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