the notion of functorial equivalence over $R$, which (in the case $R=\mathbb{Z}$) is slightly weaker than $p$-permutation equivalence, and we prove a corresponding finiteness theorem: for a given finite $p$-group $D$, there is only a finite number of pairs $(G,b)$, where $G$ is a finite group and $b$ a block idempotent of $kG$ with defect isomorphic to $D$, up to functorial equivalence over $\mathbb{F}$.

math.GR↗
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

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↗
arXiv2021

Some simple biset functors

Let $p$ be a prime number, let $H$ be a finite $p$-group, and let $\mathbb{F}$ be a field of characteristic 0, considered as a trivial $\mathbb{F} \mathrm{Out}(H)$-module. The main result of this paper gives the dimension of the evaluation $S_{H,\mathbb{F}}(G)$ of the simple biset functor $S_{H,\mathbb{F}}$ at an arbitrary finite group $G$. A closely related result is proved in the last section: for each prime number $p$, a Green biset functor $E_p$ is introduced, as a specific quotient of the Burnside functor, and it is shown that the evaluation $E_p(G)$ is a free abelian group of rank equal to the number of conjugacy classes of $p$-elementary subgroups of $G$.

math.GR↗
arXiv2020

Germs in a poset

Motivated by the theory of correspondence functors, we introduce the notion of {\em germ} in a finite poset, and the notion of {\em germ extension} of a poset. We show that any finite poset admits a largest germ extension called its {\em germ closure}. We say that a subset $U$ of a finite lattice $T$ is {\em germ extensible} in $T$ if the germ closure of $U$ naturally embeds in $T$. We show that any for any subset $S$ of a finite lattice $T$, there is a unique germ extensible subset $U$ of $T$ such that $U\subseteq S\subseteq \overline{G}(U)$, where $\overline{G}(U)\subseteq T$ is the embedding of the germ closure of $U$.

math.CO↗
arXiv2020

A functorial presentation of units of Burnside rings

Let $B^\times$ be the biset functor over $\mathbb{F}_2$ sending a finite group~$G$ to the group $B^\times(G)$ of units of its Burnside ring $B(G)$, and let $\widehat{B^\times}$ be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from $B^\times$ in the dual Burnside functor $\widehat{\mathbb{F}_2B}$, or equivalently, an explicit set of generators $\mathcal{G}_S$ of the kernel $L$ of the natural surjection $\mathbb{F}_2B\to \widehat{B^\times}$. This yields a two terms projective resolution of $\widehat{B^\times}$, leading to some information on the extension functors $\mathrm{Ext}^1(-,B^\times)$. For a finite group $G$, this also allows for a description of $B^\times(G)$ as a limit of groups $B^\times(T/S)$ over sections $(T,S)$ of $G$ such that $T/S$ is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor $B^\times$ is not finitely generated, and that its dual $\widehat{B^\times}$ is finitely generated, but not finitely presented. The last result of the paper shows in addition that $\mathcal{G}_S$ is a minimal set of generators of $L$, and it follows that the lattice of subfunctors of $L$ is uncountable.

math.GR↗
arXiv2019

Diagonal $p$-permutation functors

Let $k$ be an algebraically closed field of positive characteristic $p$, and $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $\mathbb{F}$-linear category $\mathbb{F} pp_k^Δ$ of finite groups, in which the set of morphisms from $G$ to $H$ is the $\mathbb{F}$-linear extension $\mathbb{F} T^Δ(H,G)$ of the Grothendieck group $T^Δ(H,G)$ of $p$-permutation $(kH,kG)$-bimodules with (twisted) diagonal vertices. The $\mathbb{F}$-linear functors from $\mathbb{F} pp_k^Δ$ to $\mathbb{F}\hbox{-Mod}$ are called {\em diagonal $p$-permutation functors}. They form an abelian category $\mathcal{F}_{pp_k}^Δ$. We study in particular the functor $\mathbb{F}T^Δ$ sending a finite group $G$ to the Grothendieck group $\mathbb{F}T(G)$ of $p$-permutation $kG$-modules, and show that $\mathbb{F}T^Δ$ is a semisimple object of $\mathcal{F}_{pp_k}^Δ$, equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs $(P,s)$ of a finite $p$-group $P$ and a generator $s$ of a $p'$-subgroup acting faithfully on $P$. This leads to a precise description of the evaluations of these simple functors. In particular, we show that the simple functor indexed by the trivial pair $(1,1)$ is isomorphic to the functor sending a finite group $G$ to $\mathbb{F} K_0(kG)$, where $K_0(kG)$ is the group of projective $kG$-modules.

math.GR↗
arXiv2019

Monomial $G$-posets and their Lefschetz invariants

Let $G$ be a finite group, and $C$ be an abelian group. We introduce the notions of $C$-monomial $G$-sets and $C$-monomial $G$-posets, and state some of their categorical properties. This gives in particular a new description of the $C$-monomial Burnside ring $B_C(G)$. We also introduce Lefschetz invariants of $C$-monomial $G$-posets, which are elements of $B_C(G)$. These invariants allow for a definition of a generalized tensor induction multiplicative map $\mathcal{T}_{U,λ}: B_C(G)\to B_C(H)$ associated to any $C$-monomial $(G,H)$-biset $(U,λ)$, which in turn gives a group homomorphism $B_C(G)^\times\to B_C(H)^\times$ between the unit groups of $C$-monomial Burnside rings.

math.GR↗