In order to show that X is strongly regular, we show that its regular socle is multiplicity-free.
One even may strengthen Auslander's assertion by saying every morphism in mod Λ is right determined by the isomorphism class of a multiplicity-free Λ -module.
where C runs through all the Λ -modules (or just through representatives of all multiplicity-free Λ -modules) and this is a filtered union of meet-semilattices.
In the compact case we completely characterize the subgroups that define invariant symbols that yield commuting Toeplitz operators in terms of the multiplicity-free property.
If Λ is a k -algebra, k is algebraically closed, and C is multiplicity-free (what we can assume), then the height of S Hom ( C, Y ) is the k -dimension of Hom ( C, Y ).
We prove the commutativity of the Toeplitz operators by considering the Bergman spaces as the underlying space of the holomorphic discrete series and then applying known multiplicity-free results for restrictions to certain subgroups of the holomorphic discrete series.
By Proposition 3.1(a), we know that C [ → Y 〉 only depends on add C, thus we may restrict to look at representatives of multiplicity-free Λ -modules C. Proposition 3.1(b) asserts that both C [ → Y 〉 and C ′ [ → Y 〉 are contained in C ⊕ C ′ [ → Y 〉, thus we deal with a filtered union. According to Proposition 3.2, we deal with embeddings of meet-semilattices.
When dealing with the Auslander bijections η C Y : C [ → Y 〉 → S Hom ( C, Y ), we always can assume that C is multiplicity-free and supporting, here supporting means that Hom ( C i, Y ) ≠ 0 for any indecomposable direct summand C i of C. Namely, let C ′ be the direct sum of all indecomposable direct summands C i of C with Hom ( C i, Y ) ≠ 0, one from each isomorphism class.
Unlike other statistical issues, where solutions are mostly available and agreed upon (at least at the basic levels) a methodology that is appropriate for addressing the effect of multiplicity in such free form investigations are still in the making, and the knowledge about the availability of some is not widespread.
