Publications et prépublications
en liaison avec les tresses
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| Abstract:
We define a measure of "complexity" of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators $\Delta_{ij}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators $\Delta_{ij}^k$ as $\log(|k|+1)$ instead of simply $|k|$. The geometrical complexity is some natural measure of the amount of distortion of the $n$ times punctured disk caused by a homeomorphism. Our main result is that the two notions of complexity are comparable. We also show how to recover a braid from its curve diagram in polynomial time, and we prove that every braid has a $\sigma_1$-consistent representative of linearly bounded length. The key rle in the proofs is played by a technique introduced by Agol, Hass, and Thurston.
MSC : 20F36, 20F65 |
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| Abstract:
We prove that the braid group on 4 or more strands modulo its center is co-Hopfian. We then show that any injective endomorphism of these braid groups is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from the braid group on n strands to the braid group on n+1 strands is geometric (n > 6). Additionally, we obtain related results about mapping class groups of punctured spheres. The methods use Thurston's theory of surface homeomorphisms and build upon work of Ivanov and McCarthy.
MSC : 20F36; 57M07 |
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| Abstract:
We construct a quasi-Garside monoid structure for the free group. This monoid should be thought of as a dual braid monoid for the free group, generalising the constructions by Birman-Ko-Lee and by the author of new Garside monoids for Artin groups of spherical type. Conjecturally, an analog construction should be available for arbitrary Artin groups and for braid groups of well-generated complex reflection groups.
MSC : 20F36 |
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| Abstract:
This paper provides a unified approach to results on representations of affine Hecke algebras, cyclotomic Hecke algebras, affine BMW algebras, cyclotomic BMW algebras, Markov traces, Jacobi-Trudi type identities, dual pairs (Zelevinsky), and link invariants (Turaev). The key observation in the genesis of this paper was that the technical tools used to obtain the results in Orellana and Suzuki, two a priori unrelated papers, are really the same. Here we develop this method and explain how to apply it to obtain results similar to those in Orellana and Suzuki in more general settings. Some specific new results which are obtained are the following: (a) a generalization of the results on Markov traces obtained by Orellana to centralizer algebras coming from quantum groups of all Lie types, (b) a generalization of the results of Suzuki to show that Kazhdan-Lusztig polynomials of all finite Weyl groups occur as decomposition numbers in the representation theory of affine braid groups of type A, (c) a generalization of the functors used by Zelevinsky to representations of affine braid groups of type A, (d) a definition of the affine BMW-algebra (Birman-Murakami-Wenzl) and show that it has a representation theory analogous to that of affine Hecke algebras. In particular there are ``standard modules'' for these algebras which have composition series where multiplicites of the factors are given by Kazhdan-Lusztig polynomials for Weyl groups of types A,B,and C, (e) we generalize the results of Leduc and Ram on constructing representations of centralizer algebras to affine centralizer algebras.
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