### coxeter groups - Symmetric functions in type B and type D

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 gene...Read more

### symmetric functions - Plethysm of $\mathrm{QSym}$ into $\mathrm{QSym}$: can it be defined?

I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x_1$, $x_2$, $x_3$, ..., both over $\mathbb Z$). See Victor Reiner, Hopf algebras in combinatorics, chapters 2 and 5, respectively, for the definitions.It is known that one can define a plethysm $f\circ g \in \mathrm{QSym}$ for any $f\in\Lambda$ and any $g\in\mathrm{QSym}$. This is described, e. g., in Claudia Malvenuto, Christophe Reutenauer, Plethysm and conjugation of quasi-symmetric func...Read more