Pääinvariantit

Pääinvariantit, often translated as principal invariants or fundamental invariants, are a concept in mathematics, particularly in the fields of algebraic geometry and invariant theory. They represent a minimal set of algebraically independent polynomial functions that can generate all other invariant polynomials under a given group action. In essence, they are the building blocks from which all other symmetric polynomials (or more generally, polynomials invariant under a group transformation) can be constructed.

The idea originates from studying the symmetries of mathematical objects. When a group acts on a set of variables, certain polynomials in these variables remain unchanged, or invariant, under the group's transformations. The challenge is to find a concise description of these invariants. Pääinvariantit provide this description. If a set of polynomials $p_1, p_2, \dots, p_k$ are the pääinvariantit for a group $G$ acting on a set of variables, then any other polynomial $q$ that is invariant under $G$ can be expressed as a polynomial combination of $p_1, \dots, p_k$. Furthermore, the pääinvariantit themselves are algebraically independent, meaning no one pääinvariant can be expressed as a polynomial combination of the others.

A classic example is the symmetric group $S_n$ acting on $n$ variables $x_1, \dots, x_n$. The elementary symmetric polynomials $e_1, \dots, e_n$ are the primary building blocks for all symmetric polynomials. For instance, the power sums $p_k = \sum_{i=1}^n x_i^k$ can be expressed in terms of the elementary symmetric polynomials, and the elementary symmetric polynomials themselves are algebraically independent. Therefore, the elementary symmetric polynomials are the pääinvariantit for the action of the symmetric group on $n$ variables. The concept extends to more general groups and actions, forming a cornerstone in understanding the structure of invariant rings.