Spin Invariants and Orthogonal Characters in Cyclic Sieving

Let \( \mathcal{D}_n^{(r)} \)be the set of \( r \)-fans of Dyck paths of semilength \( n \). We prove that promotion on this set exhibits cyclic sieving with the \( q \)-multi-Catalan polynomial \( g_{n,r}(q)=\prod_{1\le i\le j\le n-1}\frac{[i+j+2r]_q}{[i+j]_q} \). This proves the conjecture of Pappe, Pfannerer, Schilling and Simone, equivalently the staircase case of Hopkins' rowmotion conjecture, for every size and height. The proof uses the promotion–rotation theorem for spin invariants and a classical orthogonal-group integral. A principal-specialisation calculation evaluates every rotation trace and resolves the signs at roots of unity. The same character calculation gives the energy-polynomial congruence, explicit fixed-point products, and a cyclic-module comparison with a symplectic representation. We determine the full dihedral character of promotion and evacuation and factor both reflection counts. Combining the staircase theorem with known rowmotion embeddings and transpose symmetries proves all-height cyclic sieving for shifted staircases and determines the fixed points of all rowmotion–transpose subgroups on square plane partitions. A gap model treats propellers and dihedral root posets. For chains of \( V \)'s, we prove polynomiality, all even-order identities, and several infinite families of odd-order identities. Exact polyhedral and Ehrhart computations prove the full piecewise-linear \( E_6 \) minuscule CSP, the order-six and order-nine identities for \( E_7 \), and all root identities for \( V(4) \) and \( V(8) \). The remaining odd-order identities and uniform coefficient positivity for general \( V(n) \) are stated separately.