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
 |
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
 |
 |
If 
is a linear algebraic group and 
is its Lie algebra, then 
has semi-invariants
 |
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=12017