ABSTRACT
As said before, [24], the symmetric functions are an exceedingly fascinating object of study; they are best studied from the Hopf algebraic point of view (in my opinion), although they carry quite a good deal more important structures, indeed so much that whole books do not suffice, but see [26, 27, 31, 33, 34]. The first of the two generalizations to be discussed is the Hopf algebra, NSymm, of noncommutative symmetric functions (over the integers). As an algebra, more precisely a ring, this is simply the free associative ring over the integers, Z, in countably many indeterminates
NSymm = Z〈Z1, Z2, . . .〉 (1.1)
and the coalgebra structure is given by the comultiplication determined by
µ : Zn → ∑
Zi ⊗ Zj , where Z0 = 1 (1.2)
and i and j are in N ∪ {0} = {0, 1, 2, · · · }. The augmentation is given by
ε(Zn) = 0, n = 1, 2, 3, . . . (1.3)
(and, of course ε(Z0) = ε(1) = 1). The Hopf algebra NSymm is a noncommutative covering generalization of the Hopf algebra of symmetric functions,
Symm = Z[z1, z2, . . .] (1.4)
where the zn are seen as either the elementary symmetric functions en or the complete symetric functions hn. The interpretation of the zn as the hn seems to work out somewhat nicer, for instance in obtaining the standard inner product autoduality of Symm in terms of the natural duality between NSymm and QSymm, the Hopf algebra of quasisymmetric functions, see [24], section 6. QSymm will be described and discussed later in this paper.
