Braid moves in commutation classes of the symmetric group

Discrete Mathematics & Theoretical Computer Science, 2004

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

Arxiv preprint arXiv: …, 2008

We give a new infinite family of group homomorphisms from the braid group B k to the symmetric group S mk for all k and m ≥ 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1 ≤ l < m, we prove in particular that if m l is odd then there are 1+ m l non-conjugate homomorphisms included in our family. We define a certain natural restriction on homomorphisms B k → Sn, common to all homomorphisms in our family, which we term good, and of which there are two types. We prove that all good homomorphisms B k → S mk of type 1 are included in the infinite family of homomorphisms we gave. For m = 3, we prove that all good homomorphisms B k → S 3k of type 2 are also included in this family. Finally, we refute a conjecture made in [MaSu05] regarding permutation representations of braids and give an updated conjecture.

arXiv (Cornell University), 2010

In [1] we find certain finite homomorphic images of Artin braid group into appropriate symmetric groups, which a posteriori are extensions of the symmetric group on n letters by an abelian group. The main theorem of this paper characterizes completely the extensions of this type that are split.

We study the problem of …nding lower and upper bounds for the number of conjugacy clases in positive permutation braids. For such braids with associated type cycle (n); all possible values of their crossing numbers, the minimum and maximum crossing numbers and a more sharbend lower bound of the number of their conjugacy classes in S + n are given. Also for positive permutation braids with associate type cycle (n 1 ; n 2 ; : : : ; n k) the minimum crossing number is given..

Communications in Mathematical Physics, 2000

The main statistical characteristics of locally free groups: the growth, the drift and the entropy are considered and relations between them are established. Our results assert that: (i) the statistical properties of random walks (Markov chains) on locally free and braid groups are not the same as the uniform statistics on these groups, and (ii) the stabilization of the statistical characteristics exists when the number of generators of the group grows. Contents

Banach Center Publications, 2011

This is an introductory paper about our recent merge of a noncommutative de Finetti type result with representations of the infinite braid and symmetric group which allows us to derive factorization properties from symmetries. We explain some of the main ideas of this approach and work out a constructive procedure to use in applications. Finally we illustrate the method by applying it to the theory of group characters.

Arxiv preprint math/0409421, 2004

Mathematische Zeitschrift, 1997

The Annals of Probability, 2002

Wreath products are a type of semidirect product. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group.

The Electronic Journal of Combinatorics, 2012

In this paper we present two interesting properties of 321-avoiding Baxter permutations. The first one is a variant of refined major-balance identity for the 321-avoiding Baxter permutations, respecting the number of fixed points and descents. The second one is a bijection between the 321-avoiding Baxter permutations with the entry 1 preceding the entry 2 and the positive braid words on four strands.