Abstract: The Specht modules admit a well-known polytabloid basis parametrized by standard Young tableaux of a fixed shape. On the other hand, classical invariant theory for sl_2 and s_3 suggests that these modules also admit a diagrammatical basis, which are Temperley-Lieb diagrams and Kuperberg spiders, for two types of Specht modules, respectively. It is natural to investigate the transitioning matrix between the tableaux basis and the web basis. We developed a combinatorial formula for computing entries in these matrices. Regarding these entries we proved a positivity conjecture for the Temperley-Lieb diagrams and a non-negativity result for the Kuperberg spiders.