We provide the first effectively computable, modulo a mild conjecture, complete invariant for simultaneous conjugacy in Garside groups. This invariant generalizes the notion of super summit set of a conjugacy class, and provides a solution of the simultaneous conjugacy problem in Garside groups, of complexity a small degree polynomial in the sizes of the invariant and the input parameters. Comp...

As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel's correspondence among convex functions and Rockafellar's extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new corres...

We describe a significantly improved algorithm for computing the conjugacy classes of a finite permutation group with trivial soluble radical. We rely on existing methods for groups that are almost simple, and we are concerned here only with the reduction of the general case to the almost simple case. Dedicated to Charles Leedham-Green on the occasion of his 65th birthday

Let Γ be a co-compact Fuchsian group (with no elliptic elements) and let Σ denote the associated hyperbolic surface H/Γ. Then Γ is isomorphic to the fundamental group of Σ and each non-trivial conjugacy class in Γ contains a unique closed geodesic on Σ. If we write l[γ] for the length of the closed geodesic in the conjugacy class [γ] then it was proved in [6] (though it was already implicit in ...

Abstract We study noncommutative differential structures on permutation groups SN , defined by conjugacy classes. The 2-cycles class defines an exterior algebra ΛN which is a super analogue of the quadratic algebra EN for Schubert calculus on the cohomology of the flag variety. Noncommutative de Rahm cohomology and moduli of flat connections are computed for N < 6. We find that flat connections...

Conjugacy classes of primitive words in the free group of rank two can indexed by the rationals. A representative for each conjugacy class is given by a recursive enumeration scheme and the representative, usually called a Farey word, is denoted by Wp/q where p and q are relatively prime integers. Primitive pairs, that is, pairs of words that generate the free group, correspond to pairs of word...

We prove a version of Valiron’s conjugacy theorem for Schur class mappings of the unit ball of C . As an application we obtain a formula for the spectral radius of composition operators on the ball with Schur class symbols.

Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F ) that picks out a G(F )-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F ). This generalizes an earlier result obtained by DeBack...

Solomon’s descent algebra is used to define a family of signed measures MW,x for a finite Coxeter group W and x > 0. The measures corresponding to W of type An arise from the theory of card shuffling. Formulas for these measures are obtained and conjectured in special cases. The eigenvalues of the associated Markov chains are computed. By elementary algebraic group theory, choosing a random sem...

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group in...