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A common eigenvector of a family of endomorphisms of a vector space or module. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841801.png" /> is a set of linear mappings of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841802.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841803.png" />, a semi-invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841804.png" /> is a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841806.png" />, such that
+
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841807.png" /></td> </tr></table>
+
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|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|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|Linear representation, invariant of a]]). Let  $  G $
 +
be a [[Linear algebraic group|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 ) $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841808.png" /> is a function, called the weight of the semi-invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s0841809.png" />. A semi-invariant of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418010.png" /> is also called an invariant. The most frequently considered case is that of a [[Linear group|linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418011.png" />, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418012.png" /> is a character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418013.png" /> and may be extended to a polynomial function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418015.png" /> is a [[Linear representation|linear representation]] of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418017.png" />, then a semi-invariant of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418018.png" /> is also called a semi-invariant of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418019.png" /> (cf. also [[Linear representation, invariant of a|Linear representation, invariant of a]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418020.png" /> be a [[Linear algebraic group|linear algebraic group]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418021.png" /> a closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418023.png" /> the Lie algebras of these groups. Then there exist a faithful rational linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418024.png" /> and a semi-invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418028.png" /> are the maximal subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418030.png" /> whose images in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418031.png" /> have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418032.png" /> as semi-invariant. This implies that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418034.png" />, defines an isomorphism of the algebraic homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418035.png" /> onto the orbit of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418036.png" /> in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418037.png" />.
 
  
The term semi-invariant of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418038.png" /> is sometimes applied to a polynomial function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418039.png" /> which is a semi-invariant of the set of linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418040.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418041.png" />, where
+
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 $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418042.png" /></td> </tr></table>
+
which is a semi-invariant of the set of linear mappings $  \eta ( G ) $
 
+
of the space $  K [  \mathop{\rm End}\nolimits \  V ] $,  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418043.png" /></td> </tr></table>
+
where $$
 
+
( \eta ( g ) f \  ) ( X )  =  f ( X g ) ,
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418044.png" /> is a linear algebraic group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418045.png" /> is its Lie algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418046.png" /> has semi-invariants
+
$$
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418047.png" /></td> </tr></table>
+
g  \in  G ,  f  \in  K [  \mathop{\rm End}\nolimits \  V ] ,  X  \in    \mathop{\rm End}\nolimits \  V .
 
+
$$
of the same weight such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418049.png" /> are the maximal subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418051.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084180/s08418052.png" /> are semi-invariants (Chevalley's theorem).
+
If $  G \subset  \mathop{\rm GL}\nolimits ( V ) $
 +
is a linear algebraic group and $  \mathfrak g $
 +
is its Lie algebra, then $  G $
 +
has semi-invariants $$
 +
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C. Chevalley, "Théorie des groupes de Lie" , '''2''' , Hermann (1951) {{MR|0051242}} {{ZBL|0054.01303}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C. Chevalley, "Théorie des groupes de Lie" , '''2''' , Hermann (1951) {{MR|0051242}} {{ZBL|0054.01303}} </TD></TR></table>

Latest revision as of 21:41, 21 December 2019


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
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article