In this talk we shall be concerned with the induced simple modules of the 0-Hecke algebras of types A and B.
The irreducible representations of 0-Hecke algebras were classified and shown to be one-dimensional by Norton in 1979.
To understand the structure of a finite-dimensional module, one would ideally like to know its full submodule lattice; this is easily computable for small dimensions but much harder for larger ones. Given certain conditions, a smaller poset encoding the submodule lattice can be rather easily obtained.
We shall discuss the theory allowing us to get this smaller poset and build on results by Fayers in the type A case to state new results in type B.