Konfigurationsinvariant

Konfigurationsinvariant is a term used to describe a quantity that is assigned to a configuration of objects and remains unchanged under a specified set of transformations. In practice, it is an invariant of a configuration with respect to a group action on its configuration space. The concept appears in mathematics, particularly in geometry, topology, combinatorics, and their applications in computer science and the sciences.

Formally, take a space X and consider configurations of n points in X, denoted Config_n(X). Let G be a group that acts on X and therefore on Config_n(X). A konfigurationsinvariant is a map I: Config_n(X) → A, where A is some target set (such as numbers, polynomials, or algebraic objects), satisfying I(g·c) = I(c) for all g in G and all configurations c in Config_n(X). Different choices of G yield different invariants, reflecting the symmetries or transformations that are regarded as equivalent for the purpose at hand.

Common illustrative examples include:

- The pairwise distance multiset of a configuration in Euclidean space, which is invariant under isometries and relabeling of points.

- The Euler characteristic or other topological invariants of the configuration space Conf_n(X), which remain constant under deformations of X within a fixed class.

- Polynomial or generating-function invariants derived from combinatorial encodings of configurations, such as orbit-counting polynomials under a symmetry group.

Applications of konfigurationsinvariants span geometry, robotics (analyzing feasible configurations of mechanisms), chemistry (conformational invariants of molecules), and network science (characterizing spatial configurations of sensors or agents). The notion is closely related to the broader ideas of invariants in abstract algebra and topology, where invariants classify objects up to a prescribed notion of equivalence.