The basic reference for this section is Macdonald's book [].
Let 
an infinite set of commutative
indeterminates, which will be called an alphabet. The
elementary symmetric functions 
are defined by
the generating series
The complete homogeneous symmetric functions 
are defined by
The power sums symmetric functions 
are defined by
These generating series satisfy the following relations
Formula (1.21) is known as Newton's formula. The so-called
fundamental theorem of the theory of symmetric functions states that the
are algebraically independent. Therefore, any formal power
series 
may be considered as the
specialization of the series
to a virtual set of arguments A.
We denote by 
the algebra of symmetric functions, i.e. the
algebra generated over 
by the elementary functions. It is a graded
algebra for the weight function 
, and the dimension of its
homogeneous component of weight n, denoted by 
, is equal to 
,
the number of partitions of n. A partition is a
finite non-increasing sequence of positive integers,
.
We shall also write
, 
being the
number of parts 
which
are equal to m. The weight of 
is 
and its length is the
number of its (nonzero) parts 
.
For a partition 
, one sets
These are linear bases of 
(Top ,
Toe ,
Toh ).
For 
, not necessarily a partition,
the Schur function 
is defined by
where 
for j<0. The Schur functions indexed by partitions form a
-basis of 
(Tos ),
and one usually endows 
with a scalar product 
for which this basis is orthonormal. The 
form then an orthogonal
-basis of 
,
with 
(Zee ,
ScalarSf ).
Thus, for a
partition 
of weigth n, 
is
the cardinality of the
conjugacy class of 
whose elements have 
cycles of length k for all 
. A permutation 
in this
class will be said of type 
(Perm2Cycle ,
Perm2CycleType ),
and we shall write 
.
These definitions are motivated by the following classical results
of Frobenius. Let 
be the ring of central functions
of the symmetric group 
. The Frobenius characteristic map
associates (Sf2Char )
with any central function 
the symmetric function
where 
is the common value of the 
for all 
such that 
. We can also consider 
as a map from
the representation ring 
to 
by setting
, where 
denotes
the equivalence class of a representation 
.
Glueing these maps together, one has a linear
map
which turns out to be an isomorphism of graded rings.
We denote by 
the class of the irreducible
representation of 
associated with the partition 
, and by
its character. We have then 
.
The product 
, defined on the homogeneous component 
by
and extended to 
by defining the product of two homogeneous functions
of different weights to be zero, is called the internal product
(ITensor ).
One can identify the tensor product 
with the algebra
of polynomials which are separately symmetric in two infinite
disjoint sets of indeterminates X and Y, the correspondence
being given by 
. Denoting by X+Y
the disjoint union of X and Y, one can then define a comultiplication
on 
by setting 
. This comultiplication
endows 
with the structure of a self-dual Hopf algebra, which is very
useful for calculations involving characters of symmetric groups.
The
basic formulas are
where 
denotes the r-fold ordinary multiplication and
the iterated coproduct, and
The symmetric functions of the virtual alphabet 
are defined by the generating series
and one can more generally consider differences of alphabets. The symmetric functions of X-Y are given by the generating series
In particular, 
.
There is another coproduct 
on 
, which is obtained by considering
products instead of sums, that is
One can check that its adjoint is the internal product:
This equation can be used to give an intrinsic definition of the internal
product, i.e. without any reference to characters of the
symmetric group. Also, one can see that two bases 
of 
are adjoint to each other iff
For example, writing 
and expanding
the product, one obtains that the adjoint basis of 
is formed by the
monomial functions 
.
