Stabilizertypes

Stabilizertypes is a term used in group action theory to refer to the classification of stabilizer subgroups up to conjugacy, also called orbit-type data. Given a group G acting on a set X, the stabilizer of a point x is G_x = {g in G | g.x = x}. The stabilizer type of x is the conjugacy class [G_x] in G. The space X is partitioned into orbit-type strata X_[H] = {x in X | G_x is conjugate to H}, where H runs over subgroups of G up to conjugacy. This stratification helps describe how symmetry varies across X and how the quotient space behaves.

In the finite case, there are finitely many stabilizer types; in the smooth or algebraic setting, the corresponding strata are locally closed and fit into a partial order given by subgroup containment up to conjugacy. Points with larger stabilizers lie in the closures of strata with smaller stabilizers, reflecting symmetry enhancement on fixed sets.

Examples: consider G = S3 acting on R^3 by permuting coordinates. A generic point with all coordinates distinct has trivial stabilizer. Points with two equal coordinates have a stabilizer isomorphic to a subgroup of order 2, and the origin has the full S3 stabilizer. These orbit types illustrate how stabilizer types organize the space into strata.

Applications: stabilizertypes appear in equivariant topology, representation theory, and moduli theory, where they guide the construction of quotient spaces, inertia stacks, and orbifold structures.

See also: orbit, stabilizer, orbit-stabilizer theorem, orbit-type stratification, inertia stack.