Olcay Coşkun, Deniz Yılmaz
We determine the composition factors of the $A$-fibered Burnside functor $kB^A$ for $p$-groups over a field $k$ of characteristic $q$ with $q\neq p$ and cyclic fiber group $A$. We also show that, in this case, $kB^A$ is uniserial.
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The theory of bisets has been very useful in progress towards settling the longstanding question of determining units for the Burnside ring. In 2006 Bouc used bisets to settle the question for $p$-groups. In this paper, we provide a standard basis for the unit group of the Burnside ring for groups that contain a abelian subgroups of index two. We then extend this result to groups $G$, where $G$ has a normal subgroup, $N$, of odd index, such that $N$ contains an abelian subgroups of index $2$. Next, we study the structure of the unit group of the Burnside ring as a biset functor, $B^\times$ on …
In this article we define the $-_+$-construction and the $-^+$-construction, that was crucial in the theory of canonical induction formulas (see \cite{Boltje1998b}), in the setting of biset functors, thus providing the necessary framework to define and construct canonical induction formulas for representation rings that are most naturally viewed as biset functors. Additionally, this provides a unified approach to the study of a class of functors including the Burnside ring, the monomial Burnside ring and global representation ring.
The theory of biset functors, introduced by Serge Bouc, gives a unified treatment of operations in representation theory that are induced by permutation bimodules. In this paper, by considering fibered bisets, we introduce and describe the basic theory of fibered biset functors which is a natural framework for operations induced by monomial bimodules. The main result of this paper is the classification of simple fibered biset functors.
Essential algebra of the shifted Burnside biset functor with abelian shift
本文研究带阿贝尔移位的移位Burnside双集函子的本质代数，给出分解准则并推广互素情形，证明其可分解为矩阵代数直和并参数化不可约模。
We classify simple modules over the Green biset functor of section Burnside rings.
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)…
A Categorical Decomposition of $\mathbb C^{\times}$-fibered $p$-biset Functors
本文将Bouc在双Burnside代数中构造正交幂等元的方法推广到双ℂ×纤维化的Burnside代数中，得到有限群上ℂ×纤维化的双集函子的结构分解，并构造出完整的正交幂等元集，将该范畴分解为由无环p-群同构类索引的子范畴。
Let $A$ be an abelian group such that $\mathrm{Hom}(G,A)$ is finite for all finite groups $G$, and let $\mathbb{K}$ be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in $A$. In this paper we prove foundational properties of the $A$-fibered Burnside ring functor $B_{\mathbb{K}}^A$ as an $A$-fibered biset functor over $\mathbb{K}$. This includes the determination of the lattice of subfunctors of $B_{\mathbb{K}}^A$ and the determination of the composition factors of $B_{\mathbb{K}}^A$. The results of the paper extend results of Coşkun and Yıl…
For a non-vanishing group, we show that the evaluation functor induces an equivalence between the category of modules over the double Burnside algebra and a certain category of biset functors. Using this equivalence, we deduce that over a field of characteristic zero, the double Burnside algebra of a non-vanishing group is a quasi-hereditary algebra. We also show that the double Burnside algebra over a field is self-injective if and only if it is semisimple.
Let R be a (unital) commutative ring, and G be a finite group with order invertible in R. We introduce new idempotents in the double Burnside algebra RB(G,G), indexed by conjugacy classes of minimal sections of G, i.e. pairs (T,S) of subgroups of G, where S is a normal subgroup of T contained in the Frattini subgroup of T. These idempotents are orthogonal, and their sum is equal to the identity. It follows that for any biset functor F over R, the evaluation F (G) splits as a direct sum of specific R-modules indexed by minimal sections of G, up to conjugation. The restriction of these construct…
In this short note we investigate some consequences of the vanishing of simple biset functors. As corollary, if there is no non-trivial vanishing of simple biset functors (e.g. if the group is commutative), then we show that the double Burnside algebra is a quasi-hereditary algebra in characteristic zero. In general, this not true without the non-vanishing condition, as over a field of characteristic zero, the double Burnside algebra of the alternating group of degree 5 has infinite global dimension.
We determine the structure of the fibered biset functor sending a finite group $G$ to the complex vector space of complex valued class functions of $G$. Previously, it is studied as a biset functor by Bouc and as a $\mathbb C^\times$-fibered biset functor by Boltje and the second author. In this paper, we complete the study by considering the other choices of the fiber group. As a corollary, we obtain a (previously known) set of composition factors of the biset functor of $p$-permutation modules.
$p$-Bifree biset functors
本文引入 $p$-双自由双集函子范畴，作为经典双集函子与对角 $p$-置换函子之间的桥梁，并在特征零域上分类其简单对象，计算了若干表示论函子的合成因子。
Indecomposable factors of Fibered Burnside Rings
本文基于标准基描述了纤维Burnside环的不可分解因子，并进一步将其视为对应Weyl群纤维Burnside环的可解成分。
In this note, we present a notion of species isomorphism for fibered Burnside rings. We prove that a species isomorphism can be restricted to an isomorphism of mark tables, that it can be extended to an isomorphism between the ghost rings and we provide three additional characterizations of this concept.
On fibered Burnside rings, fiber change maps and cyclic fiber groups
本文引入了有限群纤维化Burnside环之间的纤维变换映射，研究了其函子性，并推进了循环纤维群导子的确定工作。
On the separability of some Green biset functors
研究证明复字符的Green双集函子在Z上不分离，而移位后的Burnside双集函子分离性取决于群阶在环中可逆性。
$p$-Biset Functor of Monomial Burnside Rings
本文研究了特征为零的域上的单形Burnside双集函子的结构，特别是其限制核。对于每个有限$p$-群$G$，给出了在$G$处的限制核的显式描述，并确定了该函子的全部复合因子。证明了这些复合因子的极小群$H$要么是循环$p$-群，要么是这样的群与阶为$p$的循环群的直积。进一步，识别了作为这些复合因子在极小群上的评价出现的简单$\mathbb{C}[\Aut(H)]$-模。显式分类双集函子的复合因子很少见，本文结果提供了此类分类的少数完整例子之一。
The global representation fibered ring
本文提出了一种新的有限群环结构，通过结合纤维化Burnside环和特征环，研究其环同态、谱结构及幂等元性质。
Biset functors for categories
本文将群上的双集函子推广到任意有限范畴，建立 Burnside 环、简单函子、幺半结构与纤维化等理论，并通过 Boolean 格联系对应函子。
The double Burnside $R$-algebra $\text{B}_R(G,G)$ of a finite group $G$ with coefficients in a commutative ring $R$ has been introduced by S. Bouc. It is $R$-linearly generated by finite $(G,G)$-bisets, modulo a relation identifying disjoint union and sum. Its multiplication is induced by the tensor product. B. Masterson described $\text{B}_{\mathbf{Q}}(\text{S}_3,\text{S}_3)$ as a subalgebra of $\mathbf{Q}^{8\times 8}$. We give a variant of this description and continue to describe $\text{B}_R(\text{S}_3,\text{S}_3)$ for $R\in\{\mathbf{Z},\mathbf{Z}_{(2)},\mathbf{F}_2,\mathbf{Z}_{(3)},\mathbf…
Given a finite group $G$, its double Burnside ring $B(G,G)$, has a natural duality operation that arises from considering opposite $(G,G)$-bisets. In this article, we systematically study the subgroup of units of $B(G,G)$, where elements are inverse to their dual, so called orthogonal units. We show the existence of an inflation map that embeds the group of orthogonal units of $B(G/N,G/N)$ into the group of orthogonal units of $B(G,G)$, when $N$ is a normal subgroup of $G$, and study some properties and consequences. In particular, we use these maps to determine the orthogonal units of $B(G,G)…
Groups with isomorphic fibered Burnside rings
该研究给出有限群$G$、$H$的一个条件，可推出其$A$-纤维化Burnside环同构，并证明存在不同构有限群的非平凡纤维化Burnside环同构，补充了普通Burnside环已有反例之外的相关结论。
For a finite group $G$, we define a ghost ring and a mark homomorphism for the double Burnside ring of left-free $(G,G)$-bisets. In analogy to the case of the Burnside ring $B(G)$, the ghost ring has a much simpler ring structure, and after tensoring with $\QQ$ one obtains an isomorphism of $\QQ$-algebras. As an application of a key lemma, we obtain a very general formula for the Brauer construction applied to a tensor product of two $p$-permutation bimodules $M$ and $N$ in terms of Brauer constructions of the bimodules $M$ and $N$. Over a field of characteristic 0 we determine the simple modu…
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 …
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 …
Monomial structures, I
本文旨在各种 $A$-纤维化结构上定义有限群 $G$ 的作用，在偏序集、单纯复形和拓扑空间上的 $A$-丛上引入 $A$-纤维化结构及保持同伦的函子，为后续研究 $G$-表示和 $A$-纤维化 Burnside 环奠定基础。
We examine the cokernel of the canonical homomorphism from the trivial source ring of a finite group to the ring of $p$-rational complex characters. We use Boltje and Coşkun's theory of fibered biset functors to determine the structure of the cokernel. An essential tool in the determination of this structure is Bouc's theory of rational $p$-biset functors.
We construct a well-behaved transfer map from the p-local Burnside ring of the underlying p-group S to the p-local Burnside ring of a saturated fusion system F. Using this transfer map, we give new results on the characteristic idempotent of F -- the unique idempotent in the p-local double Burnside ring of S satisfying properties of Linckelmann and Webb. We describe this idempotent explicitly both in terms of fixed points and as a linear combination of transitive bisets. Additionally, using fixed points we determine the map for Burnside rings given by multiplication with the characteristic ide…
The theory of biset functors developed by Serge Bouc has been instrumental in the study of the unit group of the Burnside ring of a finite group, in particular for the case of p-groups. The ghost ring of the Burnside ring defines an inflation functor, and becomes a useful tool in studying the Burnside ring functor itself. We are interested in studying the unit group of another representation ring: the trivial source ring of a finite group. In this article, we show how the unit groups of the trivial source ring and its associated ghost ring define inflation functors. Since the trivial source ri…
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(…
For a finite group $G$, we introduce a multiplication on the $\QQ$-vector space with basis $\scrS_{G\times G}$, the set of subgroups of $G\times G$. The resulting $\QQ$-algebra $\Atilde$ can be considered as a ghost algebra for the double Burnside ring $B(G,G)$ in the sense that the mark homomorphism from $B(G,G)$ to $\Atilde$ is a ring homomorphism. Our approach interprets $\QQ B(G,G)$ as an algebra $eAe$, where $A$ is a twisted monoid algebra and $e$ is an idempotent in $A$. The monoid underlying the algebra $A$ is again equal to $\scrS_{G\times G}$ with multiplication given by composition o…
Deflation map and the sum of inverses of the element orders in finite groups
本文利用收缩映射在有限群的Burnside环中构造元素，得到元素阶倒数和的表达式，并由此推出交叉Burnside环秩与p-局部Mackey代数Cartan矩阵行列式的另一表达式。
Let p be a prime number. This paper introduces the Roquette category R_p of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_p, every finite p-group P admits a canonical direct summand, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of R_p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from R_p to abelian groups are exactly the rational p-biset functors. This yields in particular very effi…
This paper extends the notion of $B$-group to a relative context. For a finite group $K$ and a field $\mathbb{F}$ of characteristic 0, the lattice of ideals of the Green biset functor $\mathbb{F}B_K$ obtained by shifting the Burnside functor $\mathbb{F}B$ by $K$ is described in terms of {\em $B_K$-groups}. It is shown that any finite group $(L,φ)$ over $K$ admits a {\em largest quotient $B_K$-group} $β_K(L,φ)$. The simple subquotients of $\mathbb{F}B_K$ are parametrized by $B_K$-groups, and their evaluations can be precisely determined. Finally, when $p$ is a prime, the restriction $\mathbb{F}…
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