Let
be the free Abelian group generated by
the complex irreducible representations of the symmetric group on
letters,
.
Now consider the direct sum
It is possible to define a
Hopf algebra
structure on
,
as follows. First the multiplication. Let
and
be, respectively, representations of
and
.
Taking the
tensor product
defines a representation
of
.
Consider
as a subgroup of
in the natural way. The product of
and
in
is now defined by taking the
induced representation
to
:
For the comultiplication restriction is used. Let
be a representation of
.
For every
,
,
consider the restriction of
to
to obtain an element of
.
The comultiplication of
is now defined by
There is a unit mapping
,
defined by identifying
and
,
and an augmentation
,
defined by
identity on
and
if
.
It is a theorem that
define a graded bi-algebra structure on
.
There is also an antipode, making
a
graded Hopf algebra.
This Hopf algebra can be explicitly described as follows. Consider the commutative
ring of polynomials
in infinitely many variables
,
,
,
A
co-algebra
structure is given by
and a co-unit by
,
for
.
There is also an antipode, making
also a graded Hopf algebra. Perhaps the basic result in
the representation theory of the symmetric groups is that
and
are isomorphic as Hopf algebras. The isomorphism is very nearly unique because,
[a1],
The individual components
of
are also rings in themselves under the product of representations
,
.
This defines a second multiplication on
,
which is distributive over the first, and
becomes a ring object in the category of co-algebras over
.
Such objects have been called
Hopf algebras,
[a6],
and quite a few of them occur naturally in algebraic topology. The ring
occurs in algebraic topology as
,
the cohomology of the
classifying space
of complex
-theory,
and there is a
"natural direct isomorphism"
,
[a3].
(This explains the notation used above for
:
the
"ci"
stand for Chern classes, cf.
Chern class.)
There is also an inner product on
:
counts the number of irreducible representations that
and
have in common, and with respect to this inner product
is (graded) self-dual. In particular, the multiplication
and comultiplication are adjoint to one another:
which is the same as
Frobenius reciprocity,
cf.
Induced representation,
in this case.
As a coring object in the category of algebras
,
being the representing object
of the functor of Witt vectors,
[a2],
plays an important role in formal group theory. But, so
far, no direct natural isomorphism has been found linking
with
in this manifestation.
The ring
also carries the structure of a
-ring
and it is in fact the
universal
-ring on one generator,
,
[a4],
and this gives a natural isomorphism
,
cf.
-ring
for some more details.
Finally there is a canonical notion of positivity on
:
the actual (i.e. not virtual) representations are positive and
the multiplication and comultiplication preserve positivity. This
has led to the notion of a
PSH-algebra,
which stands for positive self-adjoint Hopf algebra,
[a5].
Essentially,
is the unique PSH-algebra on one generator and
all other are tensor products of graded shifted copies of
.
This can be applied to other series of classical groups than the
,
[a5].
In combinatorics the algebra
also has a long history. In modern terminology it is at the basis of the so-called
umbral calculus,
[a7].
A modern reference to the representation theory of
the symmetric groups, both ordinary and modular, is
[a8].