Halvard Fausk
The definition and basic properties of the Burnside ring of compact Lie groups are presented, with emphasis on the analogy with the construction of the Burnside ring of finite groups.
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We introduce the concept of lattice Burnside ring for a finite group G associated to a family of nonempty sublattices of a finite G-lattice assigned to subgroups of G. The slice Burnside ring introduced by S. Bouc is isomorphic to a lattice Burnside ring. Any lattice Burnside ring is isomorphic to an abstract Burnside ring. The ring structure of a lattice Burnside ring is explored on the basis of the fundamental theorem of abstract Burnside rings. We study the units and the primitive idempotents of a lattice Burnside ring. There are certain rings called partial lattice Burnside rings. Any part…
In this paper we investigate some properties of the Burnside ring of a profinite group as defined in \cite{ds}. We introduce the notion of the crossed Burnside ring of a profinite FC-group, and generalise some results from finite to profinite (FC-)groups. In our investigations, we also obtain results on profinite FC-groups which may be of independent interest.
Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a lambda-ring. However, a priori the images of S under the lambda-operations can only be computed implicitly. In this paper we establish an explicit formula, expressing these images as linear combinations of classes of G-sets.
Let G be a finite group and let k be a field. Our purpose is to investigate the simple modules for the double Burnside ring kB(G,G). It turns out that they are evaluations at G of simple biset functors. For a fixed finite group H, we introduce a suitable bilinear form on kB(G,H) and we prove that the quotient of kB(-,H) by the radical of the bilinear form is a semi-simple functor. This allows for a description of the evaluation of simple functors, hence of simple modules for the double Burnside ring.
We describe the generalized Matsuda's theorem, and some results of a Burnside ring extend a partial Burnside ring. In particular, we give isomorphism between partial Burnside rings of different groups. Moreover, we consider the relationship between image of Frobenius-Wielandt homomorphism, a partial Burnside ring, and structure of a group.
In this note I describe the structure of the biset functor $B^\times$ sending a $p$-group $P$ to the group of units of its Burnside ring $B(P)$. In particular, I show that $B^\times$ is a rational biset functor. It follows that if $P$ is a $p$-group, the structure of $B^\times(P)$ can be read from a genetic basis of $P$: the group $B^\times(P)$ is an elementary abelian 2-group of rank equal to the number isomorphism classes of rational irreducible representations of $P$ whose type is trivial, cyclic of order 2, or dihedral.
For a saturated fusion system $\mathcal F$ on a $p$-group $S$, we study the Burnside ring of the fusion system $B(\mathcal F)$, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring $B(S)$. We give criteria for an element of $B(S)$ to be in $B(\mathcal F)$ determined by the $\mathcal F$-automorphism groups of essential subgroups of $S$. When $\mathcal F$ is the fusion system induced by a finite group $G$ with $S$ as a Sylow $p$-group, we show that the restriction of $B(G)$ to $B(S)$ has image equal to $B(\mathcal F)$. We also show that for $p=2,$ we can gain infor…
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。
This paper develops links between the Burnside ring of a finite group $G$ and the slice Burnside ring}. The goal is to gain a better understanding of ghost maps, idempotents, prime spectrum of these Burnside rings and connections between them.
This paper introduces two new Burnside rings for a finite group $G$, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of $G$-sets, and of Galois morphisms of $G$-sets, respectively. The well known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural structure of Green biset functor. The functorial structure of unit groups of these rings is also discussed.
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.
Using the Burnside ring theoretic methods a new setting and a complete description of the Artin exponent $A(G)$ of finite $p$-groups was obtained in a previous article of the first-named author. In this paper, we compute $A(G)$ for any finite group $G$ -- hence providing the global version of that result.
In a 1969 article, A. Dress described the prime ideals of the Burnside ring of a finite group G and the inclusion relations between them. One may ask whether similar results exist for the Burnside ring of a saturated fusion system F on a finite p-group S. The present paper answers positively, providing a typology for the prime ideals of A(F) and for those of A(F) (p) as a consequence.
A transitive permutation group of prime degree is doubly transitive or solvable. We give a direct proof of this theorem by Burnside which uses neither S-ring type arguments, nor representation theory.
We define and study the Burnside quotient Green ring of a Mackey functor. Some refinements of Dress induction theory are presented, together with applications to computation results for $K$-theory and $L$-theory of finite and infinite groups.
Tensor products and units of partial Burnside rings
研究有限群子群集合相关的部分伯恩赛德环张量积，给出典范同态为满射的充要条件，描述可约有限考克斯特群符号单位分解。
G.E. Wall gave two different proofs of a remarkable result about the multilinear Lie relators satisfied by groups of prime power exponent $q$. He showed that if $q$ is a power of the prime $p$, and if $f$ is a multilinear Lie relator in $n$ variables where $n\neq1\operatorname{mod}(p-1)$, then $f=0$ is a consequence of multilinear Lie relators in fewer than $n$ variables. For years I have struggled to understand his proofs, and while I still have not the slightest clue about his first proof published in the Journal of Algebra, I finally have some understanding of his second proof published in …
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…
For a finite group $G$, a Tambara functor on $G$ is regarded as a $G$-bivariant analog of a commutative ring. In this analogy, previously we have defined an ideal of a Tambara functor. In this article, we will demonstrate a calculation of the prime spectrum of the Burnside Tambara functor, when $G$ is a cyclic $p$-group for a prime integer $p$.
This note describes an application of the theory of generalised Burnside rings to algebraic representation theory. Tables of marks are given explicitly for the groups $S_4$ and $S_5$ which are of particular interest in the context of reductive algebraic groups. As an application, the base sets for the nilpotent element $F_4 (a_3)$ are computed.
The famous Burnside-Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latters we introduce Schur rings over a finite commutative ring and prove an analog of this statement for them. Besides, the finite local commutative rings are characterized in the permutation group terms.
In this note we present an algorithm for the construction of the unit group of the Burnside ring $Ω(G)$ of a finite group $G$ from a list of representatives of the conjugacy classes of subgroups of G.
The bifree double Burnside ring $B^Δ(G,G)$ of a finite group $G$ has a natural anti-involution. We study the group $B^Δ_\circ(G,G)$ of orthogonal units in $B^Δ(G,G)$. It is shown that this group is always finite and contains a subgroup isomorphic to $B(G)^\times\rtimes \Out(G)$, where $B(G)^\times$ denotes the unit group of the Burnside ring of $G$ and $\Out(G)$ denotes the outer automorphism group of $G$. Moreover it is shown that if $G$ is nilpotent then $B^Δ_\circ(G,G)\cong B(G)^\times\rtimes \Out(G)$. The results can be interpreted as positive answers to questions on equivalences of $p$-bl…
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。
Normal-Core Compression for Units in Burnside Rings
该研究定义Burnside环单位的正规核压缩映射，推导其系数性质，给出核平凡单位的刻画，证明有限群中核平凡单位群的秩无界。
Balmer and Dell'Ambrogio introduced the pseudo-functor $P$ from the bicategory of $k$-linear Mackey $2$-motives to the bicategory of $k$-linear cohomological Mackey $2$-motives over a commutative ring $k$. They showed that $P$ maps the general Mackey $2$-motives to the cohomological Mackey $2$-motives by using the ring homomorphism from the crossed Burnside ring of a finite group $G$ over $k$ to the center $ ZkG$ of group algebra $kG$ ([BD21, Theorem 5.3]). We study the behavior of motivic decomposition of cohomological Mackey $2$-motives as images by $P$ of motivic decomposition of Mackey $2$…
We consider the Burnside ring $A(\mathcal{F})$ of $\mathcal{F}$-stable $S$-sets for a saturated fusion system $\mathcal{F}$ defined on a $p$-group $S$. It is shown by S. P. Reeh that the monoid of $\mathcal{F}$-stable sets is a free commutative monoid with canonical basis $\{α_P\}$. We give an explicit formula that describes $α_P$ as an $S$-set. In the formula we use a combinatorial concept called broken chains which we introduce to understand inverses of modified Möbius functions.
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…
We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between two categorified group-sets is given by a particular equivalence between the underlying categories. We also exhibit some of the differences between the old Burnside theory and the new one. Lastly, we briefly explain how to extend these new techniques and concepts to the context of groupoids, employing the categories of (right) groupoid-sets, aiming by this to…
We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, we obtain an explicit description …
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…
In this paper, we prove that if two finite groups G and H have isomorphic Burnside rings, then G and H are the same order type groups, and give an example to show that the Burnside rings of the same order type groups are not necessarily isomorphic. This result is related to the Thompson Problem which was raised in 1987.
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…
Burnside rings for racks and quandles
本文利用Burnside环重构有限支架与拟群的分类理论，建立加法基、乘法生成元及标记理论，为分类提供新框架并加强与代数的联系。
We give a new, unexpected characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic phenomena in the language of idempotents. Interpreting our results in stable homotopy, we answer a long-standing question on stable splittings of classifying spaces of finite groups, and generalize the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. This work is partly motivated by a conjecture of Hay…
Let $G$ be a finite group and $A(G)$ its Burnside ring. For $H \subset G$ let $\mathbb{Z}_H$ denote the $A(G)$-module corresponding to the mark homomorphism associated to $H$. When the order of $G$ is square-free we give a complete description of the $A(G)$-modules $\textrm{Ext}^l_{A(G)}(\mathbb{Z}_H, \mathbb{Z}_J)$ and $\textrm{Tor}^{A(G)}_l(\mathbb{Z}_H, \mathbb{Z}_J)$ for any $H, J \subset G$ and $l \geq 0$. We show that if the order of $G$ is not square-free then there exist $H, J \subset G$ such that $\textrm{Ext}^l_{A(G)}(\mathbb{Z}_H, \mathbb{Z}_J)$ and $\textrm{Tor}^{A(G)}_l(\mathbb{Z}…
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.
On the image of the total power operation for Burnside rings
该研究证明Burnside环全幂运算的像落在组合子环内，构造了交换分次环并给出特征公式，推广了Frobenius–Wielandt同态和Burnside轨道计数引理。
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)…
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 …
On Schur Rings Over a Semi Dihedral Group
本文证明了半二面体群上仅存在平凡的本原Schur环，并由此说明每个半二面体子群都是Burnside群，即含该正则子群的本原群必为2-传递群。
Burnside type results for fusion categories
本文研究了融合范畴中Burnside型结果的扩展，证明了nilpotent单位融合范畴满足Burnside性质及其对偶，并提供了新的范畴化标准和应用。
All groups are 2-generator. For any prime-power q, Theorem 1 constructs a solvable matrix group over a quotient of a Laurent polynomial ring. This group is closely related to a group of exponent q as shown in Theorems 2 & 3 . Theorem 4 in section 5 shows that a group of prime-power exponent contains the relations of a solvable group. It follows that the Burnside groups of exponent q are solvable, and it is easy to deduce that the solvability class of these groups tends to infinity with q. Crucial to this work, especially for precise bounds on the solvability class, is an earlier paper with Hei…
Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.
Global Representation Ring and Knutson Index
本文将Knutson指数推广到一般交换环，研究其在Burnside环和全局表示环上的表现，并引入统一特征标表与Burnside标记表的全局表，探讨可恢复的群性质。
In the previous article 'A Mackey-functor theoretic interpretation of biset functors', we have constructed the 2-category $\mathbb{S}$ of finite sets with variable finite group actions, in which bicoproducts and bipullbacks exist. As shown in it, biset functors can be regarded as a special class of Mackey functors on $\mathbb{S}$. In this article, we equip $\mathbb{S}$ with a system of adjoint triplets, which satisfies properties analogous to a derivator. This system encodes the six operations for finite groups. As a corollary, we show that the associated Burnside rings satify analogous proper…
Let $G$ be a finite group and $R$ be a commutative ring. The Mackey algebra $μ_{R}(G)$ shares a lot of properties with the group algebra $RG$ however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetricity of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra. The category of Mackey functors is a closed symmetric monoidal category, so using the formalism of J.P. May for these categories, S…
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…
$p$-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。
We establish some results about large restricted Lie algebras similar to those known in the Group Theory. As an application we use this group-theoretic approach to produce some examples of restricted as well as ordinary Lie algebras which can serve as counterexamples for various Burnside-type questions.
Notes on $B$-groups
本文通过区分 $B$-群与 $BS$-群，构造了无穷多个 $B$-群但非 $BS$-群的例子，揭示了 Schur 环方法在原始群研究中的局限性。
The Artin exponent induced from cyclic subgroups of finite groups was studied extensively by T.Y. Lam. A Burnside ring theoretic version of Lam's results for $p$-groups was given by the author in an earlier paper. Here we look at the Artin exponent induced from the elementary abelian subgroups of finite $p$-groups using some results of A. Dress.
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(…
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。
If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise to isomorphic permutation representations C[X] and C[Y]. Equivalently, the map from the Burnside ring to the representation ring of G has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave-Bouc classification in the case of p-groups.
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…
A nilpotent quotient algorithm for finitely presented Lie rings over Z (LieNQ) is described. The paper studies graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. The nilpotent presentation consists of generators for the abelian group and the products---expressed as linear combinations---for pairs formed by generators. Using that presentation the word problem is decidable in $L$. Provided that the Lie ring $L$ is graded, it is possible to determine the canonical presentation for a lower central factor…
Lie rings related to the $p$-groups of maximal class
本文研究了与最大类p-群相关的李代数，证明其为有限且 nilpotent 类较小，填补了相关研究的空白。
This is the second in a series of papers investigating the space of Brauer relations of a finite group, the kernel of the natural map from its Burnside ring to the rational representation ring. The first paper classified all primitive Brauer relations, that is those that do not come from proper subquotients. In the case of quasi-elementary groups the description is intricate, and it does not specify groups that have primitive relations in terms of generators and relations. In this paper we do this for quasi-elementary groups of order p^aq.
We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free $d$-generator Lie rings $\mathfrak{f}_{c,d}$ of nilpotency class $c$ for all $c\leq2$ and for $(c,d)\in\{(3,3),(3,2),(4,2)\}$. We apply our computations to obtain information about $\mathfrak{p}$-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.
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