Planar Depth and Planar Subalgebras

We consider the notion of planar depth of a planar algebra, viz., the smallest n for which the planar algebra is generated by its ‘n-boxes’. We establish a simple result which yields a sufficient condition, in terms of the principal graph of the planar algebra, for the planar depth to be bounded by k. This suffices to determine the planar depth of the E6, E8 and the 5+132 subfactors. n consider a planar subalgebra of the ‘group planar algebra’ which is naturally associated with a group Θ of automorphisms of the given group G. We show that this planar algebra corresponds to the ‘subgroup-subfactor’ associated with the inclusion Θ⊂(G⋊Θ) (given by the semi-direct product extension). We conclude with a discussion of the planar depth of this planar algebra PΘ in some examples.