Spectral Theory of Signed Johnson Hypergraphs

Let B_n = (Z/2)^n ⋊ S_n be the hyperoctahedral group. We study the signed Johnson hypergraph SJ(n,k), whose vertices are signed k-subsets of [n] and whose hyperedges are signed (k+1)-subsets, together with its Discrete Invariant Projection Spaces (DIPS) normal operator N(k). We prove the operator identity N(k) = 2(n−k)I + R(k) for all n,k and establish the multiplicity-free decomposition of C^{Xn,k} into irreducible B_n-modules by an elementary Walsh–Fourier argument. The complete spectrum of N(1) is computed in closed form. For general k, the spectrum is obtained by a direct factorization argument: the operator R(k) decouples into independent classical Johnson adjacency operators, yielding the closed formula λ(k)a,i = 2[(n−k) + (a−i)(n−k−i) − i]. Finally, fixing k and letting n → ∞, we construct the tower of centraliser algebras, its Bratteli diagram, and the K_0-group of the resulting AF-algebra.