Invariants and conjugacy classes of triangular polynomial maps

The major index polynomial for conjugacy classes of permutations

Wachs, M.L. The major index polynomial for conjugacy classes of permutations, Discrete Mathematics 91 (1991) 283-293. Garsia (1988) gives a remarkably simple expression for the major index enumerator for permutations of a fixed cycle type evaluated at a primitive root of unity. He asks for a direct combinatorial proof of this identity. Here we give such a combinatorial derivation.

Generalized Conjugacy Classes

Generalized conjugation is the action of a group on its underlying set given by (g, x) 7→ φ(g)xg, where φ : G → G is some fixed endomorphism. Here we study combinatorial properties of the sizes of the orbits of the preceding action. In particular, we reduce the problem to a simpler case if φ has a nontrivial kernel or if it is an inner automorphism, and we give a construction that allows a part...

K-invariants of conjugacy classes of pseudo-Anosov diffeomorphisms and hyperbolic 3-manifolds

New invariants of 3-dimensional manifolds appearing in the Ktheory of certain operator algebras are introduced. First, we consider the conjugacy problem for pseudo-Anosov diffeomorphisms of a compact surface X. The operator algebra in question is an AF -algebra attached to stable (unstable) foliation of the pseudo-Anosov diffeomorphism. We prove that conjugacy classes of commensurable pseudoAno...

Differential Polynomial Rings of Triangular Matrix Rings

Simple Conjugacy Invariants for Braids

We define simple conjugacy invariants of braids, which we call turning numbers, and investigate their properties. Since our motivation comes from the investigation of periodic orbits of orientation preserving disk homeomorphisms, turning numbers work best for braids with the cyclic permutation, especially for positive permutation cyclic braids.