Web bases for sl(3) are not dual canonical

We compare two natural bases for the invariant space of a tensor product of irreducible representations of A_2, or sl(3). One basis is the web basis, defined from a skein theory called the combinatorial A_2 spider. The other basis is the dual canonical basis, the dual of the basis defined by Lusztig and Kashiwara. For sl(2) or A_1, the web bases have been discovered many times and were recently shown to be dual canonical by Frenkel and Khovanov. We prove that for sl(3), the two bases eventually diverge even though they agree in many small cases. The first disagreement comes in the invariant space Inv((V^+ tensor V^+ tensor V^- tensor V^-)^{tensor 3}), where V^+ and V^- are the two 3-dimensional representations of sl(3). If the tensor factors are listed in the indicated order, only 511 of the 512 invariant basis vectors coincide.Comment: 18 pages. This version has very minor correction

Spectral networks

We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional N=2 theories coupled to surface defects, particularly the theories of class S. In these theories spectral networks provide a useful tool for the computation of BPS degeneracies: the network directly determines the degeneracies of solitons living on the surface defect, which in turn determine the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat GL(K,C) connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover Sigma of C. This construction produces natural coordinate systems on moduli spaces of flat GL(K,C) connections on C, which we conjecture are cluster coordinate systems.Comment: 87 pages, 48 figures; v2: typos, correction to general rule for signs of BPS count

Experimental investigation into the feasibility of an extruded wing

Fabrications of extruded aluminum alloy structures are reviewed. The design criteria and the fabrication of the main structure of a sailplane wing made of a few extruded profiles longitudinally connected one to the other are illustrated. Structural tests recently carried out are reported upon

Testing Intra-Industry Trade Between Portugal and Spain [1990-1996]

This paper shows how, in the period 1990-1996, economic integration between the two Iberian economies has deepened through intra-industry trade. The enquiry was led at the level of the main forty traded products. In this context, several tests have been made. Our analysis combines first the application of the Grubel and Lloyd index of the main forty products and the marginal intra-industry trade of Brulhart, and Brulhart and Elliot. On the basis of these indexes we defined our criterion for the selection of the competitive cluster of Portugal throughout the period. We have also considered the global intra-industry trade, in nominal and real terms, according to the method of Greenaway et al. In econometric terms we have tried to know which is the relation between the intra-industry trade index and the marginal intra-industry trade at the level of the main forty products, the results however were not unequivocal. The same is true as far as concerns the relation between the intra-industry trade index and the net export position at the level of the same products. Despite the fact that some attempts have been inconclusive, knowledge about Iberian trade has developed in a field essential to its upgrading, and the way is open for further research.intra-industry trade; marginal intra-industry trade; competitive cluster; net export position.