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).

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