Identifies the full group of rigid motions carrying a fundamental alcove of an irreducible affine reflection group to itself, and builds an explicit fundamental domain for that group acting on the alcove. The linearization step is that for any simplex, projection to the linear part is an isomorphism from its isometry group onto the isometry group of the simplex of inward unit normals, a higher-dimensional analogue of angle-angle-angle congruence. Applied to the alcove this shows the isometry group is isomorphic to the automorphism group of the affine Dynkin diagram, decomposes as the fundamental group Omega extended by the finite diagram automorphisms, and is an abstract Coxeter group generated by affine involutions that are usually not hyperplane reflections. Because those involutions have high-codimension fixed sets, the fundamental domain is obtained instead by slicing the Komrakov-Premet polytope with half-spaces attached to balanced minuscule roots, sums of a diagram-automorphism-exchanged pair of halves of a set of simple roots, which guarantees the resulting polytope's vertices sit among the vertices of the polytope being sliced. The concept teaches why the alcove retains affine diagram information that the Weyl chamber discards, and it includes the failure of the stratified-centralizer property for the extended Weyl group and the resulting obstruction to equivariant triangulations.
A fundamental alcove is a single tile in the paving of a Euclidean space by an irreducible affine reflection group; when the action is irreducible the alcove is a simplex, and it is cut out by the in…