  We compare two important bases of an irreducible representation of the
symmetric group: the web basis and the Specht basis. The web basis has its
roots in the Temperley-Lieb algebra and knot-theoretic considerations. The
Specht basis is a classic algebraic and combinatorial construction of symmetric
group representations which arises in this context through the geometry of
varieties called Springer fibers. We describe a graph that encapsulates
combinatorial relations between each of these bases, prove that there is a
unique way (up to scaling) to map the Specht basis into the web representation,
and use this to recover a result of Garsia-McLarnan that the transition matrix
between the Specht and web bases is upper-triangular with ones along the
diagonal. We then strengthen their result to prove vanishing of certain
additional entries unless a nesting condition on webs is satisfied. In fact we
conjecture that the entries of the transition matrix are nonnegative and are
nonzero precisely when certain directed paths exist in the web graph.
