Compact Linear Group Research Articles

Topological and Homological Properties of the Orbit Space of a Simple Three-Dimensional Compact Linear Lie Group

Topological and Homological Properties of the Orbit Space of a Simple Three-Dimensional Compact Linear Lie Group

Representation growth of compact special linear groups of degree two

We study the finite-dimensional continuous complex representations of SL2 over the ring of integers of non-Archimedean local fields of even residual characteristic. We prove that for characteristic two, the abscissa of convergence of the representation zeta function is 1, resolving the last remaining open case of this problem. We additionally prove that, contrary to the expectation, the group algebras of C[SL2(Z/(22rZ))] and C[SL2(F2[t]/(t2r))] are not isomorphic for any r>1. This is the first known class of reductive groups over finite rings wherein the representation theory in the equal and mixed characteristic settings is genuinely different. From our methods, we explicitly obtain the primitive representation zeta polynomials of SL2(F2[t]/(t2r)) and SL2(Z/22rZ) for 1≤r≤3.

Topological and homological properties of the orbit space of a simple three-dimensional compact linear Lie group

Topological and homological properties of the orbit space of a simple three-dimensional compact linear Lie group

Local newforms for the general linear groups over a non-archimedean local field

AbstractIn [14], Jacquet–Piatetskii-Shapiro–Shalika defined a family of compact open subgroups ofp-adic general linear groups indexed by nonnegative integers and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of nonnegative integers. For the proof, we introduce the Rankin–Selberg integrals for Speh representations.

Стационарные подалгебры общего положения для тензорных произведений

The paper studies stationary subalgebras in general position of compact linear groups. We prove that, except for several specific cases, a stationary subalgebra in general position of a tensor product of real or complex compact group representations acts as a scalar on all tensor factors but possibly one. In the real case, it means that this stationary subalgebra in general position is contained in one of the direct summand subalgebras. We used the following concepts to solve this problem: conventional linear algebra arguments; theory of Lie groups, Lie algebras and their representations; and methods similar to those of solving similar problems for complex reductive linear groups.

Representation growth of compact linear groups

We study the representation growth of simple compact Lie groups and of SLn(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GLn(O). This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is r/κ, where r is the rank and κ the number of positive roots. We then show that the twist zeta function of GLn(O) exists and has the same abscissa of convergence as the zeta function of SLn(O), provided n does not divide char O. We compute the twist zeta function of GL2(O) when the residue characteristic p of O is odd and approximate the zeta function when p = 2 to deduce that the abscissa is 1. Finally, we construct a large part of the representations of SL2(Fq[[t]]), q even, and deduce that its abscissa lies in the interval [1, 5/2].

Топологические и гомологические свойства пространства орбит компактной линейной группы Ли с коммутативной связной компонентой. Выводы

Топологические и гомологические свойства пространства орбит компактной линейной группы Ли с коммутативной связной компонентой. Выводы

Топологические и гомологические свойства пространства орбит компактной линейной группы Ли с коммутативной связной компонентой

Топологические и гомологические свойства пространства орбит компактной линейной группы Ли с коммутативной связной компонентой

The existence of a polynomial factorization map for some compact linear groups

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected components, and its representation should be the direct sum of three irreducible two-dimensional real representations at least two of them being faithful.

ON THE MODEL THEORY OF THE LOGARITHMIC FUNCTION IN COMPACT LIE GROUPS

Given a compact linear Lie group G, we form a natural expansion of the theory of the reals where G and the graph of a logarithm function on G live. We prove its effective model-completeness and decidability modulo a suitable variant of Schanuel's Conjecture.

On the orbit space of a three-dimensional compact linear Lie group

We study the question of whether the topological quotient of a real linear representation of a simple three-dimensional compact Lie group is a manifold. We obtain an upper bound for the dimension of a representation whose quotient is a manifold, and examine most of the remaining cases.

Space of orbits of a three-dimensional simple compact linear Lie group

An upper estimate for the codimension of the subspace of points with finite orbits for a representation of a three-dimensional simple compact Lie group whose quotient is a manifold is obtained.

On the orbit space of a compact linear Lie group with commutative connected component

This paper is devoted to the study of topological quotients of compact linear Lie groups. More precisely, it investigates the question of when such a quotient is a topological or a smooth manifold. The topological quotient of a finite linear group was studied by Mikhailova in 1984. Here the connected component of the original Lie group G is assumed to be a torus of positive dimension.

Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups

We review the proposal of a constructive axiomatic approach to the determination of the orbit spaces of all the real compact linear groups, obtained through the computation of a metric matrix \(\widehat{P}(p)\) , which is defined only in terms of the scalar products between the gradients ∂p1(x),. . .,∂pq(x) of the elements of a minimal integrity basis (MIB) for the ring ℝ[ℝn]G of G-invariant polynomials. The domain of semi-positivity of \(\widehat{P}(p)\) is known to realize the orbit space ℝn/G of G as a semi-algebraic variety in the space ℝq spanned by the variables p1,. . .,pq.

On the local equivalence of homogeneous CR-manifolds

For certain classes of homogeneous Cauchy-Riemann manifolds it is shown that the local CR-equivalence problem is equivalent to the global CR-equivalence problem. Among these are large classes of compact linear group orbits.

Bounded symmetric domains and polynomial convexity

For every bounded symmetric domain D=G/K in a complex vector space E of finite dimension we determine all polynomially convex compact subsets of E that are invariant under the compact linear group K, where D is realized in the standard way as circular convex domain in E.

Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata

Functions which are equivariant or invariant under the transformations of a compact linear group $G$ acting in an euclidean space $\real^n$, can profitably be studied as functions defined in the orbit space of the group. The orbit space is the union of a finite set of strata, which are semialgebraic manifolds formed by the $G$-orbits with the same orbit-type. In this paper we provide a simple recipe to obtain rational parametrizations of the strata. Our results can be easily exploited, in many physical contexts where the study of equivariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau theory), in equivariant bifurcation theory, in crystal field theory and in most areas where use is made of symmetry adapted functions. A physically significant example of utilization of the recipe is given, related to spontaneous polarization in chiral biaxial liquid crystals, where the advantages with respect to previous heuristic approaches are shown.

Linear embeddings of semialgebraic G -spaces

Let G be a compact semialgebraic linear group. We prove that every regular semialgebraic G-space admits a semialgebraic G-embedding into some semialgebraic orthogonal representation space of G.

Orbit spaces of compact coregular simple Lie groups with 2, 3 and 4 basic polynomial invariants: Effective tools in the analysis of invariant potentials

The coregular real representations of the compact simple Lie groups are classified and the equalities and inequalities defining their orbit spaces are determined in the case of 2, 3 and 4 basic polynomial invariants. The results are obtained making use of the classification of the complex coregular simple Lie groups given by G. W. Schwarz and of the determination of all the allowable metric matrices for q-dimensional (q⩽4) orbit spaces of compact coregular linear groups, recently obtained by the present authors. The results are used to determine in a rigorous way the minima of two SO(10) and E6 invariant Higgs potentials.

Orbit spaces of reflection groups with 2, 3 and 4 basic polynomial invariants

Functions which are covariant or invariant under the transformations of a compact linear group can be expressed advantageously in terms of functions defined in the orbit space of the group, i.e. as functions of a finite set of basic invariant polynomials. The equalities and inequalities defining the orbit spaces of all finite coregular real linear groups (most of which are crystallographic groups) with at most four independent basic invariants are determined. For each group G acting in the Euclidean space , the results are obtained through the computation of a metric matrix , which is defined only in terms of the scalar products between the gradients of a set of basic polynomial invariants of G; the semi-positivity conditions are known to determine all the equalities and inequalities defining the orbit space of G as a semi-algebraic variety in the space spanned by the variables . In a recent paper, the -matrices, for , have been determined in an alternative way, as solutions of a universal differential equation; the present paper yields a partial, but significant, check on the correctness and completeness of these solutions. Our results can be easily exploited, in many physical contexts where the study of covariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and in most areas of solid-state theory where use is made of symmetry adapted functions.