Planar Algebra of the Subgroup-Subfactor

TL;DR

This paper establishes a correspondence between the planar algebra of a subgroup-subfactor and a G-invariant subalgebra of a bipartite graph planar algebra, providing a new model for the basic construction tower.

Contribution

It introduces a novel identification of subgroup-subfactor planar algebras with G-invariant subalgebras of bipartite graph planar algebras, using operator matrix models.

Findings

Identifies the planar algebra of subgroup-subfactors with G-invariant subalgebras of bipartite graph planar algebras.

Provides a model for the basic construction tower and standard invariant in operator matrix terms.

Establishes an identification for fixed algebra subfactors with the flip of the bipartite graph.

Abstract

We give an identification between the planar algebra of the subgroup-subfactor $R \rtimes H \subset R \rtimes G$ and the $G$-invariant planar subalgebra of the planar algebra of the bipartite graph $\star_n$, where $n = [G : H]$. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant, of $R \rtimes H \subset R \rtimes G$ in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor $R^G \subset R^H$ and the $G$-invariant planar subalgebra of the planar algebra of the `flip' of $\star_n $.