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''concomitant of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c0233301.png" /> acting on sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c0233302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c0233303.png" />''
+
{{TEX|done}}
 +
''concomitant of a group $  G $
 +
acting on sets $  X $
 +
and $  Y $ ''
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c0233304.png" /> such that
 
  
<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/c/c023/c023330/c0233305.png" /></td> </tr></table>
+
A mapping $  \phi : \  X \rightarrow Y $
 +
such that$$
 +
g ( \phi (x))  =
 +
\phi (g (x))
 +
$$
 +
for any $  g \in G $ ,
 +
$  x \in X $ .  
 +
In this case one also says that $  \phi $
 +
commutes with the action of $  G $ ,
 +
or that $  \phi $
 +
is an equivariant mapping. If $  G $
 +
acts on every set of a family $  \{ {X _{i}} : {i \in I} \} $ ,
 +
then a comitant $  \prod _ {i \in I} X _{i} \rightarrow Y $
 +
is called a simultaneous comitant of $  G $ .
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c0233306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c0233307.png" />. In this case one also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c0233308.png" /> commutes with the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c0233309.png" />, or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333010.png" /> is an equivariant mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333011.png" /> acts on every set of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333012.png" />, then a comitant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333013.png" /> is called a simultaneous comitant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333014.png" />.
 
  
The notion of a comitant originates from the classical theory of invariants (cf. [[Invariants, theory of|Invariants, theory of]]) in which, however, a comitant is understood in a narrower sense: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333015.png" /> is the general linear group of some finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333018.png" /> are tensor spaces on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333019.png" /> of specified (generally distinct) types, on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333020.png" /> acts in the natural way, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333021.png" /> is an equivariant polynomial mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333023.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333024.png" /> is a space of covariant tensors, then the comitant is called a covariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333025.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333026.png" /> is a space of contravariant tensors, the comitant is called a contravariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333027.png" />.
+
The notion of a comitant originates from the classical theory of invariants (cf. [[Invariants, theory of|Invariants, theory of]]) in which, however, a comitant is understood in a narrower sense: $  G $
 +
is the general linear group of some finite-dimensional vector space $  U $ ,  
 +
$  X $
 +
and $  Y $
 +
are tensor spaces on $  U $
 +
of specified (generally distinct) types, on which $  G $
 +
acts in the natural way, while $  \phi $
 +
is an equivariant polynomial mapping from $  X $
 +
into $  Y $ .  
 +
If, in addition, $  Y $
 +
is a space of covariant tensors, then the comitant is called a covariant of $  G $ ,  
 +
while if $  Y $
 +
is a space of contravariant tensors, the comitant is called a contravariant of $  G $ .
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333028.png" /> be a binary cubic form in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333030.png" />:
 
  
<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/c/c023/c023330/c02333031.png" /></td> </tr></table>
+
Example. Let $  f $
 +
be a binary cubic form in the variables $  x $
 +
and $  y $ :
 +
$$
 +
=   a _{0} x ^{3} +
 +
3a _{1} x ^{2} y +
 +
3a _{2} xy ^{2} +
 +
a _{3} y ^{3} .
 +
$$
 +
Its coefficients are the coordinates of a covariant symmetric tensor. The coefficients of the Hessian form of $  f $ ,
 +
that is, of the form$$
 +
H  =  {
 +
\frac{1}{36}
 +
}
 +
\left |
  
Its coefficients are the coordinates of a covariant symmetric tensor. The coefficients of the Hessian form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333032.png" />, that is, of the form
+
\begin{array}{cc}
  
<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/c/c023/c023330/c02333033.png" /></td> </tr></table>
+
\frac{\partial ^{2} f}{\partial x ^{2}}
 +
  &
 +
\frac{\partial ^{2} f}{\partial x \partial y}
 +
  \\
  
<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/c/c023/c023330/c02333034.png" /></td> </tr></table>
+
\frac{\partial ^{2} f}{\partial x \partial y}
 +
  &
 +
\frac{\partial ^{2} f}{\partial y ^{2}}
 +
  \\
 +
\end{array}
 +
\
 +
\right |  =
 +
$$
 +
$$
 +
=
 +
(a _{0} a _{2} - a _{1} ^{2} ) x ^{2}
 +
+ (a _{0} a _{3} - a _{1} a _{2} ) xy + (a _{1} a _{3} - a _{2} ^{2} ) y ^{2}
 +
$$
 +
are also the coefficients of a covariant symmetric tensor, while the mapping$$
 +
(a _{0} ,\  a _{1} ,\  a _{2} ,\  a _{3} )  \mapsto 
 +
\left (
 +
a _{0} a _{2} - a _{1} ^{2} ,\
 +
{
 +
\frac{1}{2}
 +
}
 +
(a _{0} a _{3} - a _{1} a _{2} ),\
 +
a _{1} a _{3} - a _{2} ^{2}
 +
\right )
 +
$$
 +
of the corresponding tensor spaces is a comitant (the so-called comitant of the form $  f \  $ ).  
 +
The Hessian of an arbitrary form can similarly be defined; this also provides an example of a comitant (see [[Covariant|Covariant]]).
  
are also the coefficients of a covariant symmetric tensor, while the mapping
+
In the modern geometric theory of invariants, by a comitant one often means any equivariant morphism $  X \rightarrow Y $ ,  
 
+
where $  X $
<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/c/c023/c023330/c02333035.png" /></td> </tr></table>
+
and $  Y $
 
+
are algebraic varieties endowed with a regular action of an algebraic group $  G $ .  
of the corresponding tensor spaces is a comitant (the so-called comitant of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333036.png" />). The Hessian of an arbitrary form can similarly be defined; this also provides an example of a comitant (see [[Covariant|Covariant]]).
+
If $  X $
 
+
and $  Y $
In the modern geometric theory of invariants, by a comitant one often means any equivariant morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333039.png" /> are algebraic varieties endowed with a regular action of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333040.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333042.png" /> are affine, then giving a comitant is equivalent to giving a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333044.png" />-modules of regular functions on the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333046.png" />, respectively (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333047.png" /> is the ground field).
+
are affine, then giving a comitant is equivalent to giving a homomorphism $  k [Y] \rightarrow k [X] $
 +
of $  G $ -
 +
modules of regular functions on the varieties $  Y $
 +
and $  X $ ,  
 +
respectively (where $  k $
 +
is the ground field).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.B. Gurevich,  "Foundations of the theory of algebraic invariants" , Noordhoff  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Mumford,  "Geometric invariant theory" , Springer  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Dieudonné,  "La géométrie des groups classiques" , Springer  (1955)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.B. Gurevich,  "Foundations of the theory of algebraic invariants" , Noordhoff  (1964)  (Translated from Russian) {{MR|0183733}} {{ZBL|0128.24601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Mumford,  "Geometric invariant theory" , Springer  (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Dieudonné,  "La géométrie des groupes classiques" , Springer  (1955) {{MR|}} {{ZBL|0221.20056}} </TD></TR></table>

Latest revision as of 13:16, 7 April 2023

concomitant of a group $ G $ acting on sets $ X $ and $ Y $


A mapping $ \phi : \ X \rightarrow Y $ such that$$ g ( \phi (x)) = \phi (g (x)) $$ for any $ g \in G $ , $ x \in X $ . In this case one also says that $ \phi $ commutes with the action of $ G $ , or that $ \phi $ is an equivariant mapping. If $ G $ acts on every set of a family $ \{ {X _{i}} : {i \in I} \} $ , then a comitant $ \prod _ {i \in I} X _{i} \rightarrow Y $ is called a simultaneous comitant of $ G $ .


The notion of a comitant originates from the classical theory of invariants (cf. Invariants, theory of) in which, however, a comitant is understood in a narrower sense: $ G $ is the general linear group of some finite-dimensional vector space $ U $ , $ X $ and $ Y $ are tensor spaces on $ U $ of specified (generally distinct) types, on which $ G $ acts in the natural way, while $ \phi $ is an equivariant polynomial mapping from $ X $ into $ Y $ . If, in addition, $ Y $ is a space of covariant tensors, then the comitant is called a covariant of $ G $ , while if $ Y $ is a space of contravariant tensors, the comitant is called a contravariant of $ G $ .


Example. Let $ f $ be a binary cubic form in the variables $ x $ and $ y $ : $$ f = a _{0} x ^{3} + 3a _{1} x ^{2} y + 3a _{2} xy ^{2} + a _{3} y ^{3} . $$ Its coefficients are the coordinates of a covariant symmetric tensor. The coefficients of the Hessian form of $ f $ , that is, of the form$$ H = { \frac{1}{36} } \left | \begin{array}{cc} \frac{\partial ^{2} f}{\partial x ^{2}} & \frac{\partial ^{2} f}{\partial x \partial y} \\ \frac{\partial ^{2} f}{\partial x \partial y} & \frac{\partial ^{2} f}{\partial y ^{2}} \\ \end{array} \ \right | = $$ $$ = (a _{0} a _{2} - a _{1} ^{2} ) x ^{2} + (a _{0} a _{3} - a _{1} a _{2} ) xy + (a _{1} a _{3} - a _{2} ^{2} ) y ^{2} $$ are also the coefficients of a covariant symmetric tensor, while the mapping$$ (a _{0} ,\ a _{1} ,\ a _{2} ,\ a _{3} ) \mapsto \left ( a _{0} a _{2} - a _{1} ^{2} ,\ { \frac{1}{2} } (a _{0} a _{3} - a _{1} a _{2} ),\ a _{1} a _{3} - a _{2} ^{2} \right ) $$ of the corresponding tensor spaces is a comitant (the so-called comitant of the form $ f \ $ ). The Hessian of an arbitrary form can similarly be defined; this also provides an example of a comitant (see Covariant).

In the modern geometric theory of invariants, by a comitant one often means any equivariant morphism $ X \rightarrow Y $ , where $ X $ and $ Y $ are algebraic varieties endowed with a regular action of an algebraic group $ G $ . If $ X $ and $ Y $ are affine, then giving a comitant is equivalent to giving a homomorphism $ k [Y] \rightarrow k [X] $ of $ G $ - modules of regular functions on the varieties $ Y $ and $ X $ , respectively (where $ k $ is the ground field).

References

[1] G.B. Gurevich, "Foundations of the theory of algebraic invariants" , Noordhoff (1964) (Translated from Russian) MR0183733 Zbl 0128.24601
[2] D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[3] J.A. Dieudonné, "La géométrie des groupes classiques" , Springer (1955) Zbl 0221.20056
How to Cite This Entry:
Comitant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Comitant&oldid=15732
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article