It is well known that the symmetric groups have a very nice and explicit representation theory. This is in particular true when one works with the collection of all symmetric groups simultaneously, in which case the ring of virtual representations is the ring of symmetric functions. This has many advantages: it allows one to reduce hard questions in representation theory to formal combinatorial manipulations with symmetric functions; there is a lot of extra structure like inner product, outer product, and plethysm; it leads directly to the general theory of λ-rings, etc...

From the point of view of Coxeter groups there are two other infinite families of groups that seem natural to study from the same point of view, i.e. B_{n} and D_{n}. Is there an analogue of the theory of symmetric functions in this situation, i.e. a "nice" (i.e. highly structured and purely combinatorial) description of their rings of virtual representations?