Cyclotomic Fields in Finite Geometry 2025, 31 - Group Rings & Characters

A finite group G of order v and regular action on a set P of v objects, can identify P with G by fixing a point p in P and identifying some point q with the unique element g_q in G. The blocks of the design are relative difference sets in G, which can be characterized by an equation over the internal group ring ℤ[G]. Point regular automorphism groups of designs have equivalent solutions of group ring equations. A k-subset R of group G with order mn is a (m, n, k, λ) difference set in G relative to a subgroup N iff RR^(-1) = k + λ(G - N) in ℤ[G]. A k-subset D of a group G with order v is a (v, k, λ, n) difference set in G iff DD^(-1) = n + λG in ℤ[G]. Note G* as character groups of abelian group G and U^⊥ as the subgroup of all characters trivial on U.

The ablian group G with subgroup U and W subgroup of G* has orthogonality relations Σ_g∈Uχ(g) = 0 ∀ χ∈G*\U^⊥ and Σ_χ∈Wχ(g) = 0 ∀ g∈G\W^⊥. From this follows the Fourier inversion formula. If A = Σ_g∈Ga_gg ∈ ℤ[G], then $a_g = \frac{1}{|G|}\Sigma_{\chi\in G*}\chi(Ag^{-1}) :::: \forall g\in G$ A ∈ ℤ[G], χ(A) = 0 ∀ nontrivial χ ∈ G ∃ μ ∈ ℤ: A = μG. H∈ℤ[G] with coefficients ±1, 0 is G-invariant weighing matrix W(m, n) iff HH^(-1) = n in ℤ[G].

A Hadamard difference set is a (v = 4u², k = 2u² - u, λ = u² - u, n = u²) difference set for u ∈ ℤ. They are equivalent to group invariant Hadamard matrices.