Question

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?

Answer 1

A partial answer:

If you want something like the Frobenius map from the ring of characters (with induced product) of $\bigcup_n S_n$ to the ring of symmetric functions, then something like this exists for any wreath product $G \wr S_n$, namely there is a Frobenius map from the ring of characters of $\bigcup_n G \wr S_n$ into a tensor product of copies of the ring of symmetric functions, one for each conjugacy class of $G$. And there's an inner product and Schur functions, power sum, etc. This is explained in Appendix B of Macdonald's Symmetric Functions and Hall Polynomials.

So this takes care of type B ($G = {\bf Z}/2$) but I'm not sure about type D.

Answer 2

Hi,

if you are only interested in the appropriate symmetric function ring, you might want to look here: http://arxiv.org/abs/0704.2029, it misses though the direct relation to, say inner products etc, but is quite explicite about type B,C,D (as it works in the infinite limit, such distinctions a even/odd cease to exist)

BF.