Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SL_n on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SL_n.

Let ${\displaystyle G}$ be a group, and ${\displaystyle V}$ a finite-dimensional vector space over a field ${\displaystyle k}$ (which in classical invariant theory was usually assumed to be the complex numbers). A representation of ${\displaystyle G}$ in ${\displaystyle V}$ is a group homomorphism ${\displaystyle \pi :G\to GL(V)}$, which induces a group action of ${\displaystyle G}$ on ${\displaystyle V}$. If ${\displaystyle k[V]}$ is the space of polynomial functions on ${\displaystyle V}$, then the group action of ${\displaystyle G}$ on ${\displaystyle V}$ produces an action on ${\displaystyle k[V]}$ by the following formula:

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