View source for Semi-invariant(2)

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

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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

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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

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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).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.E. Humphreys,   "Linear algebraic groups" , Springer  (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,   "Théorie des groupes de Lie" , '''2''' , Hermann  (1951)</TD></TR></table>