Showing papers on "Equivariant map published in 1975"

Reductive groups are geometrically reductive

Abstract: Let G be a semi-simple algebraic group over an algebraically closed field, k. Let G act rationally by automorphisms on the finitely generated k-algebra, R. The problem of proving that the ring of invariants, RG, is finitely generated originates with the invariant theorists of the nineteenth century. When k = C, the complex numbers, and G GL (n, C) the question is answered affirmatively by Hilbert's "fundamental theorem of invariant theory". The proof involved constructing a G equivariant projection from R to RG and then using it to prove the result algebraically. When k is of characteristic 0 and G is any semi-simple group, by a theorem of H. Weyl, every finite dimensional representation of G is completely reducible. In the 1950's D. Mumford and others (Cartier, Iwahori, Nagata) applied Weyl's theorem to construct a projection from R to RG for any semi-simple group. This made it possible to generalize Hilbert's proof to an arbitrary semi-simple group. Certain geometric applications, particularly to the theory of moduli, made a generalization to groups over fields of positive characteristic highly desirable. In positive characteristic, complete reducibility definitely fails. Hence attempts were made to replace complete reducibility with a weaker condition which would at once hold for all semi-simple groups and make a proof of finite generation of RG possible. The weakest way to state complete reducibility is the following. If V is a finite dimensional G-module containing a G-stable sub-space of co-dimension one, VT, then there is a G-stable line LC V such that V0 E L = V. Mumford conjectured a weaker version of this statement by seeking a complement only in a higher symmetric power of V, SI( V). This is the conjecture as it is stated in the preface to [16]:

Induction and structure theorems for orthogonal representations of finite groups

Abstract: The equivariant Witt ring GW(, R) of a finite group w over a Dedekind domain R is studied. It is shown that-modulo the prime 2-GW(r, Z) equals the character ring of real representations of w and GW(w, R) equals GW(w, Z) ( W(R). From this, induction theorems a la E. Artin and R. Brauer are derived for GW(-, R) and it is shown how these can be applied towards the computation of L-groups.

On equivariant vector bundles on an almost homogeneous variety

Abstract: Let k be an algebraically closed field of arbitrary characteristic. Let T be an n-dimensional algebraic torus, i.e. T = Gm × · · · × Gm n-times), where Gm = Spec (k[t, t-1]) is the multiplicative group.

Equivariant homology theories on $G$-complexes

Abstract: A definition is given for a "cellular" equivariant homology theory on G-complexes. The definition is shown to generalize to G-complexes with prescribed isotropy subgroups. A ring I is introduced to deal with the general definition. One obtains a universal coefficient theorem and studies the universal coefficients.

Characterizations of Distributions by Statistical Properties on Groups

Abstract: The article investigates some characterization problems occurring in mathematical statistics when the sample space is a homogeneous space and a transformation parameter distribution family is considered. We give general form of positive and continuous probability density admitting nontrivial sufficient statistics for this parameter. Conditions of optimality of an equivariant estimators are obtained and some corresponding characterization theorems are stated. We show also that under mild restrictions the transformation parameter family is characterized by distributions of invariant statistics.

Bayes equivariant estimators in a crossed classification random effects model

Abstract: The Bayes equivariant estimators of the variance components in the two-way crossed classification random effects model withK (K=>1) observations per cell are characterized under the usual assumptions of normality and independence of the random effects. An illustrative example of non-trivial Bayes equivariant estimators derived using a special prior distribution is provided. It is pointed out that for the squared error loss function every Bayes equivariant estimator of the residual variance component is inadmissible.

Semifree actions on homotopy spheres

Abstract: In this paper, we study the semifree Z actions on homotopy sphere pairs. We show that in some cases the equivariant normal bundle to the fixed point set is equivariantly stably trivial. We compute the rank of the torsion free part of the group of semifree actions on homotopy sphere pairs in some cases. We also show that there exist infinitely many semifree Z4s actions on even dimensional homotopy sphere pairs. 0. Introduction. Let G be a compact Lie group. A differentiable group action of G on a differentiable manifold Mn is a homomorphism i/i: GDiff (M), where Diff (M) is the group of diffeomorphisms of M. Let Fk be the submanifold of fixed points. A group action is semifree if the only isotropy subgroups are the trivial subgroup and the group itself. Under these restrictions, the action is linear in some neighborhood of Fk in M' in the sense that there is an equivariant vector bundle v normal to Fk in M' such that the action restricts on each fiber of v to a linear automorphism. In this paper, we are only interested in the differentiable semifree actions on homotopy sphere pairs, namely, the actions on homotopy spheres such that the fixed point sets are homotopy spheres. The action G = S1 defines a complex structure on v (the action on v is just that induced by the complex multiplication). In [2], Browder proved that for a semifree S1 action on a homotopy sphere pair, the normal bundle of the fixed point set is stably trivial as a complex vector bundle. The first question which interests us is the following: Problem 1. Is it true that the equivariant normal bundle to the fixed point set of a semifree Zm action on a homotopy sphere pair is equivariantly stably trivial? If G = Sl or Zm, it is known (see [2], [6]) that there are infinitely many semifree G actions on odd dimensional homotopy sphere pairs. As to semifree S1 actions on even dimensional homotopy sphere pairs, there are only finitely many [3]. Received by the editors February 22, 1974 and, in revised form, July 9, 1974. AMS (MOS) subject classifications (1970). Primary 57E25, 57E30.

Equivariant maps between representation spheres.

Abstract: Let G be a finite group, V and W be finite representations of G, S(V) and S(W) be the unit spheres in V and W. Suppose dim VH ^dim WH for every subgroup H of G. We seek to classify the G- equivariant homotopy classes of 6-equivariant maps from S(V) to S(W). Introduction. We wish to consider the following problem: Let G be a finite group, and V and W be finite dimensional orthogonal representations of G. Let 5(V) and S(W) denote the unit spheres of V and W respectively. Then S(V) and S(W) inherit G-actions. Classify the equivariant homotopy classes of G-maps from S(V) to S(W). The case where G = Zp and S(V) and S(W) have free Zp actions was done by Olum (6) and was used to give a classification of lens spaces up to homotopy equivalence. In this paper we generalize Olum's result. Our approach is to consider the behavior of an equivariant map restricted to the various fixed point sets. Explicity, if X is a space with a left action of the group G, and H is a subgroup of G, we denote by XH the set of points in X left fixed by each element of H. If /: S(V)-*S(W) is a G- equivariant map (i.e., f(gυ) = gf(v) for all g G G and veS(V)), then / induces maps /": S(VH->S(W)H for each subgroup H. Since V and W are linear representations, these fixed point sets S(V)H and S(W)H are again spheres, and we may choose an orientation for each S(V)H and S(W)H. If X is a manifold, denote by dimX the (real) dimension of X. When dimS( V)H = dimS(HOH, fH has a well-determine d degree, denoted by deg/ H. Our major theorem asserts that, under suitable hypotheses, the homotopy classes of the maps fH for all H determine the equivariant homotopy class of /: DEFINITION. Let G be a finite group. If H is a subgroup of G, denote by N(H) the normalizer of H in G. An orthogonal representa- tion V of G is completely orientable if for every subgroup H of G, the induced action of N(H) on S(V)H is orientation-p reserving.

On the homotopy groups of some equivariant automorphism groups of spheres

Abstract: © Foundation Compositio Mathematica, 1975, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

The Families with a “Universal” Location Estimator

Abstract: The problem of estimating the multidimensional location parameter is considered. The new family of multivariate distribution is characterized by the property of the independence of the best equivariant estimator on the even loss function choice. Some generalizations of known functional equations are introduced and solved for this aim.

On the extension of equivariant mappings

Abstract: It is proved that a metric space with a homeomorphism of prime period is an equivariant extensor for metric spaces with -dimensional complement if and only if the space itself and the set of fixed points of the homeomorphism are homotopically connected and locally homotopically connected in dimension .Bibliography: 9 titles.