# Semi-invariant(2)

A common eigenvector of a family of endomorphisms of a vector space or module. If $ G $
is a set of linear mappings of a vector space $ V $
over a field $ K $,
a semi-invariant of $ G $
is a vector $ v \in V $,
$ v \neq 0 $,
such that $$
g v = \chi ( g ) v , g \in G ,
$$
where $ \chi : \ G \rightarrow K $
is a function, called the weight of the semi-invariant $ v $.
A semi-invariant of weight $ 1 $
is also called an invariant. The most frequently considered case is that of a linear group $ G \subset \mathop{\rm GL}\nolimits ( V ) $,
in which case $ \chi : \ G \rightarrow K ^{*} $
is a character of $ G $
and may be extended to a polynomial function on $ \mathop{\rm End}\nolimits \ V $.
If $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $
is a linear representation of a group $ G $
in $ V $,
then a semi-invariant of the group $ \phi ( G ) $
is also called a semi-invariant of the representation $ \phi $(
cf. also Linear representation, invariant of a). Let $ G $
be a linear algebraic group, $ H $
a closed subgroup of $ G $
and $ \mathfrak h \subset \mathfrak g $
the Lie algebras of these groups. Then there exist a faithful rational linear representation $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits ( E ) $
and a semi-invariant $ v \in E $
of $ \phi ( H ) $
such that $ H $
and $ \mathfrak h $
are the maximal subsets of $ G $
and $ \mathfrak g $
whose images in $ \mathop{\rm End}\nolimits \ V $
have $ v $
as semi-invariant. This implies that the mapping $ a H \mapsto K \phi ( a ) v $,
$ a \in G $,
defines an isomorphism of the algebraic homogeneous space $ G/H $
onto the orbit of the straight line $ K v $
in the projective space $ P ( E ) $.

The term semi-invariant of a set $ G \subset \mathop{\rm End}\nolimits \ V $
is sometimes applied to a polynomial function on $ \mathop{\rm End}\nolimits \ V $
which is a semi-invariant of the set of linear mappings $ \eta ( G ) $
of the space $ K [ \mathop{\rm End}\nolimits \ V ] $,
where $$
( \eta ( g ) f \ ) ( X ) = f ( X g ) ,
$$
$$
g \in G , f \in K [ \mathop{\rm End}\nolimits \ V ] , X \in \mathop{\rm End}\nolimits \ V .
$$
If $ G \subset \mathop{\rm GL}\nolimits ( V ) $
is a linear algebraic group and $ \mathfrak g $
is its Lie algebra, then $ G $
has semi-invariants $$
f _{1} \dots f _{n} \in K [ \mathop{\rm End}\nolimits \ V ]
$$
of the same weight such that $ G $
and $ \mathfrak g $
are the maximal subsets of $ \mathop{\rm GL}\nolimits (V) $
and $ \mathop{\rm End}\nolimits \ V $
for which $ f _{1} \dots f _{n} $
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 |

**How to Cite This Entry:**

Semi-invariant(2).

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Semi-invariant(2)&oldid=44317