Intermediate planar algebra revisited

In this paper, we explicitly work out the subfactor planar algebra $P^{(N \subset Q)}$foranintermediatesubfactor$N \subset Q \subset M$ of an irreducible subfactor $N\subset M$ of finite index. We do this in terms of the subfactor planar algebra $P^{(N\subset M)}$ by showing that if $T$ is any planar tangle, the associated operator $Z_T^{(N\subset Q)}$ can be read off from $Z_T^{(N\subset M)}$ by a formula involving the so-called {\em biprojection} corresponding to the intermediate subfactor $N \subset Q \subset M$andascalar$\alpha(T)$ carefully chosen so as to ensure that the formula defining $Z_T^{(N\subset Q)}$ is multiplicative with respect to composition of tangles. Also, the planar algebra of $Q\subset M$ can be obtained by applying these results to $M\subset M_1$. We also apply our result to the example of a semi-direct product subgroup-subfactor.