Jinvariant

Jinvariant is a hypothetical mathematical invariant associated with finite combinatorial structures and their automorphism groups. It is defined to capture a notion of average symmetry by examining how many elements of the structure are fixed under its automorphisms. The term is not part of established literature and is presented here as an illustrative concept.

Definition. For a finite structure S with automorphism group Aut(S), the jinvariant J(S) is defined as the average number of elements fixed by automorphisms of S:

J(S) = (1/|Aut(S)|) sum_{g in Aut(S)} Fix_S(g),

where Fix_S(g) denotes the number of elements of S that are fixed by the automorphism g. This formulation yields a scalar value that is preserved under isomorphisms of S.

Properties. The jinvariant is invariant under isomorphism, since isomorphic structures have isomorphic automorphism groups and the fixed-point counts correspond accordingly. It lies in the interval [0, |S|], with interpretations depending on the structure: larger values indicate greater average fixed-point behavior and, loosely, higher symmetry in the action of Aut(S).

Examples. A simple graph with two vertices connected by an edge has Aut(S) of size 2 (the identity and the swap). The identity fixes both vertices, the swap fixes none, yielding J(S) = (2 + 0)/2 = 1. A highly asymmetric graph with trivial automorphism group has J(S) = |V|, since only the identity exists to fix all elements. In contrast, highly symmetric objects like complete graphs can yield smaller J(S) values due to many nontrivial automorphisms fixing few elements.

Relation to other invariants. Jinvariant is related to concepts such as fixed-point counts in Burnside’s lemma and cycle-index methods. It serves as a simple, aggregate measure of symmetry rather than a complete descriptor of structural properties.

Applications and notes. In this speculative framework, jinvariant could assist in comparing symmetry strength across structures or in heuristic approaches to symmetry-aware algorithm design. It remains a theoretical construct rather than a standard tool.