Monodromy of Rational KZ Equations and Some Related Topics

A special class of integrable Fuchsian systems on C ⁿ related to KZ equations is considered. We survey the construction of such systems and the list of the structural properties their monodromy representations. The relation of the Fuchsian systems obtained by the Veselov construction assosiated with a deformation of the A _n–1-type root system and the Gauss–Manin connection of the natural projection C ⁿ C ^n–1 is described. In this case, we prove that the monodromy representation is equivalent to the Burau representation of the Artin braid group. For a deformations of the other root system, we introduce generalized Burau representations. We conjecture that the integrable Fuchsian systems related to essential new finite sets of the vectors described by Veselov and Chalykh are the result of the Klares–Schlesinger isomonodromic deformations (or transformation) of the integrable Fuchsian system related to the Coxeter root systems.     

Conjugacy in Permutation Representations of the Symmetric Group

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S _n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S _n  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.     

Theorem about the conjugacy representation ofS n

Every group has two natural representations on itself, the regular representation and the conjugacy representation. We know everything about the construction of the regular representation, but we know very little about the conjugacy representation (for uncommutative groups). In this paper we will see that every irreducible complex character ofS _n (n>2) is a constituent of conjugacy character ofS _n .     

Alternating groups and moduli space lifting invariants

The genus of a curve discretely separates decidely different algebraic relations in two variables to focus us on the connected moduli space M _g . Yet, modern applications also require a data variable (function) on the curve. The resulting spaces are versions, depending on our needs for this data variable, of Hurwitz spaces. A Nielsen class (§1.1) consists of r >- 3 conjugacy classes C in the data variable monodromy G. It generalizes the genus.     

Using Mathematica to compute conjugacy classes

The conjugacy classes of finite groups play an important role in the representation theory of those groups, and it is useful to be able to compute the conjugacy classes quickly. A procedure is developed and then implemented with Mathematica to discover these conjugacy classes. The computations make use of the Cayley table in its regular form for the group. The conjugacy classes for C_4v, the point symmetry group of the square, are displayed.     

Minimal subsystems of affine dynamics on local fields

We describe the dynamics of an arbitrary affine dynamical system on a local field by exhibiting all its minimal subsystems. In the special case of the field \mathbbQ_p{\mathbb{Q}_p} of p-adic numbers, for any non-trivial affine dynamical system, we prove that the field \mathbbQ_p{\mathbb{Q}_p} is decomposed into a countable number of invariant balls or spheres each of which consists of a finite number of minimal subsets. Consequently, we give a complete classification of topological conjugacy for non-trivial affine dynamics on \mathbbQ_p{\mathbb{Q}_p} . For each given prime p, there is a finite number of conjugacy classes.     

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL_n(q), SL_n(q), PGL_n(q) and PSL_n(q), respectively. Using conjugacy classes of elements in GL_n(q) in terms of irreducible polynomials over the finite field GF(q) we demonstrate how the set of order elements in GL_n(q) can be obtained. This will help to find the order of elements in the groups SL_n(q), PGL_n(q) and PSL_n(q). We also show an upper bound for the order of elements in SL_n(q).     

Quantization of the Geodesic Flow on Quaternion Projective Spaces

We study a problem of the geometric quantization for the quaternionprojective space. First we explain a Kähler structure on the punctured cotangent bundleof the quaternion projective space, whose Kähler form coincides withthe natural symplectic form on the cotangent bundle and show thatthe canonical line bundle of this complex structure is holomorphicallytrivial by explicitly constructing a nowhere vanishing holomorphicglobal section. Then we construct a Hilbert space consisting of acertain class of holomorphic functions on the punctured cotangentbundle by the method ofpairing polarization and incidentally we construct an operatorfrom this Hilbert space to the L ₂ space of the quaternionprojective space. Also we construct a similar operator between thesetwo Hilbert spaces through the Hopf fiberation.We prove that these operators quantizethe geodesic flow of the quaternion projective space tothe one parameter group of the unitary Fourier integral operatorsgenerated by the square root of the Laplacian plus suitable constant.Finally we remark that the Hilbert space above has the reproducing kernel.     

The Geometry of Elation Groups of a Finite Projective Space

We study the geometry of point-orbits of elation groups with a given center and axis of a finite projective space. We show that there exists a 1-1 correspondence from conjugacy classes of such groups and orbits on projective subspaces (of a suitable dimension) of Singer groups of projective spaces. Together with a recent result of Drudge [7] we establish the number of these elation groups.     

A normal form and schemes of quadratic forms

We present a solution of the problem of construction of a normal diagonal form for quadratic forms over a local principal ideal ring R = 2R with a QF-scheme of order 2. We give a combinatorial representation for the number of classes of projective congruence quadrics of the projective space over R with nilpotent maximal ideal. For the projective planes, the enumeration of quadrics up to projective equivalence is given; we also consider the projective planes over rings with nonprincipal maximal ideal. We consider the normal form of quadratic forms over the field of p-adic numbers. The corresponding QF-schemes have order 4 or 8. Some open problems for QF-schemes are mentioned. The distinguished finite QF-schemes of local and elementary types (of arbitrarily large order) are realized as the QF-schemes of a field. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 161–178, 2007.     

On monodromy of complex projective structures

Summary We prove that for any nonelementary representation : ₁(S SL (2, )) of the fundamental group of a closed orientable hyperbolic surfaceS there exists a complex projective structure onS with the monodromy .Oblatum IV-1993 & 24-IV-1994     

On the structure of generalized symmetric spaces of SLn𝔽q)

In this paper we extend previous results regarding SL₂(k) over any finite field k by investigating the structure of the symmetric spaces for the family of special linear groups SL_n(k) for any integer n>2. Specifically, we discuss the generalized and extended symmetric spaces of SL_n(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal.     

Cocycles on a finite dimensional quotient of uq (sl(2))

We prove that certain algebra quotients of Hopf algebras are twisted Hopf algebras. On the other handu_q (sl(2)) is a crossed product of a central subalgebra with a quotient [Ubar], when q is a root of 1. Using the cocycle involved in this crossed product we construct non-trivial complex cocycles τ and we find the isomorphism classes of the corresponding twisted Hopf algebras _τ [Ubar]. These provide complex projective representations of [Ubar] which are not ordinary representations.     

Primitive cohomology and the tube mapping

Let X be a smooth complex projective variety of dimension d. We show that its primitive cohomology in degree d is generated by certain “tube classes,” constructed from the monodromy in the family of all hyperplane sections of X. The proof makes use of a result about the group cohomology of certain representations that may be of independent interest.     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.     

On products of isometries of hyperbolic space

We show that for arbitrary fixed conjugacy classes C₁,…,Cl, l?3, of loxodromic isometries of the two-dimensional complex or quaternionic hyperbolic space there exist isometries g₁,…,gl, where each gi∈Ci, and whose product is the identity. The result follows from the properness, up to conjugation, of the multiplication map on a pair of conjugacy classes in rank 1 groups.     

Target space duality of Calabi-Yau spaces with two moduli

In the context of superstring compactifications on Calabi-Yau threefolds, we consider the Picard-Fuchs equations that are obeyed by the periods of the holomorphic three-form. We review, focusing on an example with two moduli, some powerful algebro-geometric techniques for computing the monodromy group of these equations, which is closely related to the target space duality group. For the example investigated, the latter is shown to be given by a three-dimensional representation of a central extension ofB ₅, the braid group on five strands.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98. No. 3, pp. 358–374, March, 1994.     

Hilbert schemes of surfaces are dense in moduli

We prove that the locus of Hilbert schemes of n points on a projective K 3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.