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First, we present a new proof of Glaisher's formula dating from 1900 and concerning Wilson's theorem modulo p 2 . Our proof uses p-adic numbers and Faulhaber's formula for the sums of powers (17th century), as well as more recent results on Faulhaber's coefficients obtained by Gessel and Viennot. Second, by using our method, we find a simpler proof than Sun's proof regarding a formula for (p − 1)! modulo p 3 , and one that can be generalized to higher powers of p. Third, we can derive from our method a way to compute the Stirling numbers p s modulo p 3 , thus improving Glaisher and Sun's own results from 120 years ago and 20 years ago respectively. Last, our method allows to find new congruences on convolutions of divided Bernoulli numbers and convolutions of divided Bernoulli numbers with Bernoulli numbers.

The Lawrence-Krammer representation introduced by Lawrence and Krammer in order to show the linearity of the braid group is generically irreducible. We show this fact and show further that for some values of its two parameters, when these are specialized to complex numbers, the representation becomes reducible. We describe what these values are and give a complete description of the dimensions of the invariant subspaces when the representation is reducible. To cite this article: C. Levaillant, C. R. Acad. Sci. Paris, Ser. I 347 (2009). RésuméIrréductibilité de la représentation de Lawrence-Krammer de l'algèbre BMW de type A n−1 . La représentation de Lawrence-Krammer, introduite par Lawrence et Krammer pour montrer la linéarité du groupe de tresses, est génériquement irréductible. On montre ce fait et on montre également que lorsque les deux paramètres de la représentation prennent certaines valeurs complexes, la représentation devient réductible. On donne ici toutes les valeurs des paramètres pour lesquelles la repré-sentation est réductible, ainsi que les dimensions des sous-espaces stables. Pour citer cet article : C. Levaillant, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

After 100 years of effort, the classification of all the finite subgroups of SU (3) is yet incomplete. The most recently updated list can be found in [3], where the structure of the series (C) and (D) of SU (3)-subgroups is studied. We provide a minimal set of generators for one of these groups which has order 162. These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the JonesKauffman version of SU (2) Chern-Simons theory at level 4. In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product Z 9 × Z 3 S 3 with respect to conjugation.

We examine a class of operations for topological quantum computation based on fusing and measuring topological charges for systems with SU(2)4 or k = 4 Jones-Kauffman anyons. We show that such operations augment the braiding operations, which, by themselves, are not computationally universal. This augmentation results in a computationally universal gate set through the generation of an exact, topologically protected irrational phase gate and an approximate, topologically protected controlled-Z gate.

We show that the representation, introduced by Lawrence and Krammer to show the linearity of the braid group, is generically irreducible. However, for some values of its two parameters when these are specialized to complex numbers, it becomes reducible. We construct a representation of degree n(n−1) 2 of the BMW algebra of type A n−1 . As a representation of the braid group on n strands, it is equivalent to the Lawrence-Krammer representation where the two parameters of the BMW algebra are related to those appearing in the Lawrence-Krammer representation. We give the values of the parameters for which the representation is reducible and give the proper invariant subspaces in some cases. We use this representation to show that for these special values of the parameters, the BMW algebra of type A n−1 is not semisimple.

We show that, modulo some odd prime p, the powers of two weighted sum of the first p divided Bernoulli numbers equals twice the number of permutations on p − 2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p 2 divides the Fermat quotient q p (2).

Abstract. We study a subgroup F r(162 × 4) of SU (3) of order 648 which is an extension of D(9, 1, 1; 2, 1, 1) and whose generators arise from anyonic systems. We show that this group is isomorphic to a semi-direct product (Z/18Z×Z/6Z) S 3 with respect to conjugation and we give a presentation of the group. We show that the group D(18, 1, 1; 2, 1, 1) from the series (D) in the existing classification for finite SU (3)-subgroups is also isomorphic to a semi-direct product (Z/18Z × Z/6Z) S 3 , also with respect to conjugation. We show that the two groups F r(162×4) and D(18, 1, 1; 2, 1, 1) are isomorphic and we provide an isomorphism between both groups. We prove that F r(162 × 4) is not isomorphic to the exceptional SU (3) subgroup Σ(216×3) of the same order 648. We further prove that the only SU (3) finite subgroups from the 1916 classification by Blichfeldt or its extended version which F r(162 × 4) may be isomorphic to belong to the (D)-series. Finally, we show that F r(162 × 4) and D(18, 1, 1; 2, 1, 1) are both conjugate under an orthogonal matrix which we provide.