Burnside Ring Research Articles

The free βring on one generator

We provide two models for the free β-ring on one generator. One is the graded ring S( A) = ⊕ ∞ n=0 AS( n), where AS( n) denotes the Burnside ring of the symmetric group S( n) of degree n. This gives an answer to a question raised by Ochoa in [Outer plethysm, Burnside ring and β-rings, J. Pure Appl. Algebra 55 (1988) 173–195]. The second model is the ring of geometric operators on the Burnside ring functor introduced by the author in [Polynomial operations from Burnside rings to representation functors, J. Pure Appl. Algebra 65 (1990) 163–190].

A Simple Proof for a Result by Kratzer and Thévenaz Concerning the Embedding of the Burnside Ring into Its Ghost Ring

A Simple Proof for a Result by Kratzer and Thévenaz Concerning the Embedding of the Burnside Ring into Its Ghost Ring

Some remarks on projective Mackey functors

We consider certain problems in the algebra of Mackey functors for a finite group raised by equivariant topology. Our main theorems are: A Mackey functor has projective dimension 0, 1 or ∞ (Theorem 2.1). The G-theory of Mackey functors is calculable in terms of the G-theories of the Weyl groups (Theorem 3.1). For Mackey functors with the group order invertible the K-theory is the sum of the K-theories of the Weyl groups (Theorem 6.1); a similar result holds unstably (Theorem 6.4). For a group of prime order the projective class group of Mackey functors is the sum of the class group of the group ring and that of the Burnside ring (Theorem 7.1).

On the existence of Borsuk-Ulam theorems

A Borsuk-Ulam theorem for the finite group G consists of finding a function b: N → N with b( n)→ ∞ as n→ ∞ and such that the existence of a G-map SV→ SW between representation spheres without fixed points implies dim W⩾ b(dim V). We show that such a function b exists iff G is a p-group. We also prove that a G-map SV→ SW as above with W⫋ V exists. iff G is not a p-group. Similar results hold for compact Lie groups. As a corollary we obtain an algebraic characterization of p-groups. If there exists an element a ∈ A ( G) of the Burnside ring of G with | a G |=1 and a representation V of G with | a H | ·dim V H =0 for all subgroups H of G then G is not a p-group and vice versa.

Abelian and hamiltonian groups with isomorphic Burnside rings are isomorphic

Introduction. For a finite group G let f2(G) denote its Burnside ring. It is a natural question to ask whether 12(G) determines G up to isomorphism. In [8] Thrvenaz gives an example of two non-isomorphic groups of order p2 q (where p, q are primes and q I P - 1) with isomorphic Burnside rings. Deo and Varadarajan study this question in [2] and [3]. They answer it in the affirmative for a certain subclass of abelian groups; and also for groups in which every subgroup is normal if they assume the extra condition that the isomorphism f2(G) ~ f2(H) maps the semiring f2 + (G) of G-sets onto the semiring f2 § (H) of H-sets.

An application of Burnside rings in elementary finite group theory

A canonical map from the Burnside ring Ω(C) of a finite cyclic group C into the Burnside ring Ω(G) of any finite group G of the same order is exhibited and it is shown that many results from elementary finite group theory, in particular those claiming certain congruence relations, are simple consequences of the existence of this map. In addition, it is shown that this map defines an isomorphism from Ω(C) onto the subring Ω 0(G) of Ω(G), consisting of those virtual G-sets x which have the same number of invariants for every two subgroups U, V of G of the same order, if and only if G is nilpotent. Finally, a rather natural extension to profinite groups is indicated.

The equivariant Hurewicz map

Let G G be a compact Lie group, Y Y be a based G G -space, and V V be a G G -representation. If π V G ( Y ) \pi _V^G(Y) is the equivariant homotopy group of Y Y in dimension V V and H V G ( Y ) H_V^G(Y) is the equivariant ordinary homology group of Y Y with Burnside ring coefficients in dimension V V , then there is an equivariant Hurewicz map \[ h : π V G ( Y ) → H V G ( Y ) . h:\pi _V^G(Y) \to H_V^G(Y). \] One should not expect this map to be an isomorphism, since H V G ( Y ) H_V^G(Y) must be a module over the Burnside ring, but π V G ( Y ) \pi _V^G(Y) need not be. However, here it is shown that, under the obvious connectivity conditions on Y Y , this map induces an isomorphism between H V G ( Y ) H_V^G(Y) and an algebraically defined modification of π V G ( Y ) \pi _V^G(Y) . The equivariant Freudenthal Suspension Theorem contains a technical hypothesis that has no nonequivariant analog. Our results shed some light on the behavior of the suspension map when this rather undesirable technical hypothesis is not satisfied.

A survey of equivariant stable homotopy theory

EQUIVARIANT stable homotopy theory was invented by G. B. Segal in the early 1970s [45]. He was motivated by his work with Atiyah [9] on equivariant K-theory, generalizing an earlier theorem of Atiyah’s on the K-theory of classifying spaces of finite groups to compact Lie groups, and by his work on configuration space and discrete models for iterated loop spaces. His work also suggested to him the “Segal conjecture” (see 4.3), which asserts that the zero-dimensional stable cohomotopy group of the classifying space of a finite group is isomorphic to the completed Burnside ring of the group. The statement is a non-equivariant one, but the methods involved in the eventual proof of the conjecture require heavy use of the equivariant theory. In addition, J. P. May and his collaborators have pointed out that an equivariant version of spectra is the natural device for making spectrum level versions of space level constructions, such as the quadratic or p-adic construction used in defining Steenrod operations. A first attempt to construct an equivariant stable homotopy theory would define the set of stable G-maps (G is finite) from X to Y to be the direct limit l& [Z”X, x” YJG, where the G-actions on Z”X and Z”_Y are obtained by directly suspending the actions on X and Y, and c-1 1’ denotes G-homotopy classes of G-maps. Unfortunately, the theory one obtains this way does not allow for many of the familiar constructions one associates with stable homotopy theory, such as S-duality and transfer. To obtain these, one must allow the group G to act on the “suspension coordinate’*. Precisely, for any representation Vof the group G, one can form the one point compactification S” of V, which becomes a based G-space. S” is a dim( Q-sphere. The stable G-maps from X to Y are now defined to be the direct limit l$ [S’ A X, Sy A ylG, where the actions on Sy A X and S” A Y are diagonal actions, and the direct limit is taken over a certain ordering on the representations, so that V 5 W= V is a summand of IV, This construction permits S-duality and transfer. Other topologists quickly began to study the properties of this theory, notably tom Deck [21]. It gradually became clear that this theory was the natural way to stabilize G-homotopy theory; in this setting, it is possible to construct transfers and S-duality, and to prove an equivariant version of Thorn’s transversality theorem. More naive forms of stabilization do not permit these constructions. Some comments about Frank Adams’ contributions to this subject are in order. He became interested in Segal’s conjecture at an early stage, and in [l] he pointed out the central role played by the spectrum RP ?,” in the study of the conjecture for G = Z/22.

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see l2r, l3r, l4r, l5r and l10r) on the structure of the descent algebras of the Coxeter groups of type An and Bn. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in l14r, is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/WJ, where the WJ are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.

Isomorphic Burnside rings

For any finite group G, let Ω(G) denote the Burnside ring of G. Let A denote the class of finite abelian groups. Let G and H be two groups belonging to the class A . If Ω(G) and Ω(H) are isomorphic as rings, we show that for every prime p, the number of p-subgroups of G is equal to the number of p-subgroups of H. Let A 2 denote the class of finite abelian groups G with the property that for every prime p, the p-primary torsion t p ( G) of G is a direct sum of at most two cyclic groups. For groups G, H in the class A 2 we show that Ω(G) and Ω(H) are isomorphic as rings if and only if G and H are isomorphic as groups. Let Ω + (G) denote the half ring consisting of elements of Ω(G) represented by G-sets. For any prime p, let C ( p) denote the class of finite abelian p-groups. Let G and H be any two groups in C ( p). Then we show that there exists a ring isomorphism α: Ω(G) → Ω(H) further satisfying α( Ω(G) + ) = Ω(H) + if and only if G ∼- H as groups.

Equivariant Functional Duals and Completions at Ideals of the Burnside Ring

Equivariant Functional Duals and Completions at Ideals of the Burnside Ring

Multiplicative homology operations and transfer

In this work, the author presents a completely new treatment of the homology operations QS0. This approach employs the Burnside rings and produces results that are much sharper than previous results. As an application, the author improves Tsuchiya's claim and generalizes Priddy's multiplicative Kahn-Proddy theorem to apply to any prime.

Polynomial operations on stable cohomotopy

Following ideas of G. Segal [24] and [25] we give a conceptual framework for the study of a wide class of unstable operations from the 0-th stable cohomotopy πs0 to other homotopy functors, which we call polynomial operations. Using a generalization of the group completion theorem we prove that it is equivalent to study polynomial operations from πs0 as to study characteristic classes of finite coverings. Finally using the Segal conjecture we give a description of the ring of polynomial operations on πs0 in terms of polynomial operations on Burnside rings.

A categorical approach to the theory of equations

We construct a small category whose objects are monic square-free polynomials with coefficients in a field F. For a monic, irreducible, and normal polynomial, Aut ( f) is the usual Galois group of f. We prove that there exists a unique topological group G such that the category of finite discrete G-sets is equivalent to the opposite of our category. We then replace categories by commutative rings and define the Burnside ring of a field, which has Burnside rings of finite groups as its building blocks. We next extend scalars to the rationals and explicity determine the algebra that results. We find all valuations of this algebra and prove that an irreducible polynomial is completely determined by its values under these valuations.

Burnside rings of finite representation type

Let G be a finite group. It is proved that the localised Buinside ring Ωp(G) is of finite representation type if and only if for each p–perfect subgroup H of G, , where means the conjugacy class of K.

The generalized Burnside ring of a finite group

The generalized Burnside ring of a finite group

Polynomial operations from burnside rings to representation functors

We introduce the concept of polynomial operation from the Burnside ring functor A to other representation functors R, which includes operations such as symmetric powers, φ-operations and Adams operations. The set of polynomial operations from A to R, Pol( A, R), has a canonical ring structure. We give a complete description of the additive structure of this ring as well as a family of generating operations.

A multinomial identity for Witt vectors

Using the close connection between Witt vectors and Burnside rings developed in [A. W. M. Dress and Ch. Siebeneicher, Adv. in Math. 70 (1988), 87–132] we give a simple combinatorial proof of a multinomial identity for Witt vectors, established originally in the special case of universal Witt vectors by H. Gaudier (cf. [H. Gaudier, Actes du 18 ième Sém. Lotharingien Publ. I.R.M.A. Strasbourg (1988), 93–108]) , this way supporting his view that there should be a natural generalization of the theory of generating functions in the context of Witt vectors.

On the unit groups of Burnside rings

On the unit groups of Burnside rings

The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, λ-rings, and the universal ring of Witt vectors

It is shown that well-known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the Burnside ring of the infinite cyclic group on the one hand and Grothendieck's ring of formal power series with constant term 1 as well as the universal ring of Witt vectors on the other hand.