Regular diagonal subfactors
We show that a diagonal subfactor arising from a finite family of automorphisms of a $II_1$-factor $Q$ is regular precisely when the classes of the defining automorphisms occur with same cardinality and form a subgroup of $\mathrm{Out}(Q)$, a subgroup that happens to be isomorphic to the generalized Weyl group of the subfactor. Moreover, it turns out that the cleanest picture of regularity in diagonal subfactors is graph-theoretic, namely, a diagonal subfactor is regular precisely when its principal graph is a complete, regular, balanced bipartite multigraph, with the generalized Weyl group fixing its size and the common multiplicity of the defining automorphisms determining its regular edge multiplicity. Prior to the characterization of regularity, revisiting Bisch and Popa's observations on the standard invariant and depth of diagonal subfactors, we give an exact criterion for a diagonal subfactor to have any prescribed depth, in terms of a stabilizing sequence of subsets of $\mathrm{Out}(Q)$ consisting of non-reduced alternating words in the classes of the defining automorphisms, which proves useful in the characterization of regularity.