# Comitant

*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=53618