Dynamical systems defined on simplicial complexes: symmetries, conjugacies, and invariant subspaces

The paper’s Theorem 7.7 states that for a fixed partition P (resp. Q), Δ_{K_d} is invariant under G↑_d = B_{d+1} F↑_d B^T_{d+1} for all componentwise odd F↑_d constant on the blocks of P if and only if K_d is up-balanced with respect to P; and similarly for G↓_d with down-balanced and Q. The proof in the paper expands (B_{d+1} F↑_d B^T_{d+1} x)_e, groups by partition class and induced anti-colors a ∈ Z_K, and uses oddness and a generic choice of x to isolate the coefficient differences ∑_{t∈P_i, K↑_d(t)=a} I_{te} − ∑_{t∈P_i, K↑_d(t)=-a} I_{te}, which must satisfy the equal/negate/zero relations precisely defining up-balanced colorings; the down-case is analogous. This is exactly the structure of the candidate solution, which rewrites z = B^T_{d+1} x as z_t = ⟨K↑_d(t), u⟩, applies odd functions per block, and arrives at w_e = ∑_{i,a} c_{i,a}(e) f_i(⟨a,u⟩) with c_{i,a}(e) the same difference of incidence sums, then uses generic u and odd interpolation to isolate coefficients and derive the balanced conditions. The ingredients and logic match the paper’s proof step-for-step, including the zero-color case and the identical down-argument (paper: defns (58) and (63)–(69), proof equations (93)–(100), linkage to difference-of-sign-sums (72)–(74), and the theorem statement) .