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Concept
Invariant theory
Summary
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.
Let be a group, and a finite-dimensional vector space over a field (which in classical invariant theory was usually assumed to be the complex numbers). A representation of in is a group homomorphism , which induces a group action of on . If is the space of polynomial functions on , then the group action of on produces an action on by the following formula:
With this action it is natural to consider the subspace of all polynomial functions which are invariant under this group action, in other words the set of polynomials such that for all . This space of invariant polynomials is denoted .
First problem of invariant theory: Is a finitely generated algebra over ?
For example, if and the space of square matrices, and the action of on is given by left multiplication, then is isomorphic to a polynomial algebra in one variable, generated by the determinant. In other words, in this case, every invariant polynomial is a linear combination of powers of the determinant polynomial. So in this case, is finitely generated over .
If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomial relations between the basis elements (known as the syzygies) is finitely generated over .
Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main theorem on the symmetric functions that described the invariants of the symmetric group acting on the polynomial ring ] by permutations of the variables.
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