The Dihedral Stabilizer of the Lam–Postnikov Triangulation of the Hypersimplex

The standard unimodular triangulation of the hypersimplex , equivalently the Stanley–Sturmfels sorted, type A alcove, and Lam–Postnikov circuit triangulation, is a central object in algebraic and polyhedral combinatorics. Motivated by a recent conjecture of Clarke and Kölbl on equivariant Ehrhart theory, we determine its exact coordinate stabilizer. We study the triangulation through its sorted one‑skeleton and use the cyclic alternation criterion for sorted pairs of -subsets. For , this graph encodes the crossing relation of joining segments in a convex -gon with cyclically ordered vertices: two disjoint joining segments and cross exactly when  and  form a sorted edge for any admissible common core  of size . This reconstruction forces every coordinate permutation preserving the triangulation to preserve the crossing relation on disjoint joining segments. Since the boundary edges of the -gon are precisely the joining segments crossing no disjoint joining segment, such a permutation must preserve the cycle graph and is therefore dihedral. Conversely, every dihedral symmetry preserves cyclic alternation and hence the flag-sorted complex. We conclude that the coordinate stabilizer is  for , while the simplex cases  and  have full coordinate stabilizer . The result proves dihedral maximality in the non‑simplex range, supplies the necessary endpoint correction, and shows that the cyclic order is recoverable, up to reversal, from the naturally labeled sorted one‑skeleton of the canonical hypersimplex triangulation.