Semi-invariant(2)
A common eigenvector of a family of endomorphisms of a vector space or module. If is a set of linear mappings of a vector space
over a field
, a semi-invariant of
is a vector
,
, such that
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where is a function, called the weight of the semi-invariant
. A semi-invariant of weight
is also called an invariant. The most frequently considered case is that of a linear group
, in which case
is a character of
and may be extended to a polynomial function on
. If
is a linear representation of a group
in
, then a semi-invariant of the group
is also called a semi-invariant of the representation
(cf. also Linear representation, invariant of a). Let
be a linear algebraic group,
a closed subgroup of
and
the Lie algebras of these groups. Then there exist a faithful rational linear representation
and a semi-invariant
of
such that
and
are the maximal subsets of
and
whose images in
have
as semi-invariant. This implies that the mapping
,
, defines an isomorphism of the algebraic homogeneous space
onto the orbit of the straight line
in the projective space
.
The term semi-invariant of a set is sometimes applied to a polynomial function on
which is a semi-invariant of the set of linear mappings
of the space
, where
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If is a linear algebraic group and
is its Lie algebra, then
has semi-invariants
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of the same weight such that and
are the maximal subsets of
and
for which
are semi-invariants (Chevalley's theorem).
References
[1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 |
[2] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039 |
[3] | C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951) MR0051242 Zbl 0054.01303 |
Semi-invariant(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-invariant(2)&oldid=21933