Extensions and the Induced Characters of Cyclic p-Groups

For a prime p, we denote by B_n the cyclic group of order pⁿ. Let φ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character φ^G is also irreducible. The purpose of this article is to determine the subgroup N_G(N_G(B_n)) of G under the hypothesis [N_G(B_n):B_n]⁴ ≦ pⁿ.     

Algorithmic Modality Analysis for Parabolic Groups

For an algebraic group R acting morphically on an algebraic variety X the modality of the action, mod (R:X), is the maximal number of parameters upon which a family of R-orbits on X depends. Let G be a reductive algebraic group defined over an algebraically closed field K. Let P be a parabolic subgroup of G. Then P acts on its unipotent radical P_u via conjugation and on , the Lie algebra of P_u, via the adjoint action. The modality of P is defined as mod P:=mod (P: ). In this paper we discuss an algorithm which is used to compute upper bounds for mod P along with some results obtained by this algorithm. One is a classification of parabolic subgroups P of simple algebraic groups G of semisimple rank 2 and modality 0. For parabolic subgroups of semisimple rank 3 we present some partial results. This extends the results of Kashin and Popov and Röhrle, where the cases of semisimple rank 0 and 1 are handled. For exceptional groups G we show that P G has modality zero provided the class of nilpotency of P_u is at most two. The analogous result for classical groups is proved by Röhrle. For Borel subgroups B of simple groups we are able to determine the value for mod B in some small rank cases by combining lower bounds for mod B of Röhrle with upper bounds provided by the algorithm.     

Sub-principal homomorphisms in positive characteristic

 Let G be a reductive group over an algebraically closed field of characteristic p, and let uG be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism φ:SL _2/k →G whose image contains u. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group (and using, in some cases, computer calculations with explicit representatives for the unipotent orbits). The proof we give is free of case considerations (except in its dependence on the Bala-Carter theorem). Our construction of φ generalizes the construction of a principal homomorphism made by J.-P. Serre in (Invent. Math. 1996); in particular, φ is obtained by reduction modulo 𝔭 from a homomorphism of group schemes over a valuation ring 𝒜 in a number field. This permits us to show moreover that the weight spaces of a maximal torus of φ(SL _2/k ) on Lie(G) are ``the same as in characteristic 0'; the existence of a φ with this property was previously obtained, again using case considerations, by Lawther and Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G. Seitz (Invent. Math. 2000). Received: 1 February 2002; in final form: 17 June 2002 / Published online: 1 April 2003 The author was supported in part by a grant from the National Science Foundation.     

Homogeneous Vector Bundles and Koszul Algebras

Let G be a reductive algebraic group defined over an algebraically closed field of characteristic zero and let P be a parabolic subgroup of G. We consider the category of homogeneous vector bundles over the flag variety G/P. This category is equivalent to a category of representations of a certain infinite quiver with relations by a generalisation of a result in [BK]. We prove that both categories are Koszul precisely when the unipotent radical P_u of P is abelian.     

Reality properties of conjugacy classes in<Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

LetG be an algebraic group over a fieldk. We callg εG(k) real ifg is conjugate tog ⁻¹ inG(k). In this paper we study reality for groups of typeG ₂ over fields of characteristic different from 2. LetG be such a group overk. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element inG(k) is real if and only if it is a product of two involutions inG(k). Every unipotent element inG(k) is a product of two involutions inG(k). We discuss reality forG ₂ over special fields and construct examples to show that reality fails for semisimple elements inG ₂ over ℚ and ℚ_p. We show that semisimple elements are real forG ₂ overk withcd(k) ≤ 1. We conclude with examples of nonreal elements inG ₂ overk finite, with characteristick not 2 or 3, which are not semisimple or unipotent.     

The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball

We study extremal functions for a family of Poincaré-Sobolev-type inequalities. These functions minimize, for subcritical or critical p?2, the quotient ‖∇u_‖2/‖up_‖ among all u∈H¹(B)?{0} with B_∫u=0. Here B is the unit ball in RN. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of B. We also prove that, for p close to 2, minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis, and that, once the symmetry axis is fixed, they are unique (up to multiplication by a constant). In space dimension two, we prove that minimizers are not antisymmetric for large p.     

The lattice automorphisms of simple algebraic groups over\bar F_2

We prove that if u is a unipotent element of a connected reductive algebraic group G over , there exists an involution σ in G such that σuσ=u⁻¹. We use this result to determine the group of lattice automorphisms of G, when G is simple.     

Unique Node Extremal Subgroups in Chevalley Groups

Abstract

Taking G to be a Chevalley group of rank at least 3 and U to be the unipotent radical of a Borel subgroup B,an extremal subgroup A is an abelian normal subgroup of U which is not contained in the intersection of all the unipotent radicals of the rank 1 parabolic subgroups of G containing B. If there is an unique rank 1 parabolic subgroup P of G containing B with the property that A is not contained in the unipotent radical of P,then A is called a unique node extremal subgroup. In this paper we investigate the embedding of unique node extremal subgroups in U and prove that,apart from some specified cases,such a subgroup is contained in the unipotent radical of a certain maximal parabolic subgroup.     

A MAXIMAL CLASS OF ABELIAN GROUPS ARISING FROM THE ISOMORPHISM G = T[EXT[G.X)]

Abstract

We characterize the class of groups G having the property that G ? T[Ext(G,X)] for some group X. We show that for every prime p, the p-component of such a group G has the form T [π∞_n=1 Z(pn)^mpn] where m_pn is finite for every prime p and natural number n.     

Optimal Ambiguity Functions and Weil’s Exponential Sum Bound

Complex-valued periodic sequences, u, constructed by G?ran Bj?rck, are analyzed with regard to the behavior of their discrete periodic narrow-band ambiguity functions A _p (u). The Bj?rck sequences, which are defined on ?/p? for p>2 prime, are unimodular and have zero autocorrelation on (?/p?)?{0}. These two properties give rise to the acronym, CAZAC, to refer to constant amplitude zero autocorrelation sequences. The bound proven is $|A_{p}(u)| \leq2/\sqrt{p} + 4/p$ outside of (0,0), and this is of optimal magnitude given the constraint that u is a CAZAC sequence. The proof requires the full power of Weil??s exponential sum bound, which, in turn, is a consequence of his proof of the Riemann hypothesis for finite fields. Such bounds are not only of mathematical interest, but they have direct applications as sequences in communications and radar, as well as when the sequences are used as coefficients of phase-coded waveforms.     

On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups

Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms} we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F_p. Let q be a power of p and let G(q) be the finite group of F_q-rational points of G. Let F be the Frobenius morphism such that G(q) = G^F. Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) are in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).     

Controlling the space with preassigned responses

It is shown that large classes of control systems, which include certain systems of the typex+A(t)x=B(t)u, can be handled in such a way that the control functionsu(t) are actually associated with responsesx(t) that belong to known families of functions. In particular, it is possible, for a variety of perturbationsB(t)u and operatorsA(t) with convex domains, to have responses that are line segments joining the origin to the reachable states.The present approach establishes the fact that a vast number of results from functional analysis concerning ranges of operators can be effectively applied to the general theory of control. It is also rather significant that the present theory does not necessarily require the solvability of the associated Cauchy problem.The operatorsB(t)u do not have to be invertible inu. However, it is shown that continuous controlsu(t) can be obtained for a variety of problems whenB ^–1(t)u exists and is continuous int.     

Second cohomology groups for Frobenius kernels and related structures

Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U) with coefficients in k.     

A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical

LetG be a classical algebraic group defined over an algebraically closed field. We classify all instances when a parabolic subgroupP ofG acts on its unipotent radicalP _u , or onp _u , the Lie algebra ofP _u , with only a finite number of orbits.The proof proceeds in two parts. First we obtain a reduction to the case of general linear groups. In a second step, a solution for these is achieved by studying the representation theory of a particular quiver with certain relations.Furthermore, for general linear groups we obtain a combinatorial formula for the number of orbits in the finite cases.     

The Ext-group of unitary equivalence classes of unital extensions

Let A be a unital separable nuclear C*-algebra which belongs to the bootstrap category N and B be a separable stable C*-algebra. In this paper, we consider the group Ext _u (A, B) consisting of the unitary equivalence classes of unital extensions τ: A→Q(B). The relation between Ext _u (A, B) and Ext(A, B) is established. Using this relation, we show the half-exactness of Ext _u (·, B) and the (UCT) for Ext _u (A, B). Furthermore, under certain conditions, we obtain the half-exactness and Bott periodicity of Ext _u (A, ·).     

Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces

We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=G _R with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {u _t} ofG and anyg∈G, the time spent inC by the {u _t}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g ⁻¹u_tg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].     

Geometric realization of Adams maps

Let G/P be a homogenous space with G a compact connected Lie group and P a connected subgroup of G of equal rank. As the rational cohomology ring of G/P is concentrated in even dimensions, for an integer k we can define the Adams map of type k to be l _k : H*(G/P, ℚ) → H*(G/P, ℚ), l _k (u) = k ⁱ u, u ∈ H ²ⁱ (G/P, ℚ). We show that if k is prime to the order of the Weyl group of G, then l _k can be induced by a self map of G/P. We also obtain results which imply the condition that k is prime to the order of the Weyl group of G is necessary.     

Laterally closed lattice homomorphisms

Let A and B be two Archimedean vector lattices and let be a lattice homomorphism. We call that T is laterally closed if T(D) is a maximal orthogonal system in the band generated by T(A) in B, for each maximal orthogonal system D of A. In this paper we prove that any laterally closed lattice homomorphism T of an Archimedean vector lattice A with universal completion Au into a universally complete vector lattice B can be extended to a lattice homomorphism of Au into B, which is an improvement of a result of M. Duhoux and M. Meyer [M. Duhoux and M. Meyer, Extended orthomorphisms and lateral completion of Archimedean Riesz spaces, Ann. Soc. Sci. Bruxelles 98 (1984) 3-18], who established it for the order continuous lattice homomorphism case. Moreover, if in addition Au and B are with point separating order duals ^′(Au) and B^′ respectively, then the laterally closedness property becomes a necessary and sufficient condition for any lattice homomorphism to have a similar extension to the whole Au. As an application, we give a new representation theorem for laterally closed d-algebras from which we infer the existence of d-algebra multiplications on the universal completions of d-algebras.     

On the Dual of the Procesi–Formanek Lattice

Let G be the symmetric group on n letters. Procesi and Formanek have shown that C _n , the center of the generic division algebra of degree n defined over a field F, is stably isomorphic to F(B_n)^GF(B_{n})^{G} where B _n is a specific ZG-lattice. We refer to B _n as the Procesi–Formanek lattice. The question of the stable rationality of C _n is a long standing problem for which few results are known. Let F be an algebraically closed field of characteristic zero, let p be an odd prime, and let B_p^*=Hom_Z(B_p,Z)B_{p}^{*}=Hom_{Z}(B_{p},Z) be the dual of the Procesi–Formanek lattice. We show that F(B_p^*)^GF(B_{p}^{*})^{G} is stably rational over F. An interesting question is whether there exists a connection between C _p and F(B_p^*)^GF(B_{p}^{*})^{G}.     

Three-term recurrence relations with matrix coefficients. The completely indefinite case

In the space {ol _p ² of vector sequences, we consider the symmetric operatorL generated by the expression (lu)j:=Bj ^uj+1+Aj ^uj+ _B ^j−1/* uj−1, whereu−1 = 0,u ₀,u ₁, … ∈ ℂ ^p ,A _j andB _j arep × p matrices with entries from ℂ,A _j ^* =A_j, and the inversesB _j ⁻¹ (j = 0, 1, …) exist. We state a necessary and sufficient condition for the deficiency numbers of the operatorL to be maximal; this corresponds to the completely indefinite case for the expressionl. Tests for incomplete indefiniteness and complete indefiniteness forl in terms of the coefficientsA _j andB _j are derived. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 709–716, May, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00333.