Note on the Riemann-Hurwitz type formula for multiplicative sequences

We give a formula of the Riemann-Hurwitz type for classes defined by multiplicative sequences as corollaries of the Chern number formula for ramified coverings.

     

The structure of the centralizer of a permutation

Relative to the symmetric groups S_n the structure of centralizers of permutations are known as direct products of certain general wreath products. A recent generalization of the cycle notation for partial one-one transformations (charts) is applied to show that relative to the symmetric inverse semigroups C_n the structure of centralizers of permutations are also direct products of certain subsemigroups of a wreath product, and this latter wreath product includes the former as a subgroup. A necessary and sufficient condition is given for two charts to commute and the approach for the C_n-case parallels and generalizes the one for the S_n-case. As a result, the C_n-case yields the standard known characterizations of commuting permutations, as well as formulas for the orders of centralizers as corollaries. It is an open problem to extend these results to the centralizers of arbitrary charts in C_n. This research was partially supported by a Mary Washington College Faculty Development Grant.     

The centralizer of a rank-one flow

In [5], King proved that the centralizer of a rank-1 transformation equals the “weak closure” of its (positive and negative) powers (see below for a definition of the weak topology). We define rank-1 flows, and then show that simple modifications of King's proof yield an analogous statement for rank-1 flows.     

The centralizer of a semi-linear transformation

Let V be a finite-dimensional vector space over a division ring D, where D is finite-dimensional over its center F. Suppose T is a semi-linear transformation on V with associated automorphism σ of D. The centralizer of T is the ring C(T) of all linear transformations on V which commute with T. If σ^r is the identity on D for some r ? 1 and no smaller positive power of σ is an inner automorphism, then the center of C(T) is computed to be polynomials in T^r with coefficients from F₀, where F₀ is the subfield of F left elementwise fixed by σ. A matrix version of this theorem is also given.     

The correspondence formula of Dolbeault complex on pair deformation

Geometriae Dedicata - Given a holomorphic family of pairs $$\{(X_t,E_t)\}$$ where each $$E_t$$ is a holomorphic vector bundle over a compact complex manifold $$X_t$$ , we get a correspondence...     

The Cartan matrix of a centralizer algebra

The centralizer algebra of a matrix consists of those matrices that commute with it. We investigate the basic representation-theoretic invariants of centralizer algebras, namely their radicals, projective indecomposable modules, injective indecomposable modules, simple modules and Cartan matrices. With the help of our Cartan matrix calculations we determine their global dimensions. Many of these algebras are of infinite global dimension.     

On the connectedness of a centralizer

This letter finds the connected components of certain subgroups of a complex torus that arise in completing the Hamiltonian flows of the full complex Toda lattice.     

The centralizer of a set of group automorphisms

Let A be a set of automorphisms of a finite group G. The structure of the near-ring C(A) of identity preserving functions which commute with every element of A is investigated.     

The double centralizer theorem for division algebras

AssumeV is a finite-dimensional vector space over a division ringD having centerF. It is shown that ifT∈ End _D (V) is algebraic overF then the double centralizerC(C(T)) ofT is the setF[T] of all polynomials inT with coefficients fromF. Consequently, eachn×n matrix ring overD is an algebraicF-algebra if and only ifC(C(T))=F[T] for allT and all finite-dimensionalV.     

The Banach algebra induced by a double centralizer

Given a Banach algebra , R. Larsen defined, in his book ``An introduction to the theory of multipliers", a Banach algebra by means of a multiplier on , and essentially used it in the case of a commutative semisimple Banach algebra to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra induced by a bounded double centralizer of a Banach algebra . Then, our main concern is devoted to the relationships between , , and the algebras of bounded double centralizers and of and , respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.

     

The centralizer of a matrix with real quaternion elements

Let T be a Q-endomorphism on a finite dimensional left vector space V over the division ring of real quaternions Q. In this paper we show that the centralizer of T can be regarded as an algebra over the complex numbers C and the dimension of this algebra is computed in terms of the invariant factors of T. Thus the number of C-linearly independent matrices which commute with a given matrix representing T is determined.     

On the structure of the centralizer of a braid

The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on n strands, which has at most elements if n=2k, and at most elements if n=2k+1. These bounds are shown to be sharp, due to work of N.V. Ivanov and of S.J. Lee. Finally, we describe how one can explicitly compute this generating set.     

Connection between the Plancherel formula and the Howe dual pair ()

Let D be the bounded symmetric domain In this article, we describe explicitly the Plancherel formula of a weighted Bergman space on under the action of by using the Howe dual pair Received: 13 November 2000; in final form: 30 January 2002 / Published online: 6 August 2002