User:Maximilian Janisch/latexlist/Algebraic Groups/Comitant

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concomitant of a group $k$ acting on sets $x$ and $Y$

A mapping $\phi : X \rightarrow Y$ such that

\begin{equation} g ( \phi ( x ) ) = \phi ( g ( x ) ) \end{equation}

for any $g \in G$, $X \in X$. In this case one also says that $( 1 )$ commutes with the action of $k$, or that $( 1 )$ is an equivariant mapping. If $k$ 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 $k$.

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: $k$ is the general linear group of some finite-dimensional vector space $r$, $x$ and $Y$ are tensor spaces on $r$ of specified (generally distinct) types, on which $k$ acts in the natural way, while $( 1 )$ 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 $k$, while if $Y$ is a space of contravariant tensors, the comitant is called a contravariant of $k$.

Example. Let $f$ be a binary cubic form in the variables $\pi$ and $y$:

\begin{equation} f = a _ { 0 } x ^ { 3 } + 3 a _ { 1 } x ^ { 2 } y + 3 a _ { 2 } x y ^ { 2 } + a _ { 3 } y ^ { 3 } \end{equation}

Its coefficients are the coordinates of a covariant symmetric tensor. The coefficients of the Hessian form of $f$, that is, of the form

\begin{equation} H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \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| = \end{equation}

\begin{equation} = ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 } \end{equation}

are also the coefficients of a covariant symmetric tensor, while the mapping

\begin{equation} ( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } ) \end{equation}

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 $k$. If $x$ and $Y$ are affine, then giving a comitant is equivalent to giving a homomorphism $k [ Y ] \rightarrow k [ X ]$ of $k$-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 groups classiques" , Springer (1955) Zbl 0221.20056