扭型Kazhdan-Lusztig理论中的逆反多项式

本文研究扭型Kazhdan-Lusztig多项式的逆反多项式的性质及其计算方法.构造了Lusztig对偶模M的一类特异基(或D-基),获得了Hecke代数在此基上的作用公式.在有限Coxeter群情形下,获得了Lusztig-Vogan模的结构常数的关系.     

Alexander polynomials of doubly primitive knots

We give a formula for Alexander polynomials of doubly primitive knots. This also gives a practical algorithm to determine the genus of any doubly primitive knot.

     

Twisted Weyl groups of Lie groups and nonabelian cohomology

For a cyclic group A and a connected Lie group G with an A-module structure (with the additional assumptions that G is compact and the A-module structure on G is 1-semisimple if ), we define the twisted Weyl group W = W(G,A,T), which acts on T and H ¹(A,T), where T is a maximal compact torus of , the identity component of the group of invariants G ^A . We then prove that the natural map is a bijection, reducing the calculation of H ¹(A,G) to the calculation of the action of W on T. We also prove some properties of the twisted Weyl group W, one of which is that W is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.      

Geometric indices and the Alexander polynomial of a knot

It is well-known that any Laurent polynomial satisfying and is the Alexander polynomial of a knot in . We show that can be realized by a knot which has the following properties simultaneously: (i) tunnel number 1; (ii) bridge index 3; and (iii) unknotting number 1.

     

Twisted Alexander norms give lower bounds on the Thurston norm

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number greater than one, generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which are not determined by norms of McMullen and Turaev.

     

Representations of Posets and Indecomposable Torsion-Free Abelian Groups

Representations of posets in certain modules are used to find indecomposable almost completely decomposable torsion-free abelian groups. For a special class of almost completely decomposable groups we determine the possible ranks of indecomposable groups and show that the possible ranks are realized by indecomposable groups in the class.     

Representations of 2-Groups by Polynomials

It is known that any finite p-group can be represented by polynomials. However, how to represent p-groups and how to classify p-groups up to isomorphism are interesting and open questions. In this article, we investigate the 2-groups of order 8, and represent the dihedral group D_2n, the generalized quaternion group Q_2n, and the infinite dihedral group D.2000 Mathematics Subject Classification: 20C99, 20E99     

Module Correspondences for Twisted Group Algebras

Abstract

Let G and A be finite groups such that (|G|, |A|) = 1. Let K be an algebraically closed field with Char K = 0. Denote by K ^α G the twisted group algebra of G over K with factor set α. In this paper we prove that if A acts homogeneously on K ^α G, then there exists an action of A on G, and there is a one-to-one correspondence between the set of A-invariant irreducible K ^α G-modules and the set of irreducible K ^α C _G (A)-modules.     

Knot adjacency and fibering

It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of knot adjacency can be used to obtain obstructions to the fibering of knots and of 3-manifolds. As an application, given a fibered knot , we construct infinitely many non-fibered knots that share the same Alexander module with . Our construction also provides, for every , examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order .

     

The Primitive Sharp Permutation Groups of Twisted Wreath Type

A permutation group G ≤ Sym(X) on a finite set X is sharp if |G|=∏ _l?L(G)(|X| ? l), where L(G) = {|fix(g)| | 1 ≠ g ? G}. We show that no finite primitive permutation groups of twisted wreath type are sharp.     

Group Extensions and Hall Polynomials

In this paper, we give two proofs of a formula containing the numbers of automorphisms of an Abelian group, of its subgroups, and of its quotient groups. The first proof is based on the use of the theory of Hall polynomials, while the second one uses extension theory for Abelian groups.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 180–185.Original Russian Text Copyright © 2005 by G. V. Voskresenskaya.     

Twisted tensor product codes

We present two families of constacyclic linear codes with large automorphism groups. The codes are obtained from the twisted tensor product construction.      

Cyclotomic Polynomials and Finite Automorphisms of Groups

We introduce minimal polynomials for finite automorphisms of commutative groups and relate them to the exponent of the fixed points and to the reducibility of the group. Some results can be extended to the noncommutative case.     

Prym Representations of Mapping Class Groups

Let S be a closed orientable surface of genus at least 2 and let to S be a connected finite abelian covering with covering group $G$. The lifts of liftable mapping classes of S determine a central extension (by G) of a subgroup of finite index of the mapping class group of S. This extension acts on H₁( ). With a few exceptions for genus 2, we determine the Zariski closure of the image of this representation, and prove that the image is an arithmetic group.     

Strong coprimality and strong irreducibility of Alexander polynomials

A polynomial f(t) with rational coefficients is strongly irreducible if f(tk) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(tk) and g(tl) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility and prove that the Alexander polynomials of twist knots are pairwise strongly coprime and that most of them are strongly irreducible. We apply these results to describe the structure of the subgroup of the rational knot concordance group generated by the twist knots and to provide an explicit set of knots which represent linearly independent elements deep in the solvable filtration of the knot concordance group.     

Twisted Automorphisms and Twisted Right Gyrogroups

In this paper we make an attempt to study right loops (S, o) in which, for each y ∈ S, the map σ _y from the inner mapping group G _S of (S, o) to itself given by σ _y (h)(x)o h(y) = h(xoy), x ∈ S, h ∈ G _S is a homomorphism. The concept of twisted automorphisms of a right loop and also the concept of twisted right gyrogroup appears naturally and it turns out that the study is almost equivalent to the study of twisted automorphisms and a twisted right gyrogroup. We also study relationship between twisted gyrotransversals and twisted subgroups.     

Simple Twisted Group Algebras of Dimension p4 and their Semi-Centers

For a simple twisted group algebra over a group G, if G^∣ is Hall subgroup of G, then the semi-center is simple. Simple twisted group algebras correspond to groups of central type. We classify all groups of central type of order p⁴ where p is prime and use this to show that for odd primes p there exists a unique group G of order p⁴, such that there exists simple twisted group algebra over G with a commutative semi-center. Moreover, if 1 < |G| <64, then the semi-center of simple twisted group algebras over G is noncommutative and this bounds are strict.