Stable Super Summit Sets in Garside Groups
نویسندگان
چکیده
The known algorithms for solving the conjugacy problem in Garside groups involve computing a particular subset of the conjugacy class, the so-called super summit set. The super summit set [g] of an element g in a Garside group is, intuitively, the set of all conjugates of g that have the shortest normal form in the conjugacy class of g. In this paper, we define the stable super summit set [g] of g as the set of all conjugates h of g such that h ∈ [g] for all n > 1, in other words, all the powers h have the shortest normal form in the conjugacy class of g. We show that every stable super summit set is nonempty and satisfies most of the nice properties of the super summit set.
منابع مشابه
A New Approach to the Conjugacy Problem in Garside Groups
The cycling operation endows the super summit set Sx of any element x of a Garside group G with the structure of a directed graph Γx. We establish that the subset Ux of Sx consisting of the circuits of Γx can be used instead of Sx for deciding conjugacy to x in G, yielding a faster and more practical solution to the conjugacy problem for Garside groups. Moreover, we present a probabilistic appr...
متن کاملSome Power of an Element in a Garside Group Is Conjugate to a Periodically Geodesic Element Eon Kyung Lee and Sang
We show that for each element g of a Garside group, there exists a positive integer m such that gm is conjugate to a periodically geodesic element h, an element with |hn|D = |n| · |h|D for all integers n, where |g|D denotes the shortest word length of g with respect to the set D of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside g...
متن کاملSome Power of an Element in a Garside Group Is Conjugate to a Periodically Geodesic Element
We show that for each element g of a Garside group, there exists a positive integer m such that gm is conjugate to a periodically geodesic element h, an element with |hn|D = |n|·|h|D for all integers n, where |g|D denotes the shortest word length of g with respect to the set D of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside gro...
متن کاملThe cyclic sliding operation in Garside groups
We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has c...
متن کاملComplete simultaneous conjugacy invariants in Artin's braid groups
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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006