Irreducible Fuchsian system with reducible monodromy representation

We present an example of the reducible representation χ = χ₁ ? χ₂, which, on the one hand, is the monodromy representation of a Fuchsian system. On the other hand, the representation χ₂ is a counterexample to the Riemann-Hilbert problem. Using a meromorphic gauge transformation, one cannot reduce this system to the direct sum of Fuchsian systems corresponding to the subrepresentations.     

Construction of a Fuchsian equation from a monodromy representation

Translated from Matematicheskie Zametki, Vol. 48, No. 5, pp. 22–34, November, 1990.     

The annihilation theorem for the completely reducible Lie superalgebras

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1] a family of such “singular” Verma modules for osp(1,2l) cases. In this article we give a necessary and sufficient condition on the highest weight of a osp(1,2l)-Verma module for its annihilator to be generated by its intersection with the centre. This answers a question of Musson. The classical proof of the Duflo theorem is based on a deep result of Kostant which uses some delicate algebraic geometry reasonings. Unfortunately these arguments can not be reproduced in the quantum and super cases. This obstruction forced Joseph and Letzter, in their work on the quantum case (see [JL]), to find an alternativeapproach to the Duflo theorem. Following their ideas, we compute the factorization of the Parthasarathy–Ranga-Rao–Varadarajan (PRV) determinants. Comparing it with the factorization of Shapovalov determinants we find, unlike to the classical and quantum cases, that the PRV determinant contains some extrafactors. The set of zeroes of these extrafactors is precisely the set of highest weights of Verma modules whose annihilators are not generated by their intersection with the centre. We also find an analogue of Hesselink formula (see [He]) giving the multiplicity of every simple finite dimensional module in the graded component of the harmonic space in the symmetric algebra. Oblatum 1-IX-1998 & 4-XII-1998 / Published online: 10 May 1999     

On almost reducible systems with almost periodic coefficients

Let x=A(t)x be a system of two linear ordinary differential equations with almost periodic coefficients. Then there exists for any positive ε an almost reducible system of equations x=B(t)x with almost periodic coefficients and such that sup ∥A(t)?B (t)∥<ε.-∞相似文献   

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H ? GL(V) acting completely reducibly on a vector space V: if the H-orbits containing the vectors a and b have coprime lengths m and n, we prove that the H-orbit containing a + b has length mn. Such groups H are always reducible if n,m > 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor. In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O’Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.     

Measure of reducible difference systems

We consider a difference system with quasiperiodic coefficientsx _n+1=Ax _n+P(_n)x_n, _n+1=_n+ An estimate is obtained of the measure of the set of those matrices A for which the given system is reducible to a system with constant coefficients.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1572–1575, November, 1990.     

Coleman automorphisms of holomorphs of completely reducible and almost simple groups

Let H be the holomorph of a finite group G. It is proved that every Coleman automorphism of H is inner whenever G is either completely reducible or almost simple; in particular, this is the case when G is either characteristically simple or simple. As an application, we obtain the normalizer the conjecture holds for integral group rings of holomorphs of such groups in question.     

Two-dimensional Fuchsian systems and the Chebyshev property

Let (x(t),y(t))^? be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form P(t)x(t)+Q(t)y(t), where P,Q are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behavior of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a non-oscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.     

Middle convolution and deformation for Fuchsian systems

Middle convolution and addition are operations for Fuchsiansystems of differential equations which preserve the numberof accessory parameters. In this paper we show that they alsopreserve the deformation equations. Several Bäcklund transformationsof the sixth Painlevé equation are obtained from thisviewpoint.     

On completely reducible solvable subgroups of GL(n, Δ)

Let Δ be a finite field and denote by GL(n, Δ) the group ofn×n nonsingular matrices defined over Δ. LetR⊆GL(n, Δ) be a solvable, completely reducible subgroup of maximal order. For |Δ|≧2, |Δ|≠3 we give bounds for |R| which improve previous ones. Moreover for |Δ|=3 or |Δ|>13 we determine the structure ofR, in particular we show thatR is unique, up to conjugacy. This work is part of a Ph.D. thesis done at the Hebrew University under the supervision of Professor A. Mann.     

Monodromy of Fuchsian systems on complex linear spaces

On complex linear spaces, Fuchs-type Pfaffian systems are studied that are defined by configurations of vectors in these spaces. These systems are referred to as R-systems in this paper. For the vector configurations that are systems of roots of complex reflection groups, the monodromy representations of R-systems are described. These representations are deformations of the standard representations of reflection groups. Such deformations define representations of generalized braid groups corresponding to complex reflection groups and are similar to the Burau representations of the Artin braid groups.     

Topological theory for Selberg type integral associated with rigid Fuchsian systems

Using the analytic realization of middle convolution due to Dettweiler and Reiter, we show that any rigid Fuchsian system can be obtained as a subsystem of some generating system which has an integral representation of solutions of Selberg type. Twisted homology groups and twisted cohomology groups associated with such integrals are studied. In particular, contiguity relations and twisted cycles which realize local exponents are obtained.     

Transversals in completely reducible multiary quasigroups and in multiary quasigroups of order 4

An $n$-ary quasigroup $f$ of order $q$ is an $n$-ary operation over a set of cardinality $q$ such that the Cayley table of the operation is an $n$-dimensional latin hypercube of order $q$. A transversal in a quasigroup $f$ (or in the corresponding latin hypercube) is a collection of $q$$(n+1)$-tuples from the Cayley table of $f$, each pair of tuples differing at each position. The problem of transversals in latin hypercubes was posed by Wanless in 2011.An $n$-ary quasigroup $f$ is called reducible if it can be obtained as a composition of two quasigroups whose arity is at least 2, and it is completely reducible if it can be decomposed into binary quasigroups.In this paper we investigate transversals in reducible quasigroups and in quasigroups of order 4. We find a lower bound on the number of transversals for a vast class of completely reducible quasigroups. Next we prove that, except for the iterated group ${\mathbb{Z}}_4$ of even arity, every $n$-ary quasigroup of order 4 has a transversal. Also we obtain a lower bound on the number of transversals in quasigroups of order 4 and odd arity and count transversals in the iterated group ${\mathbb{Z}}_4$ of odd arity and in the iterated group ${\mathbb{Z}}_2^2.$All results of this paper can be regarded as those concerning latin hypercubes.