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 SLn 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
SLn.