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