A
linear representation
of
on a finite-dimensional vector space
over
which is a rational homomorphism of
into
.
One also says that
is a
rational
-module.
Direct sums and tensor products of a finite number of rational representations of
are rational representations. Subrepresentations and quotient representations
of any rational representation are rational representations.
Symmetric and exterior powers of any rational
representation are rational representations. The representation contragredient
to a rational representation is a rational representation.
If
is finite, then each of its linear representations will be
a rational representation, and the theory of rational representations coincides
with the theory of representations of finite groups (cf.
Representation of a group).
To a large extent, specific methods of the theory of
linear algebraic groups are used to study rational representations in
case the group under consideration is connected, and the most
thoroughly developed theory is that of rational
representations of connected semi-simple algebraic groups. Let
be such a group,
a
maximal torus,
its group of rational characters (written additively),
the
root system
of
with respect to
,
its
Weyl group,
and
a
-invariant
positive-definite non-degenerate scalar product on
.
Now let
be a rational representation. The restriction of
to
decomposes into a direct sum of one-dimensional representations; more precisely,
where
is some set of characters of
,
called the
weights
of the representation, and
The set of weights
is invariant under the action of
.
If
,
then every rational representation of
is completely reducible, but if
,
then this is not so (see
Mumford hypothesis).
Whatever the characteristic of
,
however, there is a complete description of the irreducible rational representations.
Let
be a
Borel subgroup
in
containing
and let
be the set of simple roots in
defined by
.
Identify the group
of rational characters of
with
.
In the space
,
for any irreducible rational representation
there is a unique one-dimensional weight subspace
,
,
invariant under
.
The character
is called the
highest weight of the irreducible rational representation
;
it is
dominant,
i.e.
for any
,
and every other weight
has the form
The mapping
defines a bijection between the classes
of equivalent irreducible rational representations and the dominant elements of
.
An explicit construction of all irreducible rational
representations can be obtained in the following way. Let
be the algebra of regular functions on
.
Given any
,
consider the subspace
It is finite-dimensional and is a rational
-module
under the action of
by left translation. The geometric meaning of this space is as
follows: it can be canonically identified with the set of
regular sections of the one-dimensional homogeneous vector bundle over
determined by the character
.
Let
be the element mapping positive roots into negative ones. If
,
then
is a
dominant character
and the minimal non-zero
-submodule
in
is an irreducible rational
-module
with highest weight
.
Every irreducible rational
-module
can be obtained in this way. If
,
then the
-module
is itself irreducible.
To obtain irreducible rational representations, one often applies the
above-mentioned operations to given rational representations. For example, if
is an irreducible rational representation with highest weight
,
,
then some quotient representation of
is an irreducible rational representation with highest weight
(it is called the
Cartan product
of
).
If
is an irreducible rational representation with highest weight
,
then some quotient representation of
is an irreducible rational representation with highest weight
.
Moreover
is irreducible and its highest weight is
.
Let
be the Lie algebra of
(cf.
Lie algebra of an algebraic group).
If
is a rational representation, then its differential
is a representation of the Lie algebra
.
A rational representation
is called
infinitesimally irreducible
if
is an irreducible representation of the algebra
.
An infinitesimally-irreducible rational representation is irreducible, and when
,
the converse is also true (which largely reduces the theory of rational
representations of a group to the theory of representations of its Lie algebra). But when
,
this is not so; the infinitesimally-irreducible rational representations in this
case are just those irreducible rational representations with highest weight
for which
Moreover, all the irreducible rational representations can
be constructed using the infinitesimally-irreducible ones. More precisely, if
is
simply connected,
that is, if
coincides with the lattice of weights of the root system
,
then every irreducible rational representation factors uniquely
into a tensor product of the form
where
are infinitesimally irreducible, and
is the representation obtained by applying the
Frobenius automorphism
(
,
)
to the matrix entries of the representation
.