Semigroup Actions on Sets and the Burnside Ring

Burnside rings

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.

Suppose that n is not a prime power and not twice a prime power. We prove that for any Hausdorff compactum X with a free action of the symmetric group $${\mathfrak {S}}_n$$ , there exists an $${\mathfrak {S}}_n$$ -equivariant map $$X \rightarrow {{\mathbb {R}}}^n$$ whose image avoids the diagonal $$\{(x,x,\dots ,x)\in {{\mathbb {R}}}^n\mid x\in {{\mathbb {R}}}\}$$ . Previously, the special cases of this statement for certain X were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $$\mathfrak S_n$$ -equivariant maps from the boundary $$\partial \Delta ^{n-1}$$ of $$(n-1)$$ -simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser’s conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.

Equivariant semicharacteristics and induction

Actions of inverse semigroups arising from partial actions of groups

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrix V and (tr V ) α , where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators of V and tr( V ) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.

We describe the structure of E -dense acts over E -dense semigroups in an analogous way to that for inverse semigroup acts over inverse semigroups. This is based, to a large extent, on the work of Schein on representations of inverse semigroups by partial one-to-one maps. We also study cancellative actions of semigroups as a type of generalisation of group actions and characterise locally free cancellative acts over E -dense semigroups that are also E -unitary.

A theorem of Ostrowski implies that log(x), log(x +1), … are algebraically independent over ℂ(x). More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution y and particular transformations of y, such as derivatives of y with respect to parameters, shifts of the arguments, rescaling, etc. In this paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a ûnite collection of axioms that characterizes such actions to apply them to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are ûnite free products of semigroups of the form on Tannakian categories. This is the class of semigroups that appear in many applications.

In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-nets) constitute a useful framework with connections in both group theory and graph theory. Recent attempts at formalizing K-nets in their most general form have evidenced a deeper connection with category theory. These formalizations use diagrams in sets, i.e. functors where is often a small category, providing a general framework for the known group or monoid actions on musical objects. However, following the work of Douthett–Steinbach and Cohn, transformational music theory has also relied on the use of relations between sets of the musical elements. Thus, K-net formalizations should be extended further to take this aspect into account. The present article proposes a new framework called relational PK-nets, an extension of our previous work on poly-Klumpenhouwer networks (PK-nets), in which we consider diagrams in rather than . We illustrate the potential of relational PK-nets with selected examples, by analyzing pop music and revisiting the work of Douthett–Steinbach and Cohn.

We show that for a partial action η with closed domain of a compact group G on a profinite space X the space of orbits X/~G is profinite, this leads to the fact that when G is profinite the enveloping space XG is also profinite. Moreover, we provide conditions for the induced quotient map πG : X → X / ∼G of η to have a continuous section. Relations between continuous sections of πG and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

Let G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

Let G be a compact Lie group. Let X be a Hausdorff compact G-space which is a cohomology sphere over a ring R (cf. Def. 1.1). In Section 1 we define the Euler class of a locally trivial bundle with X as fiber. Using this definition we show that the main result of [7] still holds if the sphere of an orthogonal representation is replaced by a G-space which is a cohomology sphere over R (Ths. 1.8, 1.9). In the second section we consider actions of a finite cyclic group G = Ck. For this group, and X as above, an index of X has been defined in [1] and [2]. First, with the use of the Euler class we define an index of X (cf. [7] for the case where X is the sphere of an orthogonal representation; see Def. 2.2). Next we prove that this index is equal to that introduced by Borisovich and Izrailevich (Th. 2.3), and consequently we obtain a simple geometrical interpretation of the index defined in [1]. Finally, using our definition of the index (via the Euler class) we compute its value for X = S(V ), the sphere of an orthogonal representation V of G = Ck (Th. 3.3). This allows us to find for which V this index is different from 0. From the results of [7] and those quoted above we derive a formula on the degree (modulo k) of a G-equivariant map f : S(W ) → S(V ) between the spheres of representations W and V of G = Ck (Prop. 3.11) (cf. [3], [8]).

We give a unifying description of the Dirac monopole on the 2-sphere S2, of a graded monopole on a (2, 2)-supersphere S2, 2 and of the BPST instanton on the 4-sphere S4, by constructing a suitable global projector p via equivariant maps. This projector determines the projective modules of finite type of sections of the corresponding vector bundle. The canonical connection ∇ = p ◦ d is used to compute the topological charge which is found to be equal to -1 for the three cases. The transposed projector q = pt gives the value +1 for the charges; this showing that transposition of projectors, although an isomorphism in K-theory, is not the identity map. We also study the invariance under the action of suitable Lie groups.

This paper is concerned with Higher Algebraic K-theory and actions of algebraic groups G on such ‘nice’ categories as the category of algebraic vector bundles on a scheme X. Such ‘nice’ categories are examples of ‘exact’ categories with the observation that the category of actions on G on such exact categories also form an exact category called equivariant exact categories on which one can do higher Algebraic K-theory (of Quillen) called equivariant higher Algebraic K-theory—the higher dimensional generalizations of classical equivariant K-theory which belongs to the field of representation theory. Thus, for an Algebraic group G over a number field or p-adic field F, we present constructions and computations of equivariant higher K-groups as well as ‘profinite’ or ‘continuous’ higher K-groups for some G-Scheme X. In particular, we present explicit l-completeness (l a rational prime) and finiteness computations for higher K-groups and profinite higher K-groups for twisted flag varieties.

On a categorical framework for classifying C⁎-dynamics up to cocycle conjugacy