
TL;DR
This paper explicitly computes the subfactor planar algebra for an intermediate subfactor in terms of the larger algebra, using biprojections and scalar adjustments, with applications to semi-direct product subfactors.
Contribution
It provides a new explicit formula relating intermediate subfactor planar algebras to the larger algebra's planar algebra, enhancing understanding of their structure.
Findings
Explicit formula for $Z^{(N extsubset Q)}_T$ in terms of $Z^{(N extsubset M)}_T$
Application to semi-direct product subgroup-subfactors
Method for deriving the planar algebra of $Q extsubset M$
Abstract
In this paper, we explicitly work out the subfactor planar algebra for an intermediate subfactor of an irreducible subfactor of finite index. We do this in terms of the subfactor planar algebra by showing that if is any planar tangle, the associated operator can be read off from by a formula involving the so-called {\em biprojection} corresponding to the intermediate subfactor and a scalar carefully chosen so as to ensure that the formula defining is multiplicative with respect to composition of tangles. Also, the planar algebra of can be obtained by applying these results to . We also apply our result to the example of a semi-direct product subgroup-subfactor.
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