On the modality of parabolic subgroups of linear algebraic groups

For a linear algebraic group G over an algebraically closed field k and a parabolic subgroup P of G the modality of P is defined to be the maximal number of parameters upon which a family of G-orbits on Lie P _u depends and it is denoted by mod P, where P _u is the unipotent radical of P. The principal aim of this note is a generalization of two basic “monotonicity” results from [19] to positive characteristic: (1) If Θ is a semisimple automorphism of G and P is Θ-stable, then mod P ^\Θ≤ mod P. (2) If G is reductive, char k is a good prime for G, and H is a closed reductive subgroup of G normalized by a maximal torus T⊂P of G, then mod (P∩H)≤ mod P. Received: 22 April 1998 / Revised version: 3 July 1998     

On the Isomorphism Classes of Transversals-II

Let G be a finite group and H a subgroup of G. Each right transversal of H in G has a right-quasigroup structure (induced by the binary operation of G). In this article, we prove that the index of H in G is 3 if the number of isomorphism classes of right transversals of H in G is 3, where the isomorphism classes are formed with respect to induced right-quasigroup structures.     

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191-200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L^∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.     

On parabolic subgroups in Coxeter groups of large type

In this paper, properties of the elements of parabolic subgroups in Coxeter groups of large type are considered.     

On the Isomorphism Classes of Transversals

In this article we prove that there does not exist a subgroup H of a finite group G such that the number of isomorphism classes of right transversals of H in G is two.     

Representations of parabolic subgroups of the symplectic group

It is proved that the topological space of irreducible unitary representations of a parabolic subgroup of the complex symplectic group contains an open everywhere dense set homeomorphic to the space of irreducible representations of some reductive group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 438–448, April, 1990.     

On orbit closures of spherical subgroups in flag varieties

Let be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on with finitely many orbits, and let V be an H-orbit closure in . Expanding the cohomology class of V in the basis of Schubert classes defines a union V₀ of Schubert varieties in with positive multiplicities. If G is simply-laced, we show that these multiplicities are equal to the same power of 2. For arbitrary G, we show that V₀ is connected in codimension 1. If moreover all multiplicities are 1, we show that the singularities of V are rational and we construct a flat degeneration of V to V₀ in . Thus, for any effective line bundle L on , the restriction map is surjective, and for all . Received: April 17, 2000     

A note on the modality of parabolic subgroups

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 on which a family of R-orbits on X depends upon.Let G be a simple algebraic group defined over an algebraically closed field K of characteristic 0. Let P be a parabolic subgroup of G. Then P acts on its unipotent radical P_u via conjugation. The modality of P is defined as mod P mod(P : P_u).Let r and s be the semisimple rank of G and P respectively. We show that there is a quadratic polynomial ƒ with rational coefficients such that the modality of P is at least ƒ(r − s). In particular, the modality of a Borel subgroup B of G grows at least quadratically with r. As a consequence, we obtain a finiteness result for algebraic groups from [8]: there is only a finite number of simple algebraic groups admitting parabolic subgroups of prescribed semisimple rank and prescribed modality. Combining our lower bounds with upper bounds from [6], we can compute the modality of Borel subgroups in some small rank cases.     

On a parabolic equation in a triangular domain

We prove the optimal regularity, in Sobolev spaces, of the solution of a parabolic equation set in a triangular domain T. The right-hand term of the equation is taken in Lebesgue space L^p(T). The method of operators sums in the non-commutative case is referred to.     

On SS-quasinormal subgroups and the structure of finite groups

A subgroup H of a finite group G is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B. In this paper, we investigate the structure of a group under the assumption that every subgroup with order pm of a Sylow p-subgroup P of G is SS-quasinormal in G for a fixed positive integer m. Some interesting results related to the p-nilpotency and supersolvability of a finite group are obtained. For example, we prove that G is p-nilpo...     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

On parabolic systems with variational structure

We prove global existence of strong solutions for two-dimensional parabolic systems of variational flow type subject to the one-sided condition.     

Completeness of the parabolic subgroups of projective symplectic groups

All parabolic subgroups and Borel subgroups of PSp(2m, F) over a linearable field F are shown to be complete groups.     

COMPLETENESS OF THE PARABOLIC SUBGROUPS OF PROJECTIVE ORTHOGONAL GROUPS

All parabolic subgroups and Borel subgroups of PΩ(2m 1, F) over a linear-able field F of characteristic 0 are shown to be complete groups, provided m > 3.     

Orbits of parabolic subgroups on metabelian ideals

Let k be an algebraically closed field, t∈Z?1, and let B be the Borel subgroup of GLt(k) consisting of upper-triangular matrices. Let Q be a parabolic subgroup of GLt(k) that contains B and such that the Lie algebra qu of the unipotent radical of Q is metabelian, i.e. the derived subalgebra of qu is abelian. For a dimension vector with , we obtain a parabolic subgroup P(d) of GLn(k) from B by taking upper-triangular block matrices with (i,j) block of size di×dj. In a similar manner we obtain a parabolic subgroup Q(d) of GLn(k) from Q. We determine all instances when P(d) acts on qu(d) with a finite number of orbits for all dimension vectors d. Our methods use a translation of the problem into the representation theory of certain quasi-hereditary algebras. In the finite cases, we use Auslander-Reiten theory to explicitly determine the P(d)-orbits; this also allows us to determine the degenerations of P(d)-orbits.     

On the structure of diffuse measures for parabolic capacities

Let $Q=(0,T)\times \mathrm{\Omega }$, where Ω is a bounded open subset of ${\mathbb{R}}^d$. We consider the parabolic p-capacity on Q naturally associated with the usual p-Laplacian. Droniou, Porretta, and Prignet have shown that if a bounded Radon measure μ on Q is diffuse, i.e. charges no set of zero p-capacity, $p>1$, then it is of the form $\mu =f+\text{div}(G)+g_t$ for some $f\in L^1(Q)$, $G\in {(L^{p^{\prime }}(Q))}^d$ and $g\in L^p(0,T;W_0^{1,p}(\mathrm{\Omega })\cap L^2(\mathrm{\Omega }))$. We show the converse of this result: if $p>1$, then each bounded Radon measure μ on Q admitting such a decomposition is diffuse.