The cyclic sliding operation in Garside groups

Abstract

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 conjugates which are rigid (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding.

Automorphism group of the commutator subgroup of the braid group

“…The summit length of X is defined as s (X) = min{ (Y ) | Y ∈ X G }. We denote the cyclic sliding of X and the set of sliding circuits of X by s(X) and SC(X) respectively (these notions were introduced in [8], see also [21, Definition 1.12]).…”

Discovery of cosmic rays

“…The SC of x is a subset of its SSS [8]. A rigid braid, by definition, belongs necessarily to its SC, since it is a periodic point of period 1 of the cyclic sliding operation.…”

Braid Group Cryptography

“…The appropriate prefix is: ∂(x r ) ∧ τ −p (x 1 ), which is equal to: ι(x −1 )∧ι(x). Hence, Gebhardt and González-Meneses [55] define: Definition 1.20. Given x ∈ B n , define the cyclic sliding s(x) of x as the conjugate of x by p(x) = ι(x −1 ) ∧ ι(x), that is: Given x ∈ B n , the set of sliding circuits of x, denoted by SC(x), is the set of all conjugates of x which belong to a sliding circuit.…”

“…we have that |SC(x)| = 6, but |SSS(x)| = |USS(x)| = 126498 (see [55,Section 5], based on an example from [57]). On the other hand, the size of the set SC(x) still might be exponential in the length of x (for example, if for x satisfying ℓ s (x) = 1, and in general SC(x) is a proper subset of RSSS(x) in this case.…”

“…The number of such possible minimal conjugators for a given element in SC(x) is bounded by the number of Artin generators). Hence, one can compute the whole SC(x) starting by a single elementx ∈ SC(x), and then we are done (for more information, see [55,Section 4…”