Abstract. We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defined over a finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter.

a b s t r a c tWe prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured by Broué, Malle and Rouquier and has for consequence that this algebra is a flat deformation of the group algebra of the complex reflection group G 32 , of order 155,520.

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We define a representation of the Artin groups of type ADE by monodromy of generalized KZ-systems which is shown to be isomorphic to the generalized Krammer representation originally defined by A.M. Cohen and D. Wales, and independantly by F. Digne. It follows that all pure Artin groups of spherical type are residually torsion-free nilpotent, hence (bi-)orderable. Using that construction we show that these irreducible representations are Zariski-dense in the corresponding general linear group. It follows that all irreducible Artin groups of spherical type can be embedded as Zariski-dense subgroups of some general linear group. As group-theoretical applications we prove properties of non-decomposition in direct products for several subgroups of Artin groups, and a generalization in arbitrary types of a celebrated property of D. Long for the braid groups. We also determine the Frattini and Fitting subgroups and discuss unitarity properties of the representations.

We attach to every Coxeter system (W, S) an extension C W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C W . When W is finite, we prove that this algebra is a free module of finite rank which is generically semisimple. When W is the Weyl group of a Chevalley group, C W naturally maps to the associated Yokonuma-Hecke algebra. When W = Sn this algebra can be identified with a diagram algebra called the algebra of 'braids and ties'. The image of the usual braid group in this case is investigated. Finally, we generalize our construction to finite complex reflection groups, thus extending the Broué-Malle-Rouquier construction of a generalized Hecke algebra attached to these groups. MSC 2010: 20F36; 20F55; 20C08.Proof. It is easy to prove that, if J ∼ K, then J = K . Indeed, the relation J 1 ≡ J 2 defined by J 1 = J 2 is obviously an equivalence relation, and, if K = J ∪ {sts} with {s, t} ⊂ J, we have J = K , that is J ≡ K. It follows that the relation ≡ contains all such couples (J, K), hence J ∼ K ⇒ J ≡ K.Conversely, let us assume J = K . If J = ∅ or K = ∅ then J = {1} = K implies J = K = ∅ hence J ∼ K. Otherwise, let us set W 0 = J = K . By proposition 3.1 the group W 0 is a Coxeter group with generating set S 0 ⊂ R, and W 0 ∩ R = R 0 is the set of all conjugates of elements of S 0 . Moreover, since W 0 = J , proposition 3.1 states that every element R 0 is a conjugate (inside W 0 ) of an element of J. This applies in particular to the elements of K ⊂ W 0 ∩ R = R 0 . Therefore, every x ∈ K can be written as ms 0 m −1 for some s 0 ∈ J and m ∈ W 0 = J . Writing m = s r s r−1 . . . s 1 for some s 1 , . . . , s r ∈ J we get x = s r s r−1 . . . s 1 s 0 s 1 . . . s r−1 s r for s 0 , . . . , s r ∈ J. By induction on r one gets readily that x ∈ J x , for some J x ∈ P f (R) with J x ∼ J and J ∪ J x . Since J x = J = K and K is finite, we can iterate this argument for all elements x ∈ K, and this proves that K ⊂ J ′ for some J ′ ∈ P f (R) with J ⊂ J ′ and J ∼ J ′ .Therefore, we can assume K ⊂ J. By the same argument, every element of J can be written as s r s r−1 . . . s 1 s 0 s 1 . . . s r−1 s r for s 0 , . . . , s r ∈ K ⊂ J. By induction on |J \ K| we get from this that J ∼ K.

Abstract. Let G ⊂ GL(C r ) be a finite complex reflection group. We show that when G is irreducible, apart from the exception G = S 6 , as well as for a large class of non-irreducible groups, any automorphism of G is the product of a central automorphism and of an automorphism which preserves the reflections. We show further that an automorphism which preserves the reflections is the product of an element of N GL(C r ) (G) and of a "Galois" automorphism: we show that Gal(K/Q), where K is the field of definition of G, injects into the group of outer automorphisms of G, and that this injection can be chosen such that it induces the usual Galois action on characters of G, apart from a few exceptional characters; further, replacing K if needed by an extension of degree 2, the injection can be lifted to Aut(G), and every irreducible representation admits a model which is equivariant with respect to this lifting. Along the way we show that the fundamental invariants of G can be chosen rational.

Inteins are intervening protein sequences that undergo self-excision from a precursor protein with concomitant joining of the flanking sequences. Here, we demonstrate intein trans-splicing in Nicotiana tabacum chloroplasts by using the naturally split Ssp DnaE intein. Trans-splicing occurred whether both intein fragments were encoded in the chloroplast or were separated into the chloroplast and nuclear genomes. A biolistic approach was used to integrate two fusion genes, one encoding aminoglycoside-3-adenyltransferase (aadA) and the first 123 aa of the Ssp DnaE intein (In) and the other encoding 36 C-terminal amino acid residues of the Ssp DnaE intein (Ic) and soluble modified green fluorescent protein (smGFP) into N. tabacum plastids. Expression of these gene fragments in the chloroplast resulted in ligated aadA-smGFP due to In-Ic-mediated trans-splicing. Furthermore, an N-terminal portion of the herbicide resistance gene 5-enolpyruvylshikimate-3-phosphate synthase (EPSPS) containing a chloroplast localization signal fused to In (EPSPSn-In) was integrated into the nuclear DNA of N. tabacum by using Agrobacterium tumefaciensmediated transformation. The remaining EPSPS gene fragment (EPSPSc) fused to Ic (Ic-EPSPSc) was integrated into the chloroplast genome by homologous recombination. Western blot analysis of cell extracts from these plants showed a full-length EPSPS, demonstrating that the EPSPSn-In gene product migrated to the chloroplast and underwent trans-splicing. Furthermore, these transgenic plants displayed improved resistance to the herbicide N-(phosphonomethyl)-glycine (glyphosate) when compared with wild-type N. tabacum.I nteins are intervening protein sequences that undergo an autoexcision reaction, protein splicing, that results in ligation of the flanking protein regions, termed exteins (1). Inteins are present in the genome of eubacteria, eukaryota, and archaea, and to date, Ͼ100 have been identified (InBase: www.neb.com͞neb͞inteins. html; ref.2). The blue-green algae Synechocystis sp. PCC6803 has four inteins that are encoded within the dnaE, dnaB, dnaX, and gyrB genes. The dnaE gene encodes a split copy of the replicative DNA polymerase III catalytic ␣ subunit (DnaE), with the split portion separated by 745 kb of genomic DNA. The dnaE-n gene encodes the first 774 aa of DnaE and the dnaE-c gene encodes the remaining 423 aa of the DnaE protein. These two gene portions are transcribed from opposite strands of the genome. When translated, this split gene yields one polypeptide containing the N terminus of the DnaE (N-extein) fused to an intein fragment and a second fusion peptide containing the complementary intein fragment and the C terminus (C-extein) of the polymerase. The intein fragments of the fusion proteins assemble, then mediate a protein trans-splicing reaction, resulting in cleavage of both peptide bonds at the intein͞ polymerase junction and ligation of the flanking DnaE regions, giving rise to a mature, catalytically active DNA polymerase III catalytic ␣ subunit (3, 4).The intein porti...

Abstract. Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincaré polynomial with coefficients in a finite field for one large series of such groups, and compute the second integral cohomology group for all of them. As a consequence we get non-isomorphism results for these groups.